Quelques probl`emes de la théorie des syst`emes paraboliques

Quelques problèmes
de la théorie des systèmes
paraboliques dégénérés
non-linéaires
et des lois de conservation
Boris P. ANDREIANOV
Thèse de doctorat en Mathématiques et Applications
soutenue
à l’Université de Franche-Comte, Besançon, France
le 20 janvier 2000
devant le jury composé de
Assia Benabdallah,
Philippe Bénilan,
Valery Galkin,
Raphaèle Herbin,
Louis Jeanjean,
Evgenii Radkevich,
Denis Serre,
Petra Wittbold,
Université de Franche-Comté
Université de Franche-Comté (le Directeur)
Institut de l’Energie Nucléaire, Obninsk, Russie (Rapporteur)
Université de Provence, Marseille (Rapporteur)
Université de Franche-Comté
Université Lomonossov de Moscou, Russie
Ecole Normale Supérieure de Lyon (Rapporteur)
Université Louis Pasteur de Strasbourg.
Remerciements
Je crois sincèrement avoir eu une chance extraordinaire. Pendant les six dernières
années, Stanislav Nikolaevich Kruzhkov, puis Philippe Bénilan, m’ont mené dans le monde
de la recherche. Parmi tout ce qu’ils m’ont donné, il m’est difficile de choisir le plus
précieux. Peut-être, est-ce le goût pour la beauté des mathématiques; la volonté de ne pas
s’arrêter à mi-chemin, d’aller au fond des choses. Et surtout, la confiance qu’ils m’ont
accordée, au-delà de ce que j’aurais pu espérer. A Stanislav Kruzhkov et Philippe Bénilan,
j’adresse ma première pensée et ma plus profonde gratitude.
Je remercie beaucoup Evgenii Radkevich. Son soutien a aidé à la poursuite de ce travail
en co-tutelle, les discussions avec lui ont été stimulantes.
Athanasios Tzavaras s’est intéressé à mes résultats; il m’a suggeré un des problèmes
abordés dans cette thèse. Je l’en remercie.
Denis Serre a accepté de donner son avis sur ce mémoire; son appréciation m’est
très importante. De plus, il m’a fait l’honneur de présider le jury de thèse. Je lui suis
doublement reconnaissant.
Je suis reconnaissant à Valerii Galkin, pour son rapport sur cette thèse, pour avoir
pris part à ce jury, et pour tout le soutien qu’il m’a accordé à Moscou.
Mes remerciements vont à Raphaèle Herbin, pour l’avis qu’elle a donné sur ce travail,
pour ses explications, pour avoir accepté de participer au jury.
Je remercie Brian Gilding pour avoir accepté de faire un rapport sur ce mémoire, et
pour l’attention qu’il a pretée à mes tous premiers résultats.
Assia Benabdallah m’a fait l’honneur de participer à ce jury. De longues discussions
avec elle, son encouragement constant m’ont été précieux. Je lui adresse ma profonde
reconnaissance.
Je dois une partie importante de ma thèse à la collaboration avec Petra Wittbold. Elle
a bien voulu s’intéresser aux diverses parties de ce mémoire et prendre part au jury. J’en
suis honoré. Je la remercie, et je remercie Michaël Gutnic. Notre travail à trois m’a
beaucoup appris; il a été un plaisir pour moi.
Louis Jeanjean a bien voulu s’intéresser aux questions abordées dans cette thèse et
participer à ce jury. Ses remarques me seront très utiles dans l’avenir; je le remercie de
son engagement.
La grande disponibilité de Catherine Pagani, Catherine Vuillemenot, Monique Digu,
Jacques Vernerey, Jean-Daniel Tissot, Odile Henri, Nathalie Pasquet m’a facilité la tâche;
leur gentillesse m’a touché. Je leur dis “merci”.
Le Labo de Mathématiques de Besançon m’a accueilli pendant trois années. Où que
je sois, je serai toujours nostalgique de son ambiance. J’en garderai des souvenirs, j’en
garderai des amis.
Je dis “merci” et “spasibo” à mes amis : ceux de Moscou, de Besançon, de StPeterbourg... On a partagé de bons moments; leur encouragement m’a beaucoup aidé.
Ma mère et mon père m’ont apporté un soutien inestimable. Je leur dédie cette thèse.
Résumé
Dans la première partie, on traite par l’approche de viscosité auto-similaire le problème
de Riemann pour la loi de conservation scalaire et les systèmes de type “dynamique des
gaz isentropiques” en coordonnées de Lagrange et d’Euler. Dans chacun des cas, cette
étude aboutit aux résultats d’existence et d’unicité de la solution “wave-fan admissible” pour toute fonction de flux continue. En particulier, on couvre le cas d’apparition
du vide dans la dynamique des gaz et le cas des problèmes mixtes avec transitions de
phase. D’autre part, pour une loi de conservation scalaire multi-dimensionnelle avec une
fonction de flux continue on démontre l’existence des solutions entropiques généralisées
maximum et minimum dans le cadre L1 ∩ L∞ . On étudie la nature de ces solutions à
l’aide de la théorie des semi-groupes non-linéaires; ensuite, on étend quelques résultats
d’unicité dus à Bénilan et Kruzhkov.
Dans la deuxième partie, on traite de systèmes elliptiques-paraboliques dont les coefficients peuvent dépendre de (t, x) . On démontre un théorème de continuité des solutions
variationnelles par rapport aux données et obtient ainsi le résultat d’existence de Alt et
Luckhaus sous des hypothèses plus faibles, tout en mettant en évidence l’essentiel de
leurs arguments. On applique ensuite les techniques développées pour démontrer la convergence des schémas de volumes finis pour un système modèle fortement nonlinéaire,
qui apparaı̂t dans la physique des milieux poreux. On propose ainsi une approche pour
la convergence des méthodes de volumes finis, où la preuve se fait par réduction du cas
discret au cas continu.
Mots clés :
lois de conservation, problème de Riemann, viscosité auto-similaire, dynamique des gaz
isentropiques, solutions entropiques généralisées, semi-groupes non-linéaires, systèmes
elliptiques-paraboliques, conditions de type Leray-Lions, méthodes de volumes finis.
Abstract
In the first part, one treats by the self-similar viscosity approach the Riemann problem
for a scalar conservation law and the systems of the “isentropic gas dynamics” type in
the Lagrange and Euler coordinates. In each of the cases, the study yields existence
and uniqueness of a wave-fan admissible solution for all continuous flux function. In
particular, the situation when vaccuum appears in gas dynamics is covered, as well as
the case of problems of mixed type with phase transitions. On the other hand, for a
scalar multidimensional conservation law with continuous flux function the existence of
maximum and minimum generalized entropy solutions in the L1 ∩ L∞ framework is
proved. Using the nonlinear semigroup theory, one studies the nature of these solutions;
then one extends some uniqueness results of Bénilan and Kruzhkov.
In the second part, one treats elliptic-parabolic systems with coefficients that may
depend on (t, x) . A theorem on continuity of variational solutions with respect to data
is proved. This yields the existence result of Alt and Luckhaus under weaker hypotheses,
while clarifying the essence of their arguments. Necessary techniques are developped;
next, they are applied to proving convergence of finite volume schemes for a model
strongly nonlinear system, which appears in the study of porous media. An approach for
convergence of finite volume methods is proposed, where the proof goes on by reduction
of the discret case to the continuous one.
Key words :
conservation laws, Riemann problem, self-similar viscosity, isentropic gas dynamics, generalized entropy solutions, nonlinear semigroups, elliptic-parabolic systems, Leray-Lions
type conditions, finite volume methods.
Table des matières :
Introduction
Chapitre I
Enoncés des “quelques problèmes”
et un résumé des résultats obtenus . . . . . . . . . . . . . . . . . . . . . . 13. . . 19
Chapitre II Les “quelques problèmes”
dans le contexte mathématique et physique . . . . . . . . . . . . . 21. . . 31
Part 1. Conservation Laws with Continuous Flux Function
Chapter 1.I
The Riemann Problem for Scalar Conservation Law
with Continuous Flux Function:
the Self-Similar Viscosity Approach . . . . . . . . . . . . . . . . . . . . . 35. . . 43
Chapter 1.II
The Riemann Problem for p-System
with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 45. . . 62
Chapter 1.III
On Viscous Limit Solutions to the Riemann Problem
for the Equations of Isentropic Gas Dynamics
in Eulerian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. . . 82
Chapter 1.IV
L1 -Theory of Scalar Conservation Law
with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 83. . . 98
Part 2. Weak Solutions for Elliptic-Parabolic Systems
Chapter 2.I
Elliptic-Parabolic problems: Existence and Continuity
with Respect to the Data of Weak Solutions . . . . . . . . . 101. . . 124
Chapter 2.II
Convergence of Finite Volume Approximations
for a Nonlinear Elliptic-Parabolic Problem:
a Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. . . 159
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163. . . 172
Introduction
CHAPITRE I
Enoncés des “quelques problèmes”
et un résumé des résultats obtenus†
Partie 1.
Les lois de conservation
avec une fonction de flux continue
On étudie l’admissibilité et l’unicité de solutions pour la loi de conservation scalaire et les
systèmes de type “élasticité non-linéaire” et “dynamique des gaz isentropiques”. En général,
la fonction de flux est seulement continue.
• Chapitre 1.I
Le problème de Riemann pour les lois de conservation scalaires avec une
fonction de flux continue : l’approche par viscosité auto-similaire.
· On traite le problème suivant :


 Ut + f (U)
(x = εtUxx ,
(CLε )
u− , x < 0

 U|t=0 =
u+ , x > 0,
U : IR+ × IR 7→ IR, u± ∈ IR,
avec f (·) : IR 7→ IR continue.
· Les références principales sont :
Riemann [R1860], Rayleigh [Ray10], Hopf [H50], Gelfand [G59], Kalashnikov [Ka59], Dafermos [D73a, D89], Kruzhkov [K69a, K69b, K70a], Kruzhkov, P.A.Andreyanov [KPA75],
Chapitre 1.IV.
†
On présente dans ce Chapitre l’essentiel des résultats de la thèse. Les énoncés et conditions exactes se
trouvent dans les chapitres correspondants dans les Parties 1 et 2.
14
Introduction
· Dans ce chapitre, on se donne pour objectif :
obtenir la solution admissible de (CL0 ) comme la limite des solutions de (CLε ) quand
ε ↓ 0 , en donnant ainsi une preuve directe d’unicité pour (CL0 ) ; presenter le “wave fan
admissibility criterion” comme un critère bien adapté au cas de fonction de flux qui n’est
pas régulière.
· Les principaux résultats sont :
- réduction de (CLε ) avec ε > 0 à un problème aux limites singulier, mais bien posé,
pour une équation différentielle ordinaire (Propositions 1,2);
- existence et unicité pour (CLε ) avec ε > 0 (Theorem 1);
- convergence des solutions de (CLε ) vers une limite, donnée par une formule explicite,
lorsque ε ↓ 0 , ce qui donne une démonstration directe d’unicité d’une solution “wave-fan
admissible” pour le problème de Riemann (Theorem 2);
• Chapitre 1.II
Le problème de Riemann pour les p-systèmes avec une fonction de flux
continue.
· On traite le problème suivant :


Ut − Vx = 0



 V − f (U) = εtV ,
t
x (
xx
(pSε )

(u− , v− ),



 (U, V )|t=0 = (u , v ),
+
+
x<0
x > 0,
(U, V ) : IR+ × IR 7→ IR2 ,
u± , v± ∈ IR.
Dans un premier temps, f : IR 7→ IR est supposée continue strictement croissante
(Sections 1-3), puis la condition de monotonie est supprimée.
· Les références principales sont :
Dafermos [D74, D89], Leibovich [Le74], Krejčı̀,Straškraba [KrSt93], Tzavaras [Tz96],
James [Ja80], Shearer [Sh82], Slemrod [Sl89], Chapitre 1.I.
· Dans ce chapitre, on se donne pour objectif :
obtenir la solution admissible de (pS0 ) comme la limite des solutions de (pSε ) lorsque
ε ↓ 0 ; inclure le cas hyperbolique-elliptique en toute généralité; expliciter le contexte
physique de l’admissibilité pour ce dernier problème.
· Les principaux résultats sont :
- caractérisation de solutions de (pSε ) , ε > 0 , et réduction à un problème aux limites
pour une équation différentielle ordinaire (Lemma 1, Propositions 1,2,4 ; puis Lemma 4,
Propositions 5,6);
- unicité pour (pSε ) , ε > 0 , et existence sous des hypothèses de croissance de f (·) en
±∞ (Theorem 1; puis Theorem 2);
0.I.
Les énoncés des problèmes et un résumé des résultats obtenus
15
- convergence des solutions de (pSε ) vers une limite, donnée par une formule explicite,
lorsque ε ↓ 0 (Theorem 1; puis Theorem 2);
- observation que le “Riemann solver” obtenu permet de résoudre (pS0 ) d’une manière
unique, mais ne correspond qu’à un nombre restreint de problèmes physiques avec f (·)
non-monotone (Remark 4).
• Chapitre 1.III
Sur les limites visqueuses auto-similaires comme solutions du problème
de Riemann pour les équations de la dynamique des gaz isentropiques en
coordonnées d’Euler.
· On traite le problème suivant :


ρt + (ρu)x = 0



 (ρu) + (ρu2 + p(ρ)) = εtu ,
t
x
xx
(
(GDε )

(ρ− , u− ), x < 0



 (ρ, u)|t=0 =
(ρ+ , u+ ), x > 0,
(ρ, u) : IR+ × IR 7→ IR+ × IR,
ρ± > 0, u± ∈ IR;
La fonction p : IR+ 7→ IR est supposée continue strictement croissante; en général, le
comportement de p(·) en zéro n’interdit pas l’apparition du vide dans les solutions.
· Les références principales sont :
Dafermos [D89], Kim [Kim99], Slemrod, Tzavaras [SlTz89], Rozhdestvenskii, Janenko
[RoJa], Cheng, Hsiao [ChHs], Wagner [Wa87], Chapitres 1.I et 1.II.
· Dans ce chapitre, on se donne pour objectif :
construire les solutions de (GDε ) avec ε > 0 qui peuvent contenir un point du vide, afin
de pouvoir résoudre (GD0 ) pour toutes données de Riemann; démontrer l’existence et
l’unicité pour (GDε ) , ε > 0 ; obtenir la solution admissible de (GD0 ) comme la limite
des solutions de (GDε ) lorsque ε ↓ 0 , et observer la formation du vide.
· Les principaux résultats sont :
- caractérisation des solutions de (GDε ) pour ε > 0 , description des solutions avec
et sans le vide, et réduction de (GDε ) , ε > 0 , à un problème aux limites pour une
équation différentielle ordinaire (Lemma 1, Propositions 1,2,3, Lemmae 4,5,6);
- unicité pour (GDε ) , ε > 0 et existence sous des hypotheses de croissance de p(·) en
+∞ (Theorem 1);
- convergence des solutions de (GDε ) vers une limite, donnée par une formule explicite,
lorsque ε ↓ 0 (Theorem 2); observation de la structure de la solution admissible de
(GD0 ) , conditions nécessaires et suffisantes d’existence d’une solution de (GDε ) (formule (38)) et d’apparition du vide (formule (39)).
16
Introduction
• Chapitre 1.IV
La théorie L1 des lois de conservation scalaires avec une fonction de flux
continue† .
· On traite les problèmes suivants :

 ∂u
+ divx φ(u) = g
(CP )
∂t
 u| = f,
t=0
(Eq)
u + divx φ(u) = f,
u : (0, T ) × IRN 7→ IR,
u : IRN 7→ IR,
avec φ : IR 7→ IRN continue. Le cadre fonctionnel principal est c + L1 ∩ L∞ , c ∈ IR .
L’unicité d’une solution (c’est-à-dire, solution entropique généralisée) pour (CP ) et pour
(Eq) reste un problème ouvert pour N ≥ 2 .
· Les références principales sont :
Bénilan, Kruzhkov [BK96], Kruzhkov, Panov [KP90], Kruzhkov [K69a, K69b, K70a],
Bénilan [B72], Crandall, Liggett [CL71], Bénilan, Crandall, Pazy [BCP].
· Dans ce chapitre, on se donne pour objectif :
démontrer l’existence de la solution maximum et la solution minimum pour (CP ) et
(Eq) ; établir le lien entre les solutions des deux problèmes dans le cadre de la théorie
des semi-groupes non-linéaires; établir des résultats partiels d’unicité, en particulier en
généralisant ceux de [BK96]; argumenter en faveur d’unicité en général.
· Les principaux résultats sont :
- l’existence d’une solution pour (Eq) et (CP ) dans le cadre L∞ (Lemma 2, Theorem 6);
- l’existence de la solution maximum et la solution minimum pour (Eq) et (CP ) (Theorem 1);
- la génération du semi-groupe “solution maximum de (CP)” par l’opérateur “solution
maximum de (Eq) ”, accrétif à domaine dense, et pareil pour les solutions minimum
(Proposition 2, Theorem 2, Remark 1);
- pour une fonction de flux φ(·) et une constante c données, l’équivalence de l’unicité
en général pour (Eq) et pour (CP ) (Corollary 1);
- pour une fonction de flux φ(·) donnée, l’unicité est garantie pour tout c ∈ IR , excepté
peut-être un ensemble au plus dénombrable (Propositions 1,3);
- l’unicité pour (Eq) et (CP ) dans le cas où (N − 1) composantes de φ(·) sont
monotones, par récurrence sur la dimension d’espace N (Theorem 3);
- l’unicité pour (Eq) et (CP ) dans le cas où les modules de continuité en zéro des
composantes de φ(· + c) − φ(c) satisfont la condition anisotropique de décroissance de
[KP90, BK96] (Theorem 4);
†
Il s’agit d’un travail fait en commun avec Philippe Bénilan et Stanislav N. Kruzhkov.
0.I.
Les énoncés des problèmes et un résumé des résultats obtenus
17
- malgré cela, et malgré le résultat de non-unicité dans le cadre L∞ de [KP90] qui
souligne la pertinence de la condition anisotropique dans L∞ , conclusion que la nonunicité dans le cadre L1 ∩L∞ , le cas échéant, n’est pas intrinséquement liée au caractère
non-hölderien de la fonction de flux (note en bas de page sous Theorem 4).
Partie 2.
Solutions faibles pour
les systèmes elliptiques-paraboliques
On étudie l’existence des solutions faibles et la convergence des solutions approchées pour
une classe de systèmes paraboliques dégénérés. La méthode suit et éclaircit l’approche variationnelle suggérée par Alt et Luckhaus ([AL83]).
• Chapitre 2.I
Problèmes elliptiques-paraboliques : l’existence et la continuité par rapport
aux données de solutions faibles† .
· On traite les problèmes du type suivant :

b(·, v)t = div a(·, v, Dv) + f (·, v) sur (0, T ) × Ω ⊂ IR+ × IRd



 b(·, v(·))| = u sur Ω
t=0
0
(P r)

+ conditions aux limites mixtes



de Dirichlet h(·) et de Neumann g(·, v) sur (0, T ) × ∂Ω.
Ici v : (0, T ) × Ω 7→ IRN , la fonction b : (0, T ) × Ω × IRN 7→ IRN est telle que
b(t, x, ·) est le gradient d’une fonction convexe differentiable Φ(t, x, ·) : IRN 7→ IR . On
prend des hypothèses appropriées sur les données b, a, f, g, h, u0 qui généralisent, pour
une partie, les conditions de [AL83] et [LJLL65] (les hypothèses (H1)−(H9) en général,
les hypothèses (1)−(3) dans le cas simple avec f, g, h qui s’annulent complètement et
sans dépendance en (t, x) ).
· Les références principales sont :
Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Leray, J.-L.Lions [LJLL65], Kruzhkov [K69a],
Bénilan, Wittbold [BW99].
· Dans ce chapitre, on se donne pour objectif :
obtenir un résultat d’existence et de dépendance en données de solutions faibles pour (P r)
avec les coefficients qui peuvent dépendre de (t, x) ; mettre en évidence les arguments
essentiels dans la méthode variationnelle proposée dans [ALpr, AL83]; discuter la nécessité
des conditions de structure sur a , f , Φt .
†
Il s’agit d’un travail fait en commun avec Philippe Bénilan.
18
Introduction
· Les principaux résultats sont :
- une version du Z“chain rule” lemme, c’est-à-dire
Z
d
(ChR)
B(t, ·, v(t, ·)) =< b(t, ·, v(t, ·))t , v(t, ·) >E ′ ,E − Φt (t, ·, v(t, ·)),
dt Ω
Ω
où E est l’espace
où
l’on
cherche
les
solutions
variationnelles,
E ′ est son dual et
Z
z
B(t, x, z) est
0
(b(z) − b(ζ)) dζ (Lemma 1);
- une version de l’argument de compacité de [K69a], qui suggère que, pour une famille
de solutions d’équations d’évolution, la compacité dans L1 en (t, x) se déduit d’une
estimation L1 sur les translatées en x des solutions, à condition d’avoir des estimations
uniformes sur les solutions dans L1 d’une part, et sur la partie droite des équations dans
un espace de Sobolev négatif d’autre part (Lemma 6);
- une version appropriée de l’argument de Minty-Browder (Lemma 7);
- un théorème de continuité pour les solutions faibles par rapport aux données de (P r) ,
dans une topologie naturelle vis-à-vis des hypothèses prises, sous des hypothèses de
structure supplémentaires (H11) − (H13) sur a , f et Φt et dans le cas h = 0 ;
dans la preuve de ce résultat, on applique successivement les trois arguments ci-dessus
et un argument qui combine l’équi-intégrabilité avec le théorème de Egorov (Theorem 1,
Remark 3);
- corollaires d’existence d’une solution faible pour (P r) (Corollaires 1,2,3, Remark 4);
- indications sur la non-pertinence des restrictions imposés par les conditions de structure
(H11), (H12) pour l’existence d’une solution faible.
• Chapitre 2.II La convergence des approximations par les méthodes des volumes finis
pour un problème elliptique-parabolique non-linéaire : une approche
variationnelle† .
· On traite le problème suivant :

+
d

 b(v)t = div ap (Dv) sur (0, T ) × Ω ⊂ IR × IR
(pL)
b(v)|t=0 = u0 sur Ω


v = 0 sur ∂Ω,
pour un domaine polyhédral Ω ⊂ IRd , où b est le gradient d’une fonction convexe
differentiable Φ : IRN 7→ IRN et div ap (D·) est un p-laplacien.
· Les références principales sont :
Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Eymard, Gallouët, Herbin [EyGaHe], Chapitre
2.I
†
Il s’agit d’un travail fait en commun avec Michaël Gutnic et Petra Wittbold.
0.I.
Les énoncés des problèmes et un résumé des résultats obtenus
19
· Dans ce chapitre, on se donne pour objectif :
écrire un schéma de volumes finis pour (pL) , démontrer la consistance du schéma et
proposer une approche de preuve de la convergence qui cherche à réduire le problème au
cas continu du Chapitre 2.I autant que possible.
· Les principaux résultats sont :
- proposition d’une classe de schémas de volumes finis qui approche le gradient d’une
manière satisfaisante vis-à-vis de la non-linéarite de ap , puis d’un exemple d’un tel
schéma (Définition 4, Remark 2);
- existence d’une solution discrète et estimations a priori (Theorem 2, Proposition 1);
- introduction des approximations “continues” de la solution discrète vh , du terme parabolique b(v h ) et du terme elliptique ap (“Dvh ”) ayant de “bonnes propriétés”; réécriture
du système d’équations algébriques issu du schéma des volumes finis sous une forme
d’équation dans D ′ (Lemmae 2,5,6, Proposition 2);
- consistance de l’approximation par la méthode des volumes finis de l’opérateur div ap (D·)
pour la classe de schémas proposée (Definition 5, Proposition 3, Theorem 3);
- convergence de v h , lorsque le pas de discrétisation tend vers zéro, vers une solution
faible de (pL) dans la classe de schémas proposée, qui est démontrée par la méthode
variationnelle du Chapitre 2.I appliquée aux approximations “continues” construites auparavant (Theorem 1).
CHAPITRE II
Les “quelques problèmes” dans le contexte
mathématique et physique†
1. La théorie des équations aux dérivées partielles, s’il y en a une, a pour points de départ “les
trois baleines”: les trois équations classiques. Ce sont l’équation hyperbolique (H) utt =
∆u , dite équation d’ondes; l’équation parabolique (P ) ut = ∆u , dite équation de la
chaleur; l’équation elliptique (E) ∆u = 0 , dite équation de Laplace. Il est habituel de
parler de caractère hyperbolique, parabolique ou elliptique dans des problèmes plus généraux,
en se basant sur la ressemblance des méthodes appliquées et le comportement des solutions.
Pour chacun des archétypes (H) , (P ) , (E) il existe un cadre spécifique et une théorie
mathématique à part entière, qui observe des phénomènes qui lui sont propres. Ainsi, (H)
et (P ) décrivent l’évolution en temps, tandis que (E) est relatif à la stabilité en espace.
Il y a un effet régularisant pour (P ) et (E) , c’est-à-dire, la régularité des solutions est
meilleure que celle des données; cela n’est pas le cas de (H) . Tandis que l’on observe
dans (P ) les phénomènes de propagation à vitesse infinie et de dissipation d’énergie, (H)
présente les propriétés de conservation d’énergie et de domaine de dépendance fini. On
pourrait énumérer d’autres différences substantielles entre (H) , (P ) et (E) , et pourtant
ces équations linéaires sont les modèles les plus simples de phénomènes réels qui surgissent
en physique. Quand on veut concevoir des modèles plus généraux ou plus réalistes, on
est souvent amené à considérer des problèmes qui sont, d’une part non-linéaires, et d’autre
part, qui mélangent en eux les comportements typiques pour (H) , (P ) , (E) . Le caractère
non-linéaire restreint dramatiquement les outils qui ont été conçus lors du traitement de
(H) , (P ) , (E) , et amène à en utiliser de nouveaux dans chaque cas qui se présente.
†
On se donne pour objectif de présenter ici une vue générale sur l’ensemble des sujets abordés dans cette
thèse. Deux points de vue s’imposent : celui du contexte mathématique des équations aux dérivées partielles
non-linéaires, mais aussi celui des motivations et des modèles physiques qui pèsent considérablement sur le
sujet.
Après un petit avant-propos, on fait le tour d’horizon des problèmes traités. Il est suivi par une discussion
des motivations, méthodes et résultats de chacune des deux parties de cette thèse. En conclusion, on explicite
un certain nombre des points d’interrogation que soulève le travail présenté à Votre attention.
22
Introduction
Ainsi, j’ai été confronté dans l’étude des problèmes cernés dans le Chapitre 1 ci-dessus à
un nombre de difficultés qui proviennent de la non-linéarité, de la non-régularité des données,
de changement ou d’imprécision sur le type des équations étudiées.
2. Les systèmes hyperboliques des lois de conservation font l’objet de la première partie de
la thèse. La loi scalaire (CP ) , étudiée dans les chapitres 1.I et 1.IV, est très interessante
en elle-même. Elle correspond à un certain nombre de phénomènes, elle peut modéliser le
traffic routier par exemple; et elle représente le premier pas vers l’étude de systèmes aussi
importants que ceux de la dynamique des gaz. Dans les chapitres 1.II et 1.III, on traite
le problème qui a été posé et résolu pour la première fois par B.Riemann ([R1860]) et qui
consiste à poursuivre la résolution du système de la dynamique des gaz isentropiques en
présence de discontinuités. Les systèmes (pS0 ) , (GD0 ) concernés par les chapitres 1.II et
1.III modélisent les écoulements isentropiques des gaz parfaits en coordonnées de Lagrange
et d’Euler, mais aussi l’élasticité non-linéaire et les fluides de van der Waals (cf. [R1860,
Ray10, CouFr, RoJa, Se, ChHs, Tz98, Ja80, Pe87]). Dans ce dernier cas en particulier, qu’on
traite en coordonnées de Lagrange, on est contraint à renoncer au caractère hyperbolique
du système, à cause de la présence des phases hyperboliques et elliptiques et des transitions
entre elles. Dans les coordonnées d’Euler, on considère aussi les cas d’apparition de l’état du
du vide dans les solutions; le vide est la dégénéréscence spécifique à certains systèmes de la
dynamique des gaz. L’approche employée dans les chapitres 1.I-1.III nécessite l’obtention de
résultats d’existence et d’unicité pour les problèmes de Riemann (CLε ) , (pSε ) , (GDε ) qui
sont les systèmes régularisés avec un terme parabolique lui-même dégénéré.
De plus, on traite dans (CL0 ) , (pS0 ) , (GD0 ) et (CP ) le cas de fonction de flux
seulement continue. Ceci n’est en rien pathologique; en effet, une équation aussi simple
que (u3)t + (u2 )x = 0 donne lieu à la loi de conservation scalaire wt + f (w)x = 0 avec
f : w 7→ w 2/3 , qui n’est pas une fonction localement lipschitzienne. Le manque de régularité
de la fonction de flux entraı̂ne des effets extrémement intéressants dans le cas d’une loi de
conservation scalaire multidimensionnelle. On observe le phénomène de propagation à vitesse
infinie, ce qui peut entraı̂ner la perte du caractère conservatif de l’équation et la non-unicité
d’une solution dans la classe (pourtant très bonne, cf. [K69a, K69b, K70a, PA71, KH74,
KPA75]) de solutions entropiques généralisées dans L∞ (cf. [B72, KP90, BK96]). La nonunicité n’est peut-être pas le cas dans le cadre de solutions L1 ∩ L∞ ; l’investigation de ce
dernier problème, couplée avec une étude de l’équation (Eq) , et qui développe les résultats
de [B72, KP90, BK96], fait l’objet du chapitre 1.IV.
D’un autre coté, l’unicité pour le problème de Riemann dans le cas scalaire uni-dimensionnelle (CL0 ) , pour une classe de solutions vérifiant un critère d’admissibilité usuel depuis
[D74, D89] (et qui remonte aux idées de [Ray10, H50, G59, Ka59]), est étudiée directement
dans le chapitre 1.I. Des études très similaires sont menés à bien dans chacun des chapitres 1.II
et 1.III. Ceci étend en particulier les résultats de [Ka59, Tz95, Sh82, Kim99].
0.II.
Sur le contexte mathématique et physique
23
Des problèmes liés aux dégénéréscences et singularités se présentent également dans les
équations des milieux poreux (cf. [Bear, AL83]). Les systèmes que l’on considère peuvent
en particulier contenir des phases purement elliptiques. Par exemple, pour l’équation de
Richards, qui trouve d’importantes applications dans les sciences d’environnement, le terme
parabolique atteint le régime stationnaire en 1 (l’état de saturation). Dans les régions
de dégénérescence elliptique, l’évolution en temps d’une solution n’est que partiellement
controlée, ce qui ne permet pas d’obtenir de la compacité forte des solutions là où c’était le
cas pour les systèmes paraboliques non-dégénérés. Cela complique en particulier la question
de l’existence des solutions.
On fait une étude d’existence et de dépendance continue pour les solutions faibles des
systèmes elliptiques-paraboliques (P r) dans le chapitre 2.I, en étendant en particulier le
résultat d’existence de [AL83]. On espère avoir atteint une plus grande clarté dans l’exposition
des arguments qui mènent à l’existence d’une solution faible pour (P r) , par rapport au papier
[AL83] qui est ici la référence cruciale.
Vu l’intéret que suscite le développement de schémas numériques pour la modélisation
d’écoulements dans les milieux poreux, on aborde dans le chapitre 2.II la convergence pour
les méthodes des volumes finis, employées fréquemment dans ce domaine (cf. [EyGaHe]).
On teste la convergence sur l’exemple d’une équation (en fait, d’un système) (pL) qui est
une forme simplifiée du modèle de filtration d’un fluide en régime turbulent à travers un
milieu poreux, gouvernée par une loi de Darcy non-linéaire, ou encore d’écoulement d’un gaz
turbulent dans un pipeline (cf. [DiDT94] et leurs références). A ma connaissance, il n’existe
pas d’autre résultat de convergence de schémas de volumes finis pour des équations de type
(pL) et, plus généralement, (P r) dans le cas p 6= 2 ; en étudiant (pL) , on cherche à
débroussailler le chemin pour pouvoir aborder un jour des cas plus réalistes.
3. Un problème essentiel de la théorie des systèmes hyperboliques des lois de conservation est
celui de l’unicité. C’est la théorie des solutions entropiques généralisées, dites g.e.s., qui résout
ce problème pour les lois de conservation scalaires multi-dimensionnelles. La définition et les
méthodes classiques, élaborées dans [K69a, K69b, K70a], donnent une théorie mathématique
tout à fait satisfaisante, au moins dans le cas de fonction de flux localement lipschitzienne.
On reviendra sur ce sujet dans la partie consacrée au chapitre 1.IV. Avant, on aborde une
autre facette de la question de l’unicité pour les lois et systèmes de conservation, qui est les
conditions d’admissibilité des chocs. Notons que la définition d’une g.e.s. se traduit par une
telle condition dans les solutions continues par morceaux (où, plus généralement, dans les
solutions à variation localement bornée). Mais les conditions d’admissibilité des chocs sont
antérieures à la notion d’une g.e.s. (cf. [H50, O57, Lax57, G59]), et les deux sont réliées
par une motivation commune qui est la méthode de viscosité. Ceci est important, vu les
récents développements dans le domaine des chocs non-classiques (cf. [LF98]). On peut, en
effet, construire des théories non-classiques pour certaines lois de conservation non-convexes,
24
Introduction
et où les solutions s’obtiennent comme limites d’approximations par la diffusion couplée à
une dispersion lorsque les deux tendent vers zéro. Tout ce qui a été dit exige une certaine
explication, ou plutôt un détour historique. Je ne donne pas les références à beaucoup
de travaux importants qui ont amené à une bonne position du problème mathématique de
résolution des lois de conservation; les noms des principaux intervenants et leurs contributions
sont contés dans l’introduction de l’article [D89].
La difficulté de la résolution globale en temps des systèmes de la dynamique des gaz est
connue depuis le XIX siècle. Elle surgit à cause du caractère non-linéaire de propagation
d’ondes. A la différence de l’équation d’ondes (H), où la régularité des données initiales
est preservée (faute d’effet régularisant), les solutions d’une loi de conservation non-linéaire
peuvent développer des discontinuités en temps fini même lorsque les données initiales sont
analytiques. B.Riemann ([R1860]) semble être le premier qui accepte des solutions discontinues dans la dynamique des gaz isentropiques et construit une solution pour le problème
d’évolution d’une discontinuité élémentaire, qui porte son nom. En particulier, il ne choisit
pour la construction de la solution que “la moitié” des chocs envisageables, en préssentant
ainsi la condition d’admissibilité de Lax (cf. [Lax57]). Un bref résumé du travail de Riemann
est donné dans [Se, Chap.4.6]. L’idée n’a pas fait l’unanimité. Rayleigh ([Ray10]) écrit à propos de la condition de Rankine-Hugoniot, qui décrit les chocs envisageables dans une solution
discontinue: “however valid <the Rankine-Hugoniot condition> may be, its fulfillment
does not secure that the wave so defined is possible. As a matter of fact, a whole class
of such waves is certainly impossible, and I would maintain, further, that a wave of the
kind is never possible under the conditions, laid down by Hugoniot, of no viscosity or
heat conduction.†” C’est en rétablissant une petite viscosité et/où conductivité que Rayleigh
donne un sens à la distinction entre les “régimes permanents” (les chocs) admissibles et
inadmissibles; distinction qui n’a pas été remarquée par Rankine lorsque ce dernier avait
déduit ses conditions de choc en partant des fluides conductifs. Le point de vue que défend
Rayleigh est, bien sûr, celui du mécanicien. Il ne distingue pas le problème mathématique du
phénomène physique que ce problème est appelé à décrire. La notion même d’une solution
discontinue lui est étrangère: les “régimes permanents” ne sont pour lui rien d’autre que des
ondes de compression dans une région d’espace très petite, et qui sont entretenues par la
dissipation d’énergie dans la zone de transition.
Bien sûr, aujourd’hui on peut donner très facilement un sens purement mathématique à
un système de lois de conservation. C’est le sens faible, d’ailleurs plus naturel qu’une relation
différentielle dans le contexte physique de la conservation locale des grandeurs extensives.
On peut trouver beaucoup de solutions discontinues dans ce sens pour un même problème
†
“aussi valide qu’elle < la condition de Rankine-Hugoniot > soit, rien ne garantit qu’une onde ainsi définie
soit possible dans le cas où elle < la condition de Rankine-Hugoniot > est remplie. En fait, la classe entière
de ces solutions est certainement impossible, et je maintiendrai, en outre, qu’une onde de la sorte n’est jamais
possible sous les conditions, imposées par Hugoniot, d’absence de viscosité et de transmission de la chaleur.”
0.II.
Sur le contexte mathématique et physique
25
de Riemann, de sorte qu’il n’y a pas d’unicité. Ce qui distingue les différentes solutions
dans le cas de la dynamique des gaz, c’est leurs propriétés vis-à-vis de la conservation de
l’énergie. Rayleigh semble être le premier à observer que “maintenance of type in such a
<shock> wave <of condensation> requires removal of energy from the wave, while in
the contrary case of <shock> wave of rarefaction additional energy would need to be
supplied‡ .” Ce qui permet de trancher, dans le cadre de modèle considéré, en accord avec
le deuxième principe de thermodynamique dont l’importance avait été négligée auparavant:
“although dissipative forces, such as those arising from viscosity, may possibly constitute
a machinery capable of maintaining the type of <shock> wave of condensation, in no
case they can maintain the type of <shock> wave of rarefaction§ .” Du point de vue
moderne, on peut dire que Rayleigh postule implicitement que seules les limites des solutions
du système prenant en compte de petits effets dissipatifs sont admissibles comme solutions
faibles du système de la dynamique des gaz isentropiques. C’est de là que la théorie classique
des lois de conservation est partie quarante ans plus tard, avec le papier de Hopf [H50]
sur l’équation ut + (u2 )x = 0 . Une solution faible du problème de Cauchy pour cette
équation est construite comme la limite des solutions du même problème pour l’équation
ut + (u2 )x = εuxx , que l’on peut résoudre explicitement. Il résulte du travail de Hopf que
seules les chocs qui joignent un état u− à gauche à un état inférieur u+ à droite sont
admissibles. Un moyen facile de “prédire” ce dernier résultat est de considérer la possibilité
d’approximation de chocs par les ondes planes, approche qui fait d’ailleurs le titre de ce
même papier de Rayleigh [Ray10]. Dans [G59], Gelfand montre déjà comment résoudre le
problème de Riemann pour une loi de conservation d’une manière unique, en partant de cette
caractérisation par les ondes planes. Une autre variante de l’idée de viscosité apparait chez
Kalashnikov ([Ka59]). C’est cette version, vue à travers l’intérprétation de Dafermos ([D89]),
qui donne lieu au chapitre 1.I, puis 1.II et 1.III ci-dessous.
C’est toujours l’idée de viscosité qui tend vers zéro qui suggère la pertinence de la définition
d’une g.e.s. de Kruzhkov. Cette définition propose une caractérisation intrinsèque d’une
solution; la motivation reste extérieure (Kruzhkov l’écrit très clairement dans ses notes de
cours [K70b], dont la diffusion reste malheureusement très limitée). La théorie des g.e.s.
donne une réponse parfaite aux questions d’existence, d’unicité et de dépendance continue
pour le cas d’une loi de conservation à fonction de flux localement lipschitzienne, dans la
classe de fonctions bornées, et cela en plusieurs dimensions d’espace. Hélas, la généralisation
aux systèmes de l’approche entropique sous forme de Lax (cf. [K70a, Lax71]) rencontre
‡
“la maintenance d’une telle onde < de choc de condensation > nécessite d’enlever de l’énergie à l’onde,
tandis que dans le cas contraire d’une onde < de choc > de détente il faudrait lui livrer de l’énergie additionnelle.”
§
“bien que les forces dissipatives, comme celles qui proviennent de la viscosité, pourraient constituer une
machinerie capable de maintenir le type d’onde < de choc > de condensation, en aucun cas elles ne peuvent
maintenir le type d’onde < de choc > de détente.”
26
Introduction
des difficultés importantes dans le cas des systèmes 2 × 2 , et incontournables au-delà. Ceci
incite à reprendre l’étude d’admissibilité des chocs. Or, on sait actuellement (cf. [Br96, Br99])
que la bonne condition d’admissibilité de chocs (cf. [Lax57]) pour la classe très étudiée de
systèmes vraiment nonlinéaires ou linéairement dégénérés (cf. [Lax57, Sm, Se]) peut en elle
seule donner l’unicité d’une solution. La question de l’unicité, tout comme la question de
l’existence (cf. [Gl65, Br99]), peut en fait être ramenée à l’admissibilité pour les problèmes
de Riemann. Pour les systèmes strictement hyperboliques, il existe une construction (cf.
[Liu75, Liu81]) qui permet la résolution d’une manière unique d’un problème de Riemann
lorsque le saut dans les données est petit. La résolution globale peut être beaucoup plus
compliquée dans chaque cas qui se présente.
Or, c’est une approche très bien adaptée pour la résolution du problème de Riemann que
propose Dafermos dans [D73a, D74, D89], et qui d’ailleurs permet de juger admissible ou non
une solution toute entière, plutôt que de la considérer choc par choc. L’unique solution d’un
problème de Riemann ne peut dépendre que du quotient x/t ; autrement dit, la solution est
auto-similaire. L’addition d’une viscosité comme celle dans (CLε ) , (pSε ) , (GDε ) permet de
préserver cette propriété. L’idée de l’approche par viscosité auto-similaire est donc de ramener
la discussion d’un problème de Riemann au niveau des équations différentielles ordinaires; aussi
une variété de méthodes s’applique (cf. [D74, DDp76, Tz96], et [Tz98] pour un état des lieux
dans ce domaine). Le programme habituel est de démontrer, par des méthodes topologiques,
l’existence pour le problème régularisé; obtenir des estimations sur la variation des solutions,
puis passer à la limite par compacité; enfin, étudier la structure des limites vis-à-vis de la
possibilité d’approximation des ondes de choc par les ondes planes.
Tout en profitant de la forme auto-similaire, mais également de la structure élémentaire
des systèmes particuliers étudiés, je propose dans les chapitres 1.I-1.III une approche différente
d’investigation des solutions. Les moyens mathématiques employés ici sont simples, pour
ne pas dire rudimentaires. Par une analyse de plus en plus laborieuse, on est amené dans
chacun des cas (CLε ) , (pSε ) , (GDε ) à résoudre un problème aux limites pour une équation
différentielle ordinaire. A chaque fois, cette équation admet un principe de maximum; celuici me permet d’établir des résultats très précis d’existence, d’unicité et de convergence des
solutions lorsque ε ↓ 0 . Ainsi on démontre dans chaque cas que la viscosité auto-similaire
choisit une unique solution pour les problèmes de Riemann. En plus, les formules explicites
(certes, connues) pour les solutions limites, que l’on obtient directement, permettent de
distinguer les chocs qui ne peuvent apparaı̂tre dans aucune solution et qu’on peut donc juger
comme inadmissibles; ceci remplace l’étude par les ondes planes.
Vu ce qui a été dit sur le bien-fondé de la définition d’une g.e.s. et l’origine commune
des différentes branches de la théorie des systèmes des lois de conservation, il est étonnant
qu’il existe une méthode purement mathématique qui donne un sens intrinsèque aux lois
de conservation. C’est d’ailleurs un des deux traits d’union entre les deux parties de cette
0.II.
Sur le contexte mathématique et physique
27
thèse. Cette méthode qui s’applique en même temps aux lois de conservation scalaires et
aux équations elliptiques-paraboliques est l’approche par les semi-groupes non-linéaires. D’un
point de vue pratique, elle consiste à construire les solutions des équations d’évolution par le
procédé de discrétisation implicite en temps. Le trait commun des opérateurs associés à une
loi de conservation scalaire et à une équation elliptique-parabolique coercive est la propriété
d’accrétivité, qui donne la possibilité de continuer l’approximation à chaque pas de temps
d’une manière stable. La théorie de semi-groupes non-linéaires peut permettre d’établir
l’existence; mais en outre, elle peut donner l’unicité d’une solution dans les cas où cette
unicité est un véritable problème, (CP ) par exemple. Le procédé même de construction
des solutions semi-groupes (cf. [BCP]) inscrit la condition d’admissibilité dans les solutions;
l’irréversibilité en temps, suggérée dans les modèles physiques par la deuxième loi de la
thermodynamique, est ainsi capturée (cf. [BCP88]).
Les résultats du chapitre 1.IV marquent davantage le lien qui existe entre les solutions
entropiques généralisées (cf. [K69a, K69b, K70a]) et les solutions semi-groupes (cf. [B72])
des lois de conservation. On utilise l’approche par semi-groupes non-linéaires pour établir
la génération des semi-groupes des g.e.s. maximum, minimum de (CP ) par les opérateurs
de la g.e.s. maximum, minimum de (Eq) , respectivement. On en déduit l’équivalence
entre l’unicité pour (CP ) et l’unicité pour (Eq) . Cela s’applique à la démonstration du
Théorème 3 qui se fait par récurrence sur la dimension N d’espace et qui est le résultat
le plus intéressant du chapitre 1.IV. Les autres techniques employées sont essentiellement
celles du papier [BK96]. L’existence des g.e.s. maximum et minimum, en absence présumée
d’unicité lorsque la fonction de flux n’est pas localement lipschitzienne, n’est pas un fait
trivial; on la prouve ici pour des données qui tendent vers une constante à l’infini. Comme
pour beaucoup d’autres résultats de ce travail, la démonstration est basée sur la possibilité
de prendre la fonction caractéristique de IRN tout entier comme la fonction test dans une
inégalité de type Kato (Lemma 3.1 dans [BK96]). Récemment, le résultat d’existence des
solutions maximum et minimum pour (CP ) et pour (Eq) a été étendu au cas des données
bornées générales (cf. [P..]). Cela est d’autant plus intéressant que l’on connaı̂t un exemple
de non-unicité (cf. [KP90]). Par contre, on remarque en passant que la non-unicité, le cas
échéant, d’une g.e.s. pour (CP ) et (Eq) dans le cadre L1 ∩ L∞ doit être gouvernée par
une propriété fine de la fonction de flux.
4. Jusqu’il y a quelques années, l’unicité pour les systèmes hyperboliques de lois de conservation manquait de résultats aussi importants que celui de Kruzhkov dans le cas scalaire.
C’est une analyse très engagée de la structure d’ondes qui a permis à Bressan et ses collaborateurs ([Br96, Br99]) d’apporter une lumière de compréhension à cette question. En
effet, j’ai déjà mentionné les difficultés de l’approche entropique; l’approche par les semigroupes non-linéaires est freinée par le fait que l’on ne connaı̂t pas de norme pour laquelle
l’opérateur associé au problème d’évolution soit accrétif. Cette même difficulté surgit dans
28
Introduction
l’étude d’unicité pour les systèmes elliptiques-paraboliques de type (P r) , tandis que dans le
cas scalaire une réponse satisfaisante à la question d’unicité est donnée dans [Ot96, CaW99]
et [Ca99]. Dans ce cadre, la non-unicité, le cas échéant, est due à la dégénérescence de la
partie parabolique et non à l’irrégularité générique des solutions, comme c’est le cas pour les
systèmes hyperboliques de lois de conservation.
Mais c’est encore l’existence d’une solution faible pour (P r) qui soulève des questions.
La méthode classique de construction d’une solution variationnelle (cf. [AL83]) exige les
hypothèses supplémentaires (dites conditions de structure) de la dépendance des coefficients
a, f en b(z) et non en z tout simplement. Or, l’approche par les semi-groupes non-linéaires
en particulier montre que ces conditions ne sont pas toujours indispensables (cf. [BW99]).
De plus, on trouve des indications à la même conclusion tout en restant dans le cadre de
l’approche variationnelle de Alt-Luckhaus ([AL83]), qui est la méthode du chapitre 2.I. Pas
très importante du point de vue des applications physiques, la question de pertinence des
conditions de structure n’en reste pas moins obsédante. Je n’ai pas réussi à la résoudre,
bien qu’un travail important a été fait dans le chapitre 2.I pour “minimiser” les restrictions
imposés sur (P r) par l’approche de Alt-Luckhaus.
Cette méthode prend sa première origine dans le travail [Bro63], qui se base sur la possibilité d’appliquer l’approche de monotonie introduite dans [Mi62, Mi63] à la construction de
solutions aux équations elliptiques. Cette idée à été étendue à une méthode “de monotonie
et compacité” (cf. [JLL]) qui permet de résoudre par l’approche variationnelle, à travers la
méthode de Galerkin, une classe de problèmes elliptiques ([LJLL65]) et paraboliques ([JLL]).
Les conditions de Leray-Lions réapparaissent dans tout le procédé de développement de cette
approche variationnelle.
Dans le cas très important de dégénérescence elliptique d’un système parabolique, deux
autres arguments essentiels ont été apportés dans [AL83]. Le premier (cf. aussi [Bam77])
impose à la fonction b dans (P r) d’être le gradient en z d’une fonction convexe; c’est le
“chain rule” lemme de type (ChR) . Il permet d’obtenir dans un premier temps les estimations a priori qui remplaçent celles de Leray-Lions, et de passer à la limite par l’argument de
Minty-Browder dans la phase finale de la démonstration. L’autre restriction que l’utilisation
de ce lemme impose se traduit dans les intreprétations physiques par la finitude de l’énergie
dans l’état initial. Le deuxième argument est celui de la compacité L1 en temps et en espace
non pas pour les solutions v , mais pour les termes paraboliques b(v) . C’est ce deuxième
argument qui est remplacé dans le chapitre 2.I par un lemme assez ancien ([K69a]) et très
bien adapté à la question qu’on se pose. L’utilisation de ce dernier lemme, qui est un résultat
de compacité très général pour les équations d’évolution, sert aussi de trait d’union entre les
deux parties de cette thèse. C’est en effet ce lemme qui rend facile l’existence d’une g.e.s.
bornée pour le problème de Cauchy pour une loi de conservation scalaire avec une fonction de
flux continue; une démonstration écrite de ce résultat, qui semble manquer, est donnée dans
0.II.
Sur le contexte mathématique et physique
29
les annexes du chapitre 1.IV. De la même manière, on donne dans les annexes du chapitre 2.I
les versions appropriées des trois arguments qui mènent à l’existence pour (P r) : le “chain
rule” lemme, le lemme de compacité dans L1 et l’argument de Minty-Browder.
Parallèlement à la simplification de la preuve, on couvre dans le chapitre 2.I le cas de
dépendance des coefficients dans (P r) en (t, x) , en particulier de la dépendance de b
en t (qui s’avère délicate, au point de donner lieu à un problème qui est resté ouvert).
A ce moment un quatrième argument apparaı̂t, une combinaison d’équi-integrabilité des
termes dans l’équation avec le théorème de Egorov, et qui ramène la discussion au niveau
de coefficients uniformément continus en l’ensemble des variables. Quant aux conditions
de type Leray-Lions, on essaie d’imposer les hypothèses les moins restrictives dans le cadre
Lp (0, T ; W 1,p(Ω; IRN )) , choisi pour les solutions faibles. Une étude de (P r) dans les espaces
d’Orlitz, à l’instar de [Kac90], doit permettre d’affaiblir certaines restrictions. Notons qu’on
ne traite pas dans le chapitre 2.I en toute généralité le cas où la condition au bord sur
une des composantes de la solution est une condition de Neumann pure. D’une part, cela
exigerait une hypothèse supplémentaire sur la fonction b . D’autre part, bien que l’essentiel
de nos arguments s’applique aussi dans cette situation, on ne peut pas inclure ce cas dans le
théorème principal sans nuire davantage à la lisibilité de la preuve.
Le résultat principal qu’on démontre dans le chapitre 2.I est que l’ensemble des solutions faibles de (P r) (faute d’en connaı̂tre l’unicité) est semi-continu inférieurement par
rapport aux perturbations des coefficients et des données de (P r) dans la topologie naturelle. L’existence est obtenue comme un corollaire de ce résultat, en utilisant la méthode
de Galerkin tout comme dans [JLL] and [ALpr]. C’est ce même théorème de continuité qui
donne la convergence des approximations de Galerkin.
Des problèmes elliptiques-paraboliques modélisent, en particulier, des phénomènes qui
intéressent l’industrie pétrolière. D’où l’importance des aspects numériques. Les méthodes
classiques utilisés en résolution numérique de tels systèmes, du type éléments finis, ont un
inconvénient certain. C’est l’absence du caractère conservatif local. Par contre, les méthodes
de type volumes finis sont conçus pour assurer cette conservativité (cf. [EyGaHe]). A la
différence des méthodes de Galerkin, on ne peut pas directement appliquer les résultats de
[AL83] ou du chapitre 2.I pour les volumes finis. Dans le chapitre 2.II, on s’est donné
pour objectif d’adapter le résultat du chapitre 2.I au cas d’approximation du système par les
méthodes des volumes finis.
On a choisi le système (pL) , fortement non-linéaire, pour montrer la possibilité d’une
telle adaptation. Le chemin qui est souvent emprunté pour aborder un tel problème est la
“discrétisation” des arguments de la preuve du cas “continu”; en particulier, c’est le cas de
[EGH99], où un résultat de convergence des approximations par un schéma de volumes finis
est démontré pour l’équation de Richards, qui contient (pL) au cas scalaire pour p = 2 .
On a choisi l’approche inverse, c’est-à-dire, d’appliquer les arguments du cas continu aux
30
Introduction
versions “continues” de la solution discrète et du système d’équations algébriques qui la
détermine. Le premier pas consiste donc à écrire ce système sous la forme d’une équation
dans D ′ . On arrive ensuite à garder la carcasse de la démonstration de la convergence, et
de ramener complètement cette question de la convergence à la question de la consistance de
l’approximation de l’opérateur elliptique par le schéma choisi. La démonstration de la consistance exige des nouveaux arguments, propres au cas discret, et impose des restrictions sur la
classe de schémas qu’on considère. Cela concerne tout particulièrement le choix des moyens
d’approximation de la composante tangentielle du gradient de la solution aux interfaces des
volumes de contrôle. Ce problème n’a pas de solution généralement admise; de plus, celles
que je connaı̂s ne sont pas suffisamment exactes pour nos besoins. On propose un exemple
d’approximation qui vérifie nos hypothèses. Il faut pourtant indiquer que ces hypothèses, bien
qu’elles rendent la démonstration de la consistance plutôt élégante, paraissent trop limitatives
du point de vue numérique.
5. A la fin, je citerai quelques-uns des problèmes ouverts que laisse entrevoir ce mémoire.
L’unicité d’une solution entropique généralisée pour une loi de conservation scalaire multidimensionnelle avec une fonction de flux continue dans L1 ∩ L∞ demeure inconnue.
La pertinence des conditions de structure de type a(v, Dv) = e
a(b(v), Dv) pour l’existence
d’une solution faible au problème (P r) me paraı̂t très douteuse. Cependant, je n’ai pas
réussi à démontrer qu’on peut s’en passer et avoir toutefois la convergence forte d’une suite
des solutions aux problèmes approchés, ne serait ce que dans le cadre de l’équation modèle
b(v)t = vxx + F (v)x ([BW99]) où l’existence est connue.
Le “chain rule” lemme (ChR) , qui est indispensable dans le chapitre 2.I, reste à démontrer
dans sa formulation naturelle. Cela permettrait d’avoir facilement un résultat plus général
dans le cas des conditions au bord de Dirichlet non-homogènes.
La possibilité de constuire une théorie consistante de solutions non-classiques de certaines
lois de conservation ([LF98]) soulève des questions intéressantes, de par sa relation avec les
effets dissipatifs-dispersifs (que l’on peut sans doute prendre auto-similaires) d’une part, et
de par son interpretation éventuelle dans le cadre de la théorie des semi-groupes non-linéaires
d’autre part.
L’étude précise du problème de Riemann pour des systèmes spéciaux, par les méthodes de
viscosité auto-similaire, ne s’arrête peut-être pas aux systèmes de la dynamique des gaz.
Même en présence des résultats généraux sur les systèmes 2 × 2 ([DDp76]), on peut espérer
d’obtenir plus d’information, y compris des “Riemann solvers” plus ou moins explicites, dans
des cas très particuliers. Le cas échéant, cela peut englober des systèmes qui ne sont pas
hyperboliques où qui donnent naissance aux ondes spécifiques.
Le résultat de convergence des schémas de volumes finis qu’on propose me semble amusant du point de vue purement mathématique; mais il est douteux qu’il soit accepté par la
0.II.
Sur le contexte mathématique et physique
31
communauté des numériciens tant qu’il n’est pas étendu à des systèmes plus réalistes et à des
schémas moins contraignants. En particulier, il semble ([Ey..]) que le choix d’approximation
de la composante tangentielle du gradient sur les interfaces n’est pas très important pour la
convergence des méthodes de volumes finis. D’autre part, il serait intéressant de voir ce que
pourrait apporter l’interprétation “continue” des schémas de volumes finis à l’étude d’autres
types de problèmes.
Je continuerai à chercher des réponses à ces questions.
. . . no eto uge sovsem drugaja istorija.
En concluant cette introduction, je tiens à exprimer ma profonde reconnaissance
envers Stanislav Nikolaevı̈ch Kruzhkov, qui m’a initié aux mathématiques, et je tiens
à remercier Philippe Bénilan pour tout le soutien, d’ordre scientifique et d’ordre
humain, qu’il m’a apporté tout au long de notre connaissance, et qui m’a permis de
venir à bout de ce travail.
Part 1
Conservation Laws
with Continuous Flux Function
CHAPTER 1.I
The Riemann Problem
for Scalar Conservation Law
with Continuous Flux Function:
the Self-Similar Viscosity Approach†
Introduction
Consider the following Cauchy problem:
(1ε ) Ut + f (U)x = εtUxx ,
where f : IR → IR is continuous, U maps Π+ = IR+ × IR in IR , and ε ≥ 0 ;
(
u− ,
x<0
(2) U(0, x) = U0 (x) =
u+ ,
x>0
For the sake of simplicity, assume u− < u+ .
Our main concern is the Riemann problem for the scalar conservation law (10 ) . It is
well known that, because of the non-uniqueness of a weak solution to (10 ), (2) , additional
criteria have to be introduced in order to select the admissible one. We seek to establish the
uniqueness of a solution to (10 ), (2) satisfying the wave fan admissibility criterion, proposed
by C.M.Dafermos in [D89] in the context of general hyperbolic systems of conservation laws.
More exactly, we seek to distinguish the (wave-fan admissible) solution to (10 ), (2) as the
unique a.e. limit of solutions to (1ε ), (2) as ε ↓ 0 . From this viewpoint, the term εtUxx
introduces vanishing artificial viscosity in (10 ), (2) , which we will call the self-similar viscosity.
A related approach to the admissibility for the Riemann problem has earlier been pursued
by A.S.Kalashnikov in [Ka59] (for the scalar case), V.A.Tupchiev in ([Tu64, Tu66, Tu73])
†
This chapter is an extended English version of the note [BA2], which refines the approach used in [BA1].
36
Riemann problem for Scalar Conservation Laws
and C.M.Dafermos in [D73a, D74] (for the case of systems), and many others since then
(cf. the survey paper [Tz98]). The idea is to establish the existence and a uniform BV
bound for solutions U ε , ε > 0 ; pass to the limit by the compactness argument; derive
self-contained pointwise conditions on discontinuities of a limiting function, which is a weak
solution of (10 ), (2) ; and then infer the uniqueness when possible. In this way, the problem
under consideration has already been solved by A.S.Kalashnikov in [Ka59]. In the smooth
case, he had shown that any limit of solutions of (1εn ), (2) as εn ↓ 0 fulfill conditions
that assure uniqueness for the Riemann problem (10 ), (2) (cf. (9) in Remark 3 at the end
of the chapter). As it is shown in [D73a], the most general form of such conditions is the
entropy-entropy flux admissibility (cf. [K70a, Lax71]).
For the case of scalar conservation law with continuous flux function, the wave-fan admissibility for the Riemann problem is still equivalent to the entropy admissibility as defined
by S.N.Kruzhkov (see the notion of generalized entropy solution in Definition 1 below) in
[K69a, K69b, K70a] (cf. also A.I.Vol’pert, [V67]). This is due to the fact that there is uniqueness of a generalized entropy solution to the Cauchy problem for (10 ) with general initial data
in L∞ even in the case the flux function is only continuous. Recall the following classical
definition.
Definition 1 A bounded measurable in Π+ function U(·, ·) is a generalized entropy solution
to the problem (10 ), (2) , if
+
(i) for all k ∈ IR , ψ ∈ C∞
0 (Π ) such that ψ ≥ 0 one has
ZZ
{|U(t, x) − k|ψt (t, x) + sign [U(t, x) − k][f (U(t, x)) − f (k)]ψx (t, x)} dxdt ≥ 0;
Π+
(ii) there exists a set E ⊂ IR+ of measure |E| = 0 such that for all t ∈ IR+ \ E the
function U(t, ·) is defined a.e. on IR , and for all r > 0 one has
Z
lim
|U(x, t) − U0 (x)|dx = 0.
t∈R+ \E, t→0
{|x|≤r}
While the general theory of the Cauchy problem for scalar conservation laws with Lipschitz
continuous flux function is due to S.N.Kruzhkov ([K69a, K69b, K70a]), results in the case of
only continuous flux function f (·) were first established by Ph.Bénilan ([B72]) by the nonlinear
semigroup approach, which yields the existence and uniqueness for L1 initial data in the onedimensional case. The uniqueness of a generalized entropy solution for the case including the
Riemann problem for one-dimensional scalar conservation law with continuous flux function has
first been established by S.N.Kruzhkov and P.A.Andreyanov ([KPA75]). For general L∞ initial
data, the uniqueness in this case has been shown by L.Barthélemy ([Ba88]) and S.N.Kruzhkov,
E.Yu.Panov ([KP90]). Note that in the multidimensional case, the uniqueness is false in general
for L∞ data (cf. [KP90]) and still not clear for L1 ∩ L∞ data (cf. [BK96]). Some further
1.I.0.
Introduction
37
results in this last direction are presented in Chapter 1.IV and [BABK]; while they have recently
been extended by E.Yu.Panov ([P99, P..]), the problem remains open.
It follows that for (10 ), (2) with only continuous f the approach of [Ka59] still could be
used. Arguing as in [K70a, K70b], one easily shows that a wave-fan admissible weak solution
of (10 ), (2) , whenever it exists, is a solution in the sense of Definition 1, so that it is the
unique generalized entropy solution. Note that, a fortiori, it is the unique solution to the
Riemann problem (10 ), (2) in the class of all wave-fan admissible solutions.
Here we bypass both the compactness and the entropy admissibility arguments, and prove
the uniqueness (and existence) of a wave-fan admissible weak solution independently, using
the somewhat elementary structure of the problem (10 ), (2) . The approach is essentially
one-dimensional (with respect to U as well as with respect to x ). Nevertheless, it can be
extended to special systems of conservation laws that have a kind of scalar structure (for the
system of isentropic gas dynamics, this is done in Chapters 1.II, 1.III herein) and where a
uniqueness result of general authority seems to be unknown.
The results of this chapter are summarized in Theorems 1,2 in Section 2. They state
that for all ε > 0 there exists a unique bounded self-similar solution U ε to (1ε ), (2) ,
and U ε converge as viscosity vanishes. The limiting function U , which is the wave-fan
admissible solution to (10 ), (2) , is given by an explicit formula. Curiously, we do not need
the a priori knowledge of this formula for U . Indeed, the starting point for our approach is
the observation, going back at least as far as the lecture notes [G59] of I.M.Gelfand, on the
profile of distribution solutions to (10 ), (2) obtained through approximation of shock waves
by travelling waves. It can be summarized by saying that (in case u− < u+ ) the admissible
solution of (10 ), (2) is the graph inverse function to the derivative of the convex hull of f (·)
on [u− , u+ ] † , or else by the formula
(3) U(x/t) = ∂/∂x max (xv − tf (v)).
u− ≤v≤u+
These two assertions are actually equivalent, due to the Fenchel formula. A careful construction
of the “travelling waves”-admissible solution in case of smooth f (·) can be found in lecture
notes [K70b] by S.N.Kruzhkov (see also [GR], and [ChHs] and references therein); in [GiNTe],
the profile of the solution is directly obtained from Definition 1. Writing U under the form
chosen above permits to compare the formula (3) with formulae, suggested by the convex
analysis, that were proposed in [H50, Lax57, KPe87] for different f (·) and U0 (·) . Due to
the uniqueness of a generalized entropy solution, they are all equivalent in the case of Riemann
problem with regular convex flux function.
In the lengthy Remark 3 at the end of the chapter some comments on interrelations of
the entropy admissibility, usual vanishing viscosity, self-similar viscosity and approximation by
†
We refer to the greatest convex function F (·) on [u− , u+ ] such that F ≤ f as to the convex hull of
f (·) on [u− , u+ ] .
38
Riemann problem for Scalar Conservation Laws
travelling waves are presented.
1
Restatement of the problem
First let ε > 0 be fixed. Let restrict our attention to bounded self-similar solutions of
(1ε ), (2) . For simplicity, we assume in the sequel that u− < u+ . Integrating the differential
equation for U(ξ) = U(t, x) , where ξ = x/t , we arrive to the following definition.
Definition 2 Let ε > 0 . A solution of (1ε ), (2) is a function U(t, x) = U(x/t) , U ∈
C 1 (IR) , that verifies
Z ξ
′
(4) εU (ξ) = −
ζU ′ (ζ)dζ + f (U(ξ)) + K with some ξ0 , K ∈ IR, and U(±∞) = u± .
ξ0
The strict monotony property for solutions of (4) , trivial in the case of smooth flux
function, can fail if f (·) is only continuous (cf. Remark 2). Nevertheless, we have the
following result.
Lemma 1 All solution U(ξ) of the problem (4) is non-decreasing on IR .
Proof: Assume the contrary. Then there exists c a point of extremum of U(·) on IR such
that U(c) = u0 6= u± . For definiteness, assume that c is a point of local maximum. Consider
the greatest segment [c1 , c2 ] containing c such that U|[c1 ,c2 ] ≡ u0 . For all α > 0 small
enough there exist ξ1 = ξ1 (α) , ξ2 = ξ2 (α) such that ξ1 < c1 ≤ c2 < ξ2 , U(ξ1 ) = U(ξ2 ) =
u0 − α , and the distance between ξ2 and ξ1 is the least possible. Since U ∈ C1 (IR) ,
there exists maxξ∈[ξ1 ,ξ2 ] |U ′ (ξ)| = |U ′ (ξ)| = M > 0 with some ξ ∈ [ξ1 , ξ2 ] . For definiteness,
assume that c2 < ξ ≤ ξ2 Consider separately the two possibilities.
a) U ′ (ξ) = −M . In this case U(ξ1 ) = U(ξ2 ) ≤ U(ξ) < u0 = U(c1 ) . Take ξ2 = ξ
and ξ1 = max{ξ| ξ1 ≤ ξ < c1 ,U(ξ) = U(ξ)} . One has U > U(ξ1 ) on [ξ1 , c1 ] , therefore
U ′ (ξ1 ) = m ≥ 0 . Taking into account that U ′ |[c1,c2 ] ≡ 0 , and using (4) with ξ0 = c and
the corresponding constant K = Kc , one obtains
Z ξ1
′
εU (ξ1 ) = −
ζU ′ (ζ)dζ + f (U(ξ1 )) + Kc ,
c1
′
εU (ξ2 ) = −
Z
ξ2
ζU ′ (ζ)dζ + f (U(ξ2 )) + Kc .
c2
Subtracting these two equalities, one finds that ε(M + m) ≤ M · (|ξ1| + |ξ2 |) · ((c1 − ξ1 ) +
(ξ2 − c2 )) .
b) U ′ (ξ) = M > 0 . According to the choice of ξ2 , one has ξ < ξ2 and U(ξ) > U(ξ2 ) , so
that there exists ξ2 = min{ξ| ξ < ξ < ξ2 ,U(ξ) = U(ξ)} . It follows that U ′ (ξ2 ) = −m ≤ 0 .
Set ξ1 = ξ and argue as in case a).
1.I.1.
Restatment of the problem
39
In the two cases, we infer that ε ≤ (|ξ1 | + |ξ2|) · ((c1 − ξ1 ) + (ξ2 − c2 )) . As α → +0 , one
has ξ1 → c1 , ξ2 → c2 , so that ε ≤ 0 at the limit. This contradiction proves the lemma.
⋄
By Lemma 1, the function Ξ(u) = [U(ξ)]−1 is defined a.e. on [u− , u+ ] and monotone.
Proposition 1 Assume
Z uthat a function U is a solution of (1ε ), (2) in the sense (4) . Then
the function Φ(u) =
Ξ(v)dv − K on [u− , u+ ] is a solution of the following problem :
U (0)


Φ ∈ C[u− , u+ ] and Φ is convex





Φ(u) ≤ f (u) on [u− , u+ ] and Φ(u± ) = f (u± )


ε
∈ Lloc
(5)
1 (u− , u+ )
f −Φ
)

ε


Φ̈(u)
≥
in the sense of measures

f (u)−Φ(u)



ε
 (f (u) − Φ(u)) Φ̈(u) −
=0
on (u− , u+ )
f (u)−Φ(u)
(in the rest of the chapter, ˙ stands for d/du ).
Conversely, assume Φ(·) is a solution of (5) . Then the function
U(t, x) = U(x/t) = [Φ̇(u)]−1 ≡ ∂/∂x max (xv − tΦ(v))
u− ≤v≤u+
is a solution of (1ε ), (2) in the sense (4) .
Proof: (4) ⇒ (5) Rewrite the equation (4) under the form
(6) εU ′ (ξ) = f (U(ξ)) − Φ(U(ξ));
it follows that Φ ∈ C(u− , u+ ) and Φ ≤ f . Since Φ̇(u) = Ξ(u) is an a.e. continuous,
non-decreasing function, Φ is convex. Set Ω = {u| ∃ξ : U(ξ) = u, U ′ (ξ) = 0} . The
Lebesgue measure |Ω| is zero, by the Sard lemma, and Ω ≡ {u| Φ(u) = f (u)} . For all
ε
u ∈ [u− , u+ ] \ Ω there exists Φ̈(u) = Ξ̇(u) = 1/U ′ (ξ) > 0 ; we have Φ̈(u) = f (u)−Φ(u)
ε
and Φ(u) < f (u) . Since (f − Φ)Φ̈ = 0 on Ω , it follows that (f − Φ) Φ̈ − f −Φ
=0
ε
in the sense of measures on (u− , u+ ) . Since |Ω| = 0 and Φ̈ ≥ 0 , one has Φ̈ ≥ f −Φ
on (u− , u+ ) in the same sense. Consequently, for all segment [a, b] ⊂ (u− , u+ ) one has
Rb
ε
ε
du ≤ Φ̇(b + 0) − Φ̇(a − 0) < ∞ , so that f −Φ
∈ Lloc
1 (u− , u+ ) . The equation
a f (u)−Φ(u)
(6) , together with U(±∞) = u± , imply that the limits lim U ′ (ξ) exist and are zero, so
ξ→±∞
that Φ(u± ) = f (u± ) and Φ ∈ C[u− , u+ ] .
(5) ⇒ (4) Define the multivalued function Ξ(·) by Ξ : u ∈ [u− , u+ ] 7→ [Φ̇(u −0), Φ̇(u +
0)] , with Φ̇(u± ± 0) = ±∞ . Set Ω := {u| Φ(u) = f (u)} . It is clear that Ξ is strictly
increasing and single-valued on the complementary of Ω . Let U = [Ξ]−1 ; one has U ∈
ε
C(IR) . If u0 ∈
/ Ω , there exists Ξ̇(u) = f (u)−Φ(u)
> 0 in a neighbourhood of u0 , so that
(6) is satisfied at the point ξ 0 = Ξ(u0 ) . If u0 ∈ Ω and ξ 0 ∈ Ξ(u0 ) , then for all α > 0
40
Riemann problem for Scalar Conservation Laws
there exists a neighbourhood of u0 such that f (u) − Φ(u) < εα ; consequently, Φ̈ > 1/α
in this neighbourhood. It follows that |Ξ(u0 + δ) − ξ 0 | ≥ |δ|/α for all δ small enough;
therefore there exists U ′ (ξ 0 ) = 0 and (6) is satisfied in all the cases. Thus U ∈ C1 (IR)
and (6) implies (4) , by the definition of U(·) . Clearly, U(±∞) = u± . By the Fenchel
formula, it follows that the other representation of U holds.
⋄
Remark 1 It is interesting to observe that , according to Proposition 1, the function w(t, x) =
Φ̇(U(t, x)) satisfies the Hopf equation ([H50]) wt + wwx = 0 if U is a solution of (1ε ), (2)
in the sense (4) .
Results and proofs
2
In this section we investigate solvability and convergence properties for the problem (5) and
deduce the corresponding results for the Riemann problems (1ε ), (2) .
Proposition 2 For all ε > 0 , u− < u+ and f ∈ C[u− , u+ ] there exists a unique solution
to the problem (5) .
Proof: Let argue by reductio ad absurdum in order to prove the uniqueness of a solution to
(5) . Assume Φ, Ψ are two different solutions, and c be a point of (positive) local maximum
of (Φ − Ψ) on [u− , u+ ] . In fact, in case Φ(c) < f (c) one could get a contradiction
from the standard maximum principle. In the general case, one can find ∆ > 0 such that
ε
ε
[c, c + ∆] ⊂ [c, b) and Φ̈ ≥ f −Φ
> f −Ψ
= Ψ̈ in the sense of measures on [c, c + ∆] .
Indeed, one has Ψ(c) < Φ(c) ≤ f (c) , so that it can be assumed that Ψ ∈ C2 [c, c + ∆] and
satisfies (5) with equality on [c, c + ∆] . Moreover, the assumption of maximality above,
together with the convexity of Φ , imply that there exists Φ̇(c) = Ψ̇(c) . It follows that
Φ̇(c + δ ± 0) > Ψ̇(c + δ ± 0) for all δ ∈ (0, ∆) , which contradicts to the choice of c .
In order to prove the existence, introduce the penalized problem∗
(
ε
Φ̈n (u) = Gn (u, Φn (u)) = f (u)−Φ
∧ n, n = 1, 2, ...
n (u)
(7)
Φn (u± ) = f (u± ), Φn ∈ C2 [u− , u+ ].
Since Gn (u, Φ) is continuous on u and Φ is bounded, there exists a solution of (7) .
The maximum principle is verified for equations of type (7) , because Gn (u, Φ) is inp
creasing in Φ . Set G = ε(u − u− )(u+ − u) and denote by F the convex hull of f on
[u− , u+ ] . One has (F − G)¨ ≥ −G̈ ≥ Gε ≥ f −(Fε −G) , so that (F − G) is a subsolution of
ε
. One finds for n ≥ m that G∞ (u, Φ) ≥
the problem (7) corresponding to G∞ = f (u)−Φ
≥ Gn (u, Φ) ≥ Gm (u, Φ) ; by the maximum principle, it follows that Φm (u) ≥ Φn (u) ≥
≥ F (u) − G(u) on [u− , u+ ] .
∗
Let a ∧ b denote min{b, max{a, 0}} for a, b ∈ IR .
1.I.2.
Results and proofs
41
Thus Φn (u) ↓ Φ(u) ∈ IR for all u ∈ [u− , u+ ] ; in addition, Φ(u± ∓ 0) = Φ(u± ) = f (u± )
+
ε
∈ IR
so that Φ ∈ C[u− , u+ ] and Φ is convex. Moreover, Gn (u, Φn (u)) tends to f (u)−Φ(u)
on [u− , u+ ] . Take a test function ϕ ∈ C∞
0 (u− , u+ ) such that ϕ ≥ 0 ; by (7) and the
Fatou lemma, one has
(8)
Zu+
ϕ̈Φ(u) du = lim
u−
= lim
Zu+
n→∞
u−
Zu+
n→∞
u−
ϕ̈Φn (u) du =
ϕGn (u, Φn (u)) du ≥
Zu+
ϕ
u−
ε
du.
f (u) − Φ(u)
ε
ε
Therefore Φ̈ ≥ f −Φ
in the sense of measures on (u− , u+ ) , and f −Φ
∈ Lloc
1 (u− , u+ ) .
∞
Now take a test function ϕ ∈ C0 (u− , u+ ) with supp ϕ ⊂ {u| Φ < f } . In this
case (8) becomes an equality, because there exists N = N(ϕ) such that for all n ≥ N ,
Gn (u, Φn (u)) ≤ G
on supp ϕ . Since f − Φ = 0 on [u− , u+ ] \ {u| Φ < f } ,
N (u, ΦN (u))
ε
⋄
one has (f − Φ) Φ̈ − f −Φ = 0 in the sense of measures on (u− , u+ ) .
Remark 2 It is easy to show that for a Lipschitz continuous flux function f the solution of
ε
(5) is a classical solution to the equation Φ̈ = f −Φ
on (u− , u+ ) . Nevertheless, it can be
√
shown using the maximum principle that for f (u) = u and for all interval (u− , u+ ) ∋ 0
sufficiently small, the derivative of the solution to (5) has a positive jump at 0 . This jump
corresponds to an interval of ξ where the solution U(·) of (1ε ), (2) is constant.
Propositions 1 and 2 yield the following result.
Theorem 1 For all ε > 0 , u− < u+ and f ∈ C[u− , u+ ] there exists a unique solution U ε
to the problem (1ε ), (2) in the sense of Definition 2. This solution is given by the formula
analogous to (3) :
U ε (t, x) = U ε (x/t) = ∂/∂x max (xv − tΦε (v)),
u− ≤v≤u+
where Φε (·) is the unique solution of (5) .
Since we are interested in passing to the limit as ε → 0 , let introduce the subscript ε in
the notation for solutions of (4) and (5) . We need the following two lemmae.
Lemma 2 Let F (·) be the convex hull of f (·) on [u− , u+ ] . Then Φε (·) converge to F (·)
uniformly on [u− , u+ ] as ε → +0 .
Proof: The proof is based upon a kind of maximum principle argument.
First note that Φε ≤ F . On the other hand, for all α > 0 there exists a function
G ∈ C2 [u− , u+ ] such that 0 < F −G < α and G̈ ≥ C(α) > 0 on [u− , u+ ] . Let c(ε, α) be a
point of global maximum of G−Φε on [u− , u+ ] ; assume that this maximum is positive. Then
42
Riemann problem for Scalar Conservation Laws
Φε (c) ≤ G(c) < F (c) ≤ f (c) , so that there exists Φ̈ε (c) ≥ G̈(c) ≥ C(α) . Consequently,
ε
≥ C(α) , and for all ε small enough one has G − Φε < α on [u− , u+ ] . It follows
f (c)−Φε (c)
that F − Φε < 2α on [u− , u+ ] in this case; besides, this last inequality is evident in case
the maximum is nonpositive, whence the claim of the lemma.
⋄
Lemma 3 Let F n (·) , n = 1, 2, ... be a sequence of convex functions converging to F 0 (·)
on [u− , u+ ] . Then the sequence U n (·) tends to U 0 (·) at the points ξ of continuity of
U 0 (·) , where U n (·) , n = 0, 1, ... are the functions constructed by the formula (3) applied
to the functions F n .
Lemma 3 is a corollary of the Fenchel formula and general theorems of the convex analysis
and basic probability theory. An elementary proof can be found in [BA1].
Finally, we establish the relation between the problem (10 ), (2) and the self-similar viscosity
regularized problems (1ε ), (2) , ε > 0 .
Theorem 2 Let U ε be the solution to the problem (1ε ), (2) in the sense of Definition 2
(which exists and is unique, due to Theorem 1). Then U ε converge a.e. on Π+ to the
function
U(t, x) = U(x/t) = ∂/∂x max (xv − tF (v)) ≡ ∂/∂x max (xv − tf (v))
u− ≤v≤u+
u− ≤v≤u+
as ε → +0 , and U is a generalized entropy solution of the problem (10 ), (2) .
Proof: The convergence of U ε to U follows readily from Lemmae 2,3 and the Fenchel
formula. It is easy to see that U(t, x) → u± as x/t → ±∞ , which implies that (i) of
Definition 1 holds. Moreover, U is a limit of viscous approximations of (10 ), (2) (i.e., a
wave-fan admissible weak solution). Using Kruzhkov’s techniques (cf. [K69a, K69b, K70a]),
one easily deduces that U satisfies (i) of Definition 1 as well.
⋄
Remark 3 As it is underlined in [Tz96], the wave fan admissibility criterion is different from
usual admissibility conditions for the Riemann problem in that it penalizes the whole fan of
shocks and rarefactions in the solution and not the shocks one by one. In this sense, it is
closer to the global criteria such as the Kruzhkov entropy admissibility criterion (Definition 1)
or the Dafermos entropy rate admissibility criterion (cf. [D73b]).
In piecewise continuous solutions of scalar conservation laws with continuous flux function,
the Kruzhkov criterion still decides on admissibility of each shock in a solution separately; it
induces the conditions
(9) S(u) := sign (ur −ul ){s(u−ul ) − (f (u)−f (ul))} ≤ 0 for all u between ul and ur ,
and S(ur ) = 0 if the shock joins ul at the left to ur at the right and propagates with
the speed s . This property is directly motivated by the wave fan admissibility criterion for
1.I.2.
Results and proofs
43
the Riemann problem, on account of the explicit representation for the solution. At the same
time, motivation by the travelling waves condition, which is the most common outcome of the
self-similar viscosity approach, becomes more delicate when f is not Lipschitz continuous.
Indeed, this condition requires that an admissible shock could be approximated by solutions
Uε to the equation
(10) Ut + f (U)x = εUxx
of the form Uε (t, x) = Uε ((x − st)/ε) † (in case of multiple inflexion points, one admits
also that it be a limit of such shocks with respect to infinitesimal perturbations of f or/and
ul , ur ). Such approximation is possible if and only if there exists a solution to the problem
(11)
d
U(ζ) = S(U(ζ)),
dζ
U(−∞) = ul , U(+∞) = ur .
In case f is Lipschitz continuous, this yields the condition (9) with strict inequality, by virtue
of uniqueness for the ODE Cauchy problem. In case of f continuous, analysis of stationary
points of the equation (11) shows that one still can deduce (9) from the travelling waves
condition.
The relation between the usual and self-similar viscosity limits for the Riemann problem
(10 ), (2) becomes less clear in absence of regularity of f . Kalashnikov in [Ka59] proves the
equivalence, starting from the uniqueness of a self-similar viscous limit and using the maximum
principle to compare primitives of solutions to (1ε ) and (10) . In absence of regularity of
f , this comparison becomes delicate† . A reason for the equivalence remains the uniqueness
theory for generalized entropy solutions (e.g., cf. [KPA75] and Chapter 1.IV).
In conclusion, the wave fan admissibility criterion directly selects a unique solution to the
Riemann problem for a scalar conservation law with continuous flux function. As it is shown
in Chapters 1.II and 1.III, the same is true for the p-system and the corresponding system in
Eulerian coordinates, even in the case vaccuum appears; for the p-system, the hyperbolicity
condition can be omitted.
†
Clearly, both travelling waves and self-similar viscosity approaches to the Riemann problem are motivated
by introducing the vanishing viscosity of type (10), which is the original idea came from the fluid mechanics
(cf. [Ray10]). Even if the rigorous mathematical study of discontinuous solutions of conservation laws had
started from such a description for Burgers’ equation, given by E.Hopf in his pioneering work [H50], the limits
of solutions of (10) remain difficult to describe directly, especially in case of systems. The recent success in
proving uniqueness for the Glimm scheme and the wave-tracking algorithm, due to A.Bressan and collaborators
(cf. [Br99] for a survey of results), does not promise the uniqueness of solutions that are limits of vanishing
viscosity unless uniform BV estimates on viscous solutions are obtained.
On the other hand, approximation by travelling waves or self-similar viscosity permits to pursue the analysis
within the field of ordinary differential equations, where a variety of tools apply. From the physical viewpoint,
both seem acceptable but not natural.
†
Maximum principle is an argument of the same order as the monotony of U ε in Lemma 1. While the
assertion of Lemma 1 is evident when f is smooth, the author was unable to establish it in the general case
by a simpler method than that exposed in the proof.
CHAPTER 1.II
The Riemann Problem for p-System
with Continuous Flux Function†
Introduction
Consider the Riemann problem for a so-called p-system, i.e. the initial-value problem
(
Ut − Vx = 0
,
(U, V ) : (t, x) ∈ IR+ × IR 7→ IR2 ;
Vt − f (U)x = 0
U(0, x) =
(
u+ ,
u− ,
x>0
,
x<0
V (0, x) =
(
v+ ,
v− ,
x>0
x<0
u± , v± ∈ IR.
(1)
(2)
The flux function f : IR 7→ IR is assumed to be continuous and strictly increasing (except in
Section 5, where the monotony assumption is relaxed).
In the case of piecewise smooth flux function the problem (1),(2) was treated by L.Leibovich,
[Le74] (cf. also [ChHs] and references therein). By analyzing the wave curves on the plane
(u, v) it has been shown that a self-similar distribution solution that is consistent with a
certain admissibility criterion (cf. B.Wendroff, [We72]; also I.Gelfand, [G59] and S.Kruzhkov,
[K70b] for the original idea carried out in the case of scalar conservation laws) may be explicitly
constructed through convex and concave hulls of the flux function f . It has been noticed by
C.Dafermos in [D74] that the same solution satisfies the wave fan admissibility criterion, i.e.,
it can be obtained as limit of self-similar viscous approximations as viscosity goes to 0 . Here
we follow this last idea.
Let introduce some notation. For given [a, b] ⊂ IR and fn: u ∈ [a, b] →
7 IR continuous,
the convex hull of f on [a, b] is the function u ∈ [a, b] 7→ sup φ(u) | φ is convex and φ ≤
†
This chapter extends the author’s graduate paper [BA0] written at the Chair of Differential Equations
under the supervision of S.N.Kruzhkov. The contents of this chapter, excluding Sections 4 and 5, will be
published in [BA3].
46
Riemann problem for p-Systems
o
f on [a, b] . Respectively, the concave hull of f on [a, b] is the function u ∈ [a, b] 7→
n
o
inf φ(u) | φ is concave and φ ≥ f on [a, b] . Take u0 in IR ; by F+ (·; u0 ) denote the
convex hull of f on [u0 , u+ ] if u0 ≤ u+ , and the concave hull of f on [u+ , u0] if
u0 ≥ u+ . Replacing u+ by u− , define F− (·; u0) in the same way. Let shorten F± (·; u0)
to F± when no confusion can arise.
h
i−1
dF+
dF+
Since f is strictly increasing, the inverse of du , denoted by du
, is well defined in
the graph sense as function from [0, +∞) to [u0 , u+ ] if u0 < u+ (respectively, to [u+ , u0 ]
i−1
h
+
mean the function on [0, +∞) identically
if u0 > u+ ). In the case u0 = u+ let dF
du
equal to u0 . With the same notation for F− , u− in place of F+ , u+ and Ḟ± standing for
dF±
, which are non-negative, the self-similar solution of the problem (1),(2) constructed in
du
[Le74] may be written as
 h
i−1
 Ḟ+ (·; u0)
(x2 /t2 ),
h
i
U(t, x) =
 Ḟ (·; u ) −1 (x2 /t2 ),
−
0
V (t, x) = v− −
Z
x≥0
,
(3)
x≤0
x/t
ζdU(ζ),
(4)
−∞
dU(ζ) being regarded as measure; and, for a bijective flux function f , the value u0 is
uniquely determined by
Z u+ q
Z u− q
v− − v+ =
Ḟ+ (u; u0)du +
Ḟ− (u; u0)du.
(5)
u0
u0
In the case of bijective locally Lipschitz continuous flux function f , the same formulae (3)(5) were obtained by P.Krejčı́, I.Straškraba ([KrSt93]) for the unique solution to satisfy their
“maximal dissipation” condition. This solution was also shown to be the unique a.e-limit as
ε → 0 of solutions to Riemann problem for the p-system regularized by means of infinitesimal
!
0
parameter ε > 0 , introduced into the flux function f , and the viscosity
.
εtVxx
In this chapter a refinement of these results is presented. The techniques employed are
those used by the author while treating the Riemann problem for a scalar conservation law with
continuous flux function (cf. Chapter 1.I and [BA1],[BA2]). In the general case of continuous
strictly increasing flux function f , the Riemann problem (2) for the p-system (1) and the
regularized system
(
Ut − Vx = 0
(6)
Vt − f (U)x = εtVxx
are treated. The main result is the following theorem:
1.II.1.
Restatement of the problem
47
Theorem 1 Suppose f : IR → IR is increasing and bijective. Then for all u± , v± ∈ IR ,
ε > 0 there exists a unique bounded self-similar distribution solution (U ε , V ε ) of the problem
(6),(2).
Besides, as ε ↓ 0 , (U ε , V ε )(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by the
formulae (3)-(5), so that (U, V ) is a self-similar distribution solution of the problem (1),(2).
The bijectivity condition is only needed for the existence of solutions and cannot be omitted
(see Remark 7.6 in [KrSt93]), though it can be relaxed (see Remark 2 in Section 3).
The chapter is organized as follows. In the first section the problem (6),(2) is reduced
to a pair of boundary-value problems for a second-order ordinary differential equation on the
domains (min{u0 , u± }, max{u0 , u± }) ; u0 is a priori unknown and satisfies an additional
algebraic equation. In Section 2 existence, uniqueness and convergence (as ε → 0 ) results
are obtained for the ODE problem stated in Section 1, with u0 ∈ IR fixed. Then it is shown in
Section 3 that u0 is in fact uniquely determined by the flux function f , ε , and the Riemann
data u± , v± ; finally, Theorem 1 above is proved† .
Restatement of the problem
1
We start by fixing ε > 0 . Consider the problem (6),(2) in the class of bounded distribution
solutions (U, V ) of (6) such that (U, V )(t, ·) tends to (U, V )(0, ·) in L1loc (IR) × L1loc (IR)
as t tends to +0 essentially. Moreover, since both the initial data (2) and the system (6)
are invariant under the transformations (t, x) → (kt, kx) with k in IR (here is the reason
to introduce the viscosity with factor t ), it is natural to seek for self-similar solutions, i.e.
(U, V ) depending solely on the ratio x/t . By abuse of notation, let write (U, V )(t, x) =
(U, V )(x/t) . Let ξ denote x/t and use U ′ , V ′ for dU/dξ, dV /dξ and so on.
Lemma 1 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution
solution of (6),(2) if and only if U, V, ξU ′ and V ′ are continuous on IR , the equations
Z ξ
′
εξU (ξ) = −
ζ 2 U ′ (ζ)dζ + f (U(ξ)) + C
(7)
0
V (ξ) = −
Z
ξ
ζU ′ (ζ)dζ + K
(8)
0
are fulfilled with some constants C,K, and also
U(±∞) = u± ,
V (±∞) = v± .
(9)
Besides, there exist ξ± in IR± , ξ− ≤ ξ+ , such that U, V are strictly monotone on each
of (−∞, ξ− ) , (ξ+ , +∞) , with U ′ 6= 0 , and U, V are constant on (ξ− , ξ+ ) .
†
In Section 4 we present some comments on the results obtained in Section 3. Section 5 is devoted to the
case where the monotonicity assumption on f is relaxed, which gives rise to a hyperbolic-elliptic problem.
48
Riemann problem for p-Systems
Proof: Let (U, V ) be bounded self-similar distribution solution
h of the system (6).
i′ Then
−ξU ′ −V ′ = 0 and −ξV ′ −f (U)′ = εV ′′ in D ′ (IR) ; therefore ξ 2 U −f (U)+εξU ′ = 2ξU
in D ′ (IR) . Since U ∈ L∞ (IR) , it follows that
2
′
ξ U − f (U) + εξU =
Z
ξ
2ζU(ζ)dζ + C
0
∈
C(IR)
(10)
with some C in IR . Hence one deduce consecutively that ξU ′ ∈ L∞
loc (IR) , U ∈ C(IR\{0})
1
and finally, U ∈ C (IR\{0}) . Thus for all ξ 6= 0 (7) holds.
Now let prove the monotony property stated. For (ξ− , ξ+ ) take the largest interval in IR
containing ξ = 0 such that U = U(0) on (ξ− , ξ+ ) . For instance, let ξ+ be finite and
therefore U not constant on (0, +∞) ; suppose U is not strictly monotone on (ξ+ , +∞) .
Since U ′ ∈ C(ξ+ , +∞) , it follows that there exists c > ξ+ such that U ′ (c) = 0 and
U ′ is non-zero in some left neighbourhood of c . For instance, assume U ′ > 0 in this
neighbourhood. Clearly, there exists a sequence {ξn } ⊂ IR increasing to c such that for all
n ∈ IN the maximum of U ′ on [ξn , c] is attained at the point ξn . Since f is increasing,
it follows that f (U(ξn )) < f (U(c)) . Take (7) at the points ξ = ξn and ξ = c ; subtraction
yields
Z c
Z c
′
2 ′
′
εξn U (ξn ) − εc · 0 ≤
ζ U (ζ)dζ + f (U(ξn )) − f (U(c)) ≤ U (ξn )
ζ 2dζ.
ξn
ξn
As n → ∞ , one deduces that ε ≤ 0 , which is impossible.
Thus U , and consequently V , are indeed monotone on (−∞, 0) and (0, +∞) ; therefore there exist U(±0) = limξ→±0 U(ξ) . Hence by (10) there exist limξ→±0 ξU ′ (ξ) , which
are necessarily zero since U ∈ L∞ (IR) . Thus (10) yields f (U(+0)) = f (U(−0)) , so that
U ∈ C(IR) . Consequently, ξU ′ ∈ C(IR) , V ′ ∈ C(IR) , and V ∈ C(IR) . It follows that
(7),(8) hold for all ξ in IR .
The converse assertion, i.e. that (7),(8) imply (6) in the distribution sense, is trivial.
Finally, since U and V are shown to be monotone on IR± whenever (7),(8) hold, it is
evident that (9) is fulfilled if and only if self-similar U, V satisfy (2) in L1loc -sense as t → 0
essentially.
⋄
Let use this result to obtain another characterisation of self-similar solutions to (6),(2).
The idea is to seek for solutions of the same form as in formulae (3)-(5), substituting F± by
appropriate functions depending on ε . One thus has to “inverse” (3)-(5).
Set u0 := U(0) and consider (7) separately on (−∞, ξ− ) , (ξ− , ξ+ ) , and (ξ+ , +∞) ,
where ξ± are defined in Lemma 1. Assume u0 6= u− , u0 6= u+ . Let introduce the
notation I(a, b) for the interval between a and b in IR . One has U(ξ) = u0 for all
ξ ∈ (ξ− , ξ+ ) ; besides, the inverse functions U+−1 : I(u0 , u+ ) 7→ (ξ+ , +∞) and U−−1 :
1.II.1.
Restatement of the problem
49
I(u0 , u− ) 7→ (−∞, ξ− ) are well defined. For all u ∈ I(u0, u+ ) (respectively, u ∈ I(u0, u− ) )
set
Z u
Z u
2
2
ε
−1
ε
−1
Φ+ (u; u0) :=
U+ (w) dw − C
resp., Φ− (u; u0) :=
U− (w) dw − C (11)
u0
u0
with C taken from (7). The shortened notation Φ± (u) will be used for Φε± (u; u0 ) whenever
ε, u0 are fixed. Now (7) can be rewritten as εξU ′ (ξ) = f (U(ξ)) − Φ± (U(ξ)) for ξ ∈
I(ξ± , ±∞) . The reasoning in the proof of Lemma 1 shows that U is not only monotone,
but also U ′ is different from 0 outside of [ξ− , ξ+ ] . It follows that for all u in I(a, b) ,
where a = u0 , b = u+ (resp., for all u in I(a, b) , where a = u0 , b = u− ), the function
Φ+ (resp., Φ− ) is twice differentiable and satisfies the equation
Φ̈(u) =
2εΦ̇(u)
,
f (u) − Φ(u)
with Φ̇(u) > 0 and Φ̈(u) · (b − a) > 0.
(12)
Hence Φ+ < f ( Φ+ > f ) if u0 < u+ (if u0 > u+ ), and the same for Φ− , u− in place
of Φ+ , u+ .
Note that one can extend the functions Φ+ , Φ− to be continuous on I(u0, u+ ) , I(u0 , u− )
respectively, and in this case one has
Φ+ (u0 ) = f (u0 ), Φ+ (u+ ) = f (u+ )
resp., Φ− (u0 ) = f (u0 ), Φ− (u− ) = f (u− ) . (13)
Indeed, one gets Φ± (u0 ) = f (u0) directly from (11) and (7). Besides, for ξ ∈ IR± , εξU ′ (ξ)
is equal to f (U(ξ))−Φ± (U(ξ)) , which has finite limits as ξ → ±∞ because U(±∞) = u±
and Φ± are convex and bounded on I(u0 , u± ) . The limits of εξU ′ (ξ) cannot be non-zero
since U is bounded, thus one naturally assign Φ± (u± ) := f (u± ) .
Now from (8)-(11) it follows that
Z u+ q
Z
ε
Φ̇+ (u; u0)du +
v− − v+ =
u−
u0
u0
q
Φ̇ε− (u; u0)du.
(14)
Note that in the case u0 = u+ ( u0 = u− ), (12)-(14) formally make sense, with Φ+
defined at u = u0 = u+ by f (u+ ) (resp., with Φ− defined at u = u0 = u− by f (u− ) ).
Finally, the reasoning above is inversible. More presisely, for given u0 ∈ IR and Φε± (·; u0) ∈
C 2 (I(u0 , u± )) ∩ C(I(u0, u± )) such that (12)-(14) hold, define U, V by
 h
i−1
 Φ̇ε+ (·; u0)
(ξ 2), ξ ≥ 0
h
i−1
U(ξ) =
(15)
 Φ̇ε (·; u )
2
(ξ ), ξ ≤ 0
0
−
V (ξ) = v− −
Z
ξ
ζdU(ζ),
(16)
−∞
with [Φ̇ε+ (·; u0)]−1 (and [Φ̇ε− (·; u0)]−1 ) taken in the graph sense and equal to u+ (to u− )
identically whenever u0 = u+ ( u0 = u− ). Then (U, V ) satisfy (7)-(9). Indeed, U is
50
Riemann problem for p-Systems
continuous, Φε+ (u0; u0 ) = Φε− (u0 ; u0) , and the equation εξU ′ (ξ) = f (U(ξ)) − Φε± (U(ξ); u0)
holds for all ξ ∈ IR± . Hence ξU ′ ∈ C(IR) and (7) is true. Therefore V ′ , V are continuous
and (8),(9) are easily checked.
We collect the results obtained above in the following proposition:
Proposition 1 Let ε, f, u± , v± be fixed. Formulae (15),(16) provide a one-to-one correspondence between the sets A and B defined by
n
A :=
u0 , Φ± (·) | u0 ∈ IR, Φ± : I(u0 , u± ) 7→ IR, Φ± ∈ C 2 (I(u0 , u± )) ∩ C(I(u0 , u± ))
o
and (12) − (14) hold
n
o
B := (U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)
In fact, it will be shown in Section 3 that A and thus B are one-element or empty sets.
The resemblance of formulae (3),(4),(5) and (15),(16),(14) permits to get the convergence
result of Theorem 1 if one has convergence of Φε± to F± as ε → 0 .
2
The problem (12),(13) with fixed domain
Let fix a, b ∈ IR and consider the equation (12) on the interval I(a, b) , with the boundary
conditions as in (13). For instance, suppose a ≤ b .
Proposition 2 For all continuous strictly increasing f , ε > 0 , and a, b ∈ IR there exists a
unique Φ in C 2 (I(a, b)) ∩ C(I(a, b)) satisfying (12) such that Φ(a) = f (a) and Φ(b) =
f (b) .
For f and [a, b] fixed, let Φε denote the function Φ from Proposition 2 corresponding
to ε , ε > 0 .
Proposition 3 With the notation above, Φε converge in C[a, b] , as ε → 0 , to the convex
hull F of the function f on the segment [a, b] .
Remark 1 In the case a ≥ b , the corresponding limit is the concave hull of f on [b, a] .
The following two assertions will be repeatedly used in the proofs in Sections 2,3:
Lemma 2 (Maximum Principle) Let Φ, Ψ ∈ C2 (a, b) ∩ C[a, b] and satisfy, for all u ∈
(a, b) , the equations Φ̈(u) = G(u, Φ(u), Φ̇(u)) and Ψ̈(u) = H(u, Ψ(u), Ψ̇(u)) , respectively,
with G, H : (a, b) × IR × (0, +∞) 7→ (0, +∞] .
a) Assume that G(u, z, w) < H(u, ζ, w) for all u ∈ (a, b) such that Φ(u) < Ψ(u)
and all z, ζ, w such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and
Φ(b) ≥ Ψ(b) .
b) Assume that G(u, z, w) ≡ H(u, z, w) , increases in w and strictly increases in z ; let
Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ − Ψ) is monotone on [a, b] .
1.II.2.
Problem (12),(13) with fixed domain
Proof: The proof is straightforward.
51
⋄
Lemma 3 Let functions F, Fn , n ∈ IN , be continuous and convex (or concave) on [a, b] .
Assume that Fn (u) converge to F (u) for all u ∈ [a, b] . Then this convergence is uniform
on all [c, d] ⊂ (a, b) and
a) Ḟn converge to Ḟ a.e. on [a.b] ;
Z bq
Z bq
b) if Fn , F are increasing, then
Ḟn (u)du converge to
Ḟ (u)du ;
a
a
h i−1 h i−1
c) let Ḟ , Ḟn
denote the graph inverse functions of F, Fn respectively; then
h i−1
h i−1
h i−1
Ḟn
(ξ) tends to Ḟ
(ξ) for all ξ such that Ḟ
is continuous at the point ξ .
Proof: An elementary proof of a),c) is given in [BA1]. Besides, the assumptions of the
Lemma imply that for
δ > 0 Ḟn are bounded uniformly
Z all
in n ∈ IN for u ∈ [a + δ, b − δ] .
Z b q
a+δ q
Since, in addition, Ḟn (u)du +
Ḟn (u)du → 0 uniformly in n ∈ IN as δ →
b−δ
a
0 , the conclusion b) follows from the Lebesgue Theorem.
⋄
Proof of Proposition 2: There is nothing to prove if a = b ; let a 0 (17)
n, otherwise
for all u ∈ [a, b] . Since Gn is continuous in all variables and bounded, the existence of
solution follows for arbitrary boundary data such that Φ(a) < Φ(b) ; in particular, a solution
Φn exists such that Φn (a) = f (a) , Φn (b) = f (b) . The Maximum Principle yields that Φn
decrease to some convex non-decreasing function Φ on [a, b] as n → ∞ .
Further, there exists a solution Ψ of (12) on [a, b] with any assigned value of Ψ(a) less
than f (a) , or any assigned value of Ψ(b) less than f (b) . In fact, in the first case one
takes Ψ(u) ≡ Ψ(a) ; in the second case there exists a solution on the whole of [a, b] to
the equation (12) with the Cauchy data Ψ(b) (fixed) and Ψ̇(b) sufficiently large. By the
Maximum Principle Φn ≥ Ψ on [a, b] ; therefore Φ(a + 0) = f (a) and Φ(b − 0) = f (b) .
Consequently Φ is continuous on [a, b] .
Now if for all [c, d] ⊂ (a, b) there exists m0 > 0 such that f − Φ ≥ m0 on [c, d] , then
the functions Gn (u, Φn (u), Φ̇n (u)) are bounded uniformly in n ∈ IN for u ∈ [c, d] ; indeed,
on [c, d] , by convexity, Φ̇n are uniformly bounded and Φn converge to Φ uniformly, so
Φ̇n
that f2ε−Φ
≤ M(c, d) for all n large enough. Hence it will follow by Lemma 3a) and the
n
2εΦ̇(u)
Lebesgue Theorem that Φ̈(u) = f (u)−Φ(u)
for all u ∈ [c, d] , and consequently Φ ∈ C2 [c, d] .
Thus the existence of solution to problem (12),(13) will be shown.
First let show
n that Φ̇(u ± 0) > 0 for
o all u > a . It suffices to prove that û = a ,
where û := sup u ∈ [a, b] | Φ(u) = f (a) . Note that û f (a) .
52
Riemann problem for p-Systems
Φ̇
Assume û > a ; by the Lebesgue Theorem Φ̈ = f2ε−Φ
in some neighbourhood of û . Since
Φ̇(û − 0) = 0 , by the uniqueness theorem for the Cauchy problem Φ is constant in this
neighbourhood. Therefore necessarily û = b , which is impossible.
Further, by Lemma 3a), (17), and the Fatou Lemma one has
2εΦ̇
f −Φ
∈ L1loc (a, b) . Hence
Φ̇
Φ ≤ f and f2ε−Φ
≤ Φ̈ on (a, b) in measure sense. Now take [c, d] ∈ (a, b) and ũ ∈ [c, d] ;
− 0) > 0 , B := Φ̇(d − 0) > 0 . For all
set m := f (ũ) − Φ(ũ) ≥ 0 . Set A := Φ̇( a+c
2
a+c
u ∈ [ 2 , ũ] , f (u) − Φ(u) ≤ m + B(ũ − u) and Φ̇(u ± 0) ≥ A since Φ is convex and f
increasing. Hence
Z ũ
Z ũ
Z ũ
2εΦ̇(u)
2εA
B−A≥
Φ̈du ≥
du ≥
du = K1 − K2 ln m,
a+c
a+c f (u) − Φ(u)
a+c m + B(ũ − u)
2
2
2
with some positive constants K1 , K2 depending only on c, d . Thus m ≥ m0 (c, d) > 0 and
the proof of existence is complete.
The uniqueness is clear from the Maximum Principle for solutions of (12).
⋄
Proof of Proposition 3: Let a 0 and a barrier function Ψα such that
α/2 ≤ F − Ψα ≤ α and Ψ̈α ≥ m(α) > 0 on [a, b] . Such a function can be constructed
through the Weierstrass Theorem.
By the Maximum Principle Φε increase as ε decrease. Therefore there exists [c, d]
ε
inside
(a, b) such that
n
o for all ε in (0, 1) Φ ≥ Ψα on [a, b] \ [c, d] . It follows that
u | Φε (u) < Ψα (u) ⊂ [c, d] and thus Φ̇ε ≤ M(α) on this set uniformly in ε . Now
one may apply the Maximum Principle to Φε and Ψα , hence
for all ε less than α·m(α)
2M (α)
0 ≤ F − Φε ≤ α for all ε small enough.
⋄
3
Solutions of the problem (6),(2) and the proof of Theorem
1
Proposition 2 above implies that for all f , ε , u± fixed, for all u0 ∈ IR there exist unique
Φε+ (·; u0) and Φε− (·; u0) satisfying (12),(13); thus by Proposition 1, for an arbitrary v− in
IR and v+ obtained from (14), (U, V ) provided by (15),(16) is a self-similar solution to the
Riemann problem (6),(2). Now since not u0 but v± are given by (2), one needs to find u0
in IR such that (14) holds with these assigned values of v± .
Proposition 4 a) Assume f (±∞) = ±∞ . Then for all u± , v± ∈ IR , ε > 0 there exists a
unique u0 such that (14) holds, with Φε+ , Φε− the (unique) solutions to (12),(13).
Z ±∞ q
1
b) Assume f ∈ W1 locally in IR and
f˙(u)du = ±∞ . Then for all u± , v± ∈ IR
0
and ε < ε0 = ε0 (u± , v+ − v− ) there exists a unique u0 such that (14) holds, with the same
Φε± .
1.II.3.
Problem (6),(2) and proof of Theorem 1
53
Let F± (·; u0) be, as in the Introduction, the convex (concave) hulls of f on I(u0 , u± )
according to the sign of (u± − u0 ) . Set
Z u+ q
Z u+ q
ε
0
ε
Φ̇± (u; u0)du,
∆± (u0 ) :=
Ḟ± (u; u0)du.
∆± (u0 ) :=
u0
u0
Φε± (·; u0)
It will be convenient to extend
, F± (·; u0) to continuous functions on IR by setting
each of them constant on (−∞, min{u0 , u± }] and [max{u0 , u± }, +∞) . In the lemma below
a few facts needed for the proofs of Proposition 4 and Theorem 1 are stated.
Lemma 4 With the notation above, and u0 running through IR , the following properties
hold.
a) For all u ∈ IR and ε > 0 , u0 7→ Φε± (u; u0) do not decrease; nor do u0 7→ F± (u; u0) .
b) For all u ∈ IR and ε > 0 , u0 7→ sign(u± −u0 )Φ̇ε± (u; u0) do not increase; nor do
u0 7→ sign(u± −u0 )Ḟ± (u; u0 ) .
c) For all ε > 0 the maps u0 7→ Φε± (·; u0) are continuous for the L∞ (IR) topology; so
do u0 7→ F± (·; u0) .
d) For all ε ≥ 0 , u0 7→ ∆ε± (u0 ) are continuous and strictly decreasing.
Proof: Combining the continuity and monotony of f with a),b) of the Maximum Principle
for solutions of (12),(13), one gets a)-c) for Φε± . The same assertions for F± follow now
from Proposition 3 and Lemma 3a); they can also be easily derived from the definition of
convex hull. Finally, d) results from c), Lemma 3b), b) and the strict monotony of f . ⋄
Proof of Proposition 4: a) By Lemma 4d), it suffices to prove that ∆ε± (±∞) = ∓∞ .
Assume the contrary, for instance that ∆ε+ (−∞) = M < +∞ .
Consider u0 < u+ ; Φε+ is convex, therefore for all u0 there exists c = c(u0 ) ∈ [u0 , u+ ]
such that Φ̇ε+ (·; u0) ≥ 1 on [c, u+ ) and Φ̇ε+ (·; u0 ) ≤ 1 on (u0 , c] . By Lemma 4b) c(u0)
increase with u0 . Obviously, for all u0 , M > ∆ε+ (u0 ) ≥ [Φε+ (c; u0 ) − f (u0 )] + [u+ − c] .
Set d := u+ − M ; clearly, c(u0 ) ≥ d for all u0 . Considering the functions Φε (·; u0 ) with
u0 → −∞ , one obtains a sequence {Ψn } such that Ψn satisfy (12) on [d, u+ ) , Ψ̇n (d) ≤
1 , Ψn (u+ ) = f (u+ ) , and finally, Ψn (d) → −∞ (this last since Ψn (d) ≤ f (u0) + M →
f (−∞) + M = −∞ , as u0 → −∞ ). On the other hand, for n large enough, the unique
solution Ψ to the equation (12) with the Cauchy data Ψ(d) = Ψn (d) , Ψ̇(d) = 2 is defined
on the whole of [d, u+ ] , which means that Ψ(u+ ) < f (u+ ) . Now by b) of the Maximum
Principle, (Ψ − Ψn ) is increasing and thus positive. Hence Ψn (u+ ) ≤ Ψ(u+ ) < f (u+ ) ,
which is a contradiction.
b) Take u0 < u+ . First suppose f ∈ C2 [u0 , u+ ] and has a finite number of points
of inflexion; denote by F the corresponding convex
hull. The segment [u0 , u+ ] can be
n
decomposed into the three disjoint sets: M1 := u | ∃δ > 0 s.t. Ḟ ≡ const on (u − δ, u +
o
n
o
δ) ∩ [a, b] , M2 := u | Ḟ (u) = f˙(u) \ M1 , and M3 finite. Using the Cauchy-Schwarz
Z u+ q
Z u+ q
0
inequality on every (c, d) ⊂ M1 , one gets
Ḟ (u)du ≡ ∆+ (u0 ) ≥
f˙(u)du .
u0
u0
54
Riemann problem for p-Systems
In the general case, let proceed with the density argument, choosing a sequence
{f
q
qn }
such that fn are increasing and smooth as above, fn → f in C[u0 , u+ ] with
f˙n → f˙
in L1 [u0 , u+ ] as n → ∞ . Denote the convex hull of fn on [u0 , u+ ] by Fn ; it is easy
to see that kFZ
as n → ∞ . By Lemma 4b),
n − F kC[u0 ,u+ ] ≤ kfn − f kC[u0 ,u+ ] →
Z u0+ q
u+ q
∆0+ (u0 ) = lim
f˙(u)du in the general case as
Ḟn (u)du , so that ∆0+ (u0 ) ≥
well. Thus
n→∞ u
0
∆0+ (−∞)
u0
= +∞ by the assumption on f .
Now Proposition 3 and Lemma 3b) imply that for given v± in IR , there exists ε0 =
ε0 (v+ − v− ) such that one has ∆ε+ (−L) > |v− − v+ | (and in the same way, ∆ε+ (L) <
−|v− − v+ | ) for all ε < ε0 whenever L is large enough. Lemma 4d) yields now the required
fact.
⋄
Finally, here is the proof of the result announced in the Introduction.
Proof of Theorem 1: The existence and uniqueness of a bounded self-similar distribution
solution to the Riemann problem (6),(2) follow immediately from Propositions 1, 2 and 4.
Now let ε decrease to 0 . Take uε0 , Φε± (·; uε0 ) corresponding to the unique solution
of (6),(2) in the sense of Proposition 1. Take u0 a limit point in IR of {uε0}ε>0 . Suppose
first uε0k → u0 ∈ IR , εk → 0 as k → ∞ ; let show that, with the notation as in Lemma
4, Φε+ (·; uε0 ) converge to F+ (·; u0) in L∞ (IR) . Indeed, take α > 0 ; |uε0k − u0 | < α for
all k large enough. By Proposition 3 and Lemma 4a), there exists ε0 > 0 such that, for all
εk < ε0 , F+ (·; u0 −α)−α ≤ Φε+k (·; u0 −α) ≤ Φε+k (·; uε0k ) ≤ Φε+k (·; u0 +α) ≤ F+ (·; u0 +α)+α .
Thus the required result follows from Lemma 4c); clearly, it also holds for Φε−k , F− in place
of Φε+k , F+ .
Now by Lemma 3b) ∆0+ (u0 ) + ∆0− (u0 ) is the limit of ∆ε+k (uε0k ) + ∆ε−k (uε0k ) ≡ v− − v+ ;
hence by Lemma 4d), u0 is unique if it is finite. Besides if, for instance, u0 = −∞ , then
for all L ∈ IR , v− − v+ = limεk →0 [∆ε+k (uε0k ) + ∆ε−k (uε0k )] ≥ ∆0+ (L) + ∆0− (L) by Lemma
4d) and Lemma 3b). It is a contradiction; indeed, it is easy to see that ∆0± (L) → +∞ as
L → −∞ .
Thus in fact uε0 → u0 as ε → 0 , u0 ∈ IR and (5) holds. Further, let u0 < u± ; the
other cases are similar and those of u0 = u− or u0 = u+ are trivial. For all α > 0 there
exists ε0 = ε0 (α) > 0 such that for all ε < ε0 [uε0 , u± ] ⊂ [u0 − α, u± ] . The functions U ε
in the statement of Theorem 1 are given by formula (15), when applied to Φε± (·; u0) with
their natural domains [uε0 , u± ] . Taking for the domains [u0 − α, u± ] , one do not change
U ε (ξ) for ξ 6= 0 and ε < ε0 . The same being valid for U given by (3), one may use the
fact, proved above, that kΦε± (·; uε0) − F± (·; u0)kC[u0 −α,u± ] → 0 as ε → 0 , and conclude
by Lemma 3c) that U ε (ξ) → U(ξ) for a.a. ξ ∈ IR . Hence it follows by (4),(16) that
V ε → V a.e., so that (U, V ) given by (3)-(5) is the unique a.e.-limit of self-similar bounded
distribution solutions of the problem (6),(2). Thus (U, V ) is a distribution solution of the
Riemann problem (1),(2).
⋄
1.II.4.
Comments
55
Remark 2 Note that using b) of Proposition 4 instead
a), one gets a result similar to the
Z ±∞of
q
Theorem 1 in the case of f ∈ W11 locally in IR ,
f˙(u)du = ±∞ ; in fact, the exact
0
condition is the bijectivity of the functions u0 7→ ∆0± (u0) for continuous strictly increasing
flux function f . Under each of this conditions the existence of bounded self-similar solution
of (6),(2) is guaranteed for all ε < ε0 = ε0 (u± , v+ − v− ) .
Note
After this paper had been completed, the author had an opportunity to meet Prof. A.E.Tzavaras
and get acquanted with his papers on viscosity limits for the Riemann problem; in particular,
in [Tz95] very close results
! were obtained for p-systems regularized by viscosity terms of the
0
form
, without involving the explicit formulae for the limiting solution.
εt(k(U)Vx )x
For results on self-similar viscous limits for general strictly hyperbolic systems of conservation laws, refer to the survey paper [Tz98] and literature cited therein. Let only note that the
structure of wave fans in self-similar viscous limits remains the same as in the case of scalar
conservation laws ([G59],[K70b]) and in the case of p-systems, where it can be easily observed
through the formulae (3),(4).
On the other hand, Prof. B.Piccoli turned my attention to Riemann solvers for hyperbolicelliptic systems (1) (i.e. the case of non-monotone f ). The global explicit Riemann solver
extends to this case (see Krejčı́,Straškraba, [KrSt97],[KrSt93]); it can be proved, with the
techniques used here and in [BA1],[BA2], that this solver is the unique limit of self-similar
bounded solutions to the problem (6),(2).
Precise results on hyperbolic-elliptic p-systems and a discussion of other viscosity terms
are given in Section 5 below.
4
Comments
It is possible to!treat, in almost the same way, the case of (by no means physical) viscosity
0
term
. Lemma 1 still holds; moreover, in this case U, V ∈ C 1 (IR) . For Φε± (·; u0)
εtUxx
defined by (11), one gets instead of (12) the equation
q
2ε Φ̇(u)
Φ̈(u) =
with Φ̇ > 0 and Φ̈ > 0 or Φ̈ < 0;
f (u) − Φ(u)
this problem shares the properties of (12) that were important for us.
!
!
0
0
Further, the more general viscosity terms
or
, with
εt(k(U)Vx )x
εt(k(U)Ux )x
k ∈ C(IR; (0, +∞)) , also yield unique approximate solutions, which converge to (U, V ) given
56
Riemann problem for p-Systems
by (3)-(5). The only difference is the factor k(u) in the right-hand side of the equation in
(12).
Finally, the classical example of p-system with f (u) = 1/u , k(u) = 1/u on the domain
{u > 0} , given by the isentropic gas dynamics in Lagrangian
coordinates, can be included; as
Z q
in b) of Proposition 4, the divergence of the integral
f˙(u)du at 0 and +∞ guarantees
existence of solution to (6),(2) for all data u± > 0 , v± ∈ IR .
On the contrary, in the case of γ -pressure
laws with γ > Z1 (i.e. f (u) = 1/uγ ) we
Z +∞
q
+∞ q
˙
have to impose the restriction v+ − v− <
f (u)du +
f˙(u)du in order to
u−
u+
have existence of a bounded self-similar solution to the approximating system. Otherwise, we
have to deal with solutions where u is unbounded, which corresponds to vanishing density
ρ . Still the vaccuum remains invisible in Lagrangian representation† , so that the appropriate
description of such problems has to be given in the Eulerian one. In Chapter 1.III, we will
carry out a thoroughful investigation of the Riemann problem for the system of isentropic
gas dynamics in Eulerian coordinates. Similar results will be obtained for general continuous
pressure laws and arbitrary Riemann data, including those that give rise to vaccuum in the
solutions.
Another extension of techniques applied above is presented in Section 5 below, where we
treat the Riemann problem for hyperbolic-elliptic systems of form (1). Indeed, as it is shown in
†
Unless one considers measure-valued solutions. In [Wa87] the equivalence of equations of gas dynamics
in Eulerian and Lagrangian coordinates is shown for a very large class of weak solutions. Moreover, the
transformation between the two representations preserves the class of convex entropies, so that it preserves
the entropy admissibility in the sense of Lax (cf. [K70a, Lax71]).
In particular, the result of [Wa87] applies to the Riemann problem with the two initial states different from
vaccuum. It would be interesting to check whether Wagner’s arguments apply to viscosity regularized systems
(which is, clearly, the case when there is no vaccuum). If yes, the wave-fan admissibility is preserved under the
change of representation. As show the results of Chapter 1.III, the solutions to the Riemann problem (6),(2) are
Z +∞ q
Z +∞ q
˙
still ordinary functions, though may be unbounded when the assumption
f (u)du+
f˙(u)du =
1
1
+∞ fails. But we have to pass to the limit in the sense of Radon measures, since the Lagrangian equivalent
of the solution (40)-(45) in Chapter 1.III can contain a Dirac mass (a so-called δ -wave).
There is a connection with another interesting question. Namely, one system where δ -waves naturally arise
in solutions to the Riemann problem is the nonstrictly hyperbolic system
(
ut + (u2 /2)x = 0
(∗)
vt + (uv)x = 0
considered by K.T.Joseph in [Jo93]. This is the simplest representative of another class of systems where the
idea of the present work could be applied for solving the Riemann problem. Complications will arise of the
same order as indicated above. Still it indicates that there could be an explicit formula for admissible solutions
of the Riemann problem. Not surprisingly, this
! formula does exist for the system (∗) : it is obtained in [Jo93],
εuxx
through the viscous regularisation
and the Hopf-Cole-Lax transformation (cf. [H50, Lax57]).
εvxx
1.II.5.
Hyperbolic-elliptic case
57
[KrSt93] (cf. [KrSt97] for a detailed exposition), the monotony of f is not essential; under the
assumption that there exist (finite or infinite) limits f± =: lim f (u) and f− < f (u) < f+
u→±∞
for all u , one can justify the appropriate version of formulae (3)-(5) through the maximal
dissipation principle of Krejčı́-Straškraba. We give a justification of this formulae by the selfsimilar vanishing viscosity approach.
5
The hyperbolic-elliptic case
Let consider the problems (1),(2) and (6),(2) without the monotony assumption on f . Instead, we require that
f− < f (u) < f+
for all u,
where f− =: lim inf f (u), f+ =: lim sup f (u). (18)
u→−∞
u→+∞
Self-similar viscosity limits for a class of hyperbolic-elliptic p-systems modelling van der Waals
fluids have been constructed in [Sl89] and [Fan92], in case of the identity viscosity matrix. It
generates solutions that have the same structural properties as the solution (19)-(22) obtained
below as the limit of solutions of (6),(2). In particular, only stationary phase transitions (i.e.,
jumps across elliptic regions) occur in the two cases. This is not always satisfactory; some
comments on this issue are presented in Remark 4 at the end of this section.
As before, let restrict our attention to bounded self-similar solutions. Let (U, V ) be such
a solution to (6),(2) with some ε > 0 . In general, U will not be continuous at ξ = 0 , since
the continuity of f (U) at 0 yields it no more. Neither will U be strictly monotone outside
of a neighbourhood of ξ = 0 . In turn, this implies that the functions Φ± (·) , defined as in
(11), do not necessarily verify (12) in the classical sense. Whence the two modifications we
have to perform, comparing to the case of strictly increasing f .
First, we take f0 = f (U)(0) , which is well defined, for the parameter in our study
of viscous solutions corresponding to given u± and v− . It will be shown that the limits
u0,± := lim U(ξ) and the value of v− − v+ are uniquely defined by f0 . The relation with
ξ→±0
the problem (12)-(13) on the intervals I(u0,± , u± ) , now understood in a weakened sense
similar to (5) in Chapter 1.I, will be established, followed by the existence and uniqueness result
for this problem. Second, in order to overcome the loss of regularity relative to possible sharper
singularities in f , we will follow the ideas carried out in the case of scalar equation (Chapter
1.I). Beyond some minor modifications in other arguments, we have to upgrade Lemma 1 (the
monotony part), Proposition 1 and the Maximum Principle; in turn, this simplifies the proof
of the version of Proposition 2.
Let f0 ∈ (f− , f+ ) . Under the assumption (18), there exist (unique) u0,± that satisfy


f0 < f (u± )
 max{u < u± | f (u) = f0 },
(19)
u0,± :=
u± ,
f0 = f (u± )


min{u > u± | f (u) = f0 },
f0 > f (u± ).
58
Riemann problem for p-Systems
Denote by F+ = F+ (·; u0,+ ) the convex hull of f on [u0,+ , u+ ] if u0,+ ≤ u+ , and the
concave hull of f on [u+ , u0,+ ] if u0,+ ≥ u+ . Replacing u+ by u− and u0,+ by u0,− ,
define F− = F− (·; u0,− ) in the same way. We will prove that the unique viscous limit solution
to (1),(2) is given by the formulae
 h
i−1
 Ḟ+ (·; u0,+ )
(x2 /t2 ),
h
i
U(t, x) =
 Ḟ (·; u ) −1 (x2 /t2 ),
−
0,−
V (t, x) = v− −
Z
x≥0
,
(20)
x≤0
x/t
ζdU(ζ),
(21)
−∞
dU(ζ) being regarded as measure; (19) and the formula
Z u+ q
Z u− q
v− − v+ =
Ḟ+ (u; u0,+)du +
Ḟ− (u; u0,−)du.
u0,+
(22)
u0,−
provide a one-to-one correspondence between f0 ∈ (f− , f+ ) and an interval of admissible
values of v− − v+ . Additional conditions are required in order to have all values of v− − v+
admissible; for example, f± = ±∞ is sufficient (see [KrSt93]).
Let start with
Lemma 5 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution
solution of (6),(2) if and only if:
· V, V ′ , ξU ′ are continuous on IR , U is continuous on IR \ {0} and admits limits u0,±
as ξ → ±0 such that f (u0,− ) = f (u0,+ ) ;
· the equations
′
εξU (ξ) = −
V (ξ) = −
Z
Z
ξ
ζ 2U ′ (ζ)dζ + f (U(ξ)) + C,
(23)
0
ξ
ζU ′ (ζ)dζ + K
(24)
0
are fulfilled with some constants C, K ;
·
U(±∞) = u± ,
V (±∞) = v± .
Besides, U, V are monotone on each of (−∞, 0) , (0, +∞) .
In addition, u0,± are connected to f0 := f (u0,− ) = f (u0,+ ) through (19).
(25)
1.II.5.
Hyperbolic-elliptic case
59
Proof: Just as in the proof of Lemma 1, we first deduce (23) for ξ > 0 and ξ < 0 .
We would arrive to U ∈ C(IR \ {0}) , ξU ′ (ξ) ∈ C(IR) with limξ→±0 ξU ′ (ξ) = 0 , f (U) ∈
C(IR) , and V ∈ C 1 (IR) , if we only could show that there exist u0,± = limξ→±0 U(ξ) .
Again, it follows from the monotony of U on both sides from the singularity point ξ = 0 .
The monotony property can be proved using (23) and the reasoning carried out in Lemma 1
of Chapter 1.I (see also [BA1]). We complete the proof of the equivalence as in Lemma 1.
Further, note that C = −f0 . For instance, let f (u− ) < f0 . The formula (23) together
with the monotony of U imply that, first, sign (u− − u0,− ) = sign (f (u− ) − f (u0,− )) = −1 ;
second, for all u ∈ [u− , u0,− ) , f (u) < f0 . These two conditions yield u0,− = min{u >
u− | f (u) = f0 } . Thus (19) holds for this case; the other cases are similar.
⋄
Let (U, V ) be a bounded self-similar solution of (6),(2). Using the monotony of U on
(−∞, 0) and (0, +∞) , one can inverse it on each of these intervals. The resulting functions
U±−1 are defined a.e. on I(u0,± , u± ) † and monotone. For all u ∈ I(u0,± , u± ) , set
Z u 2
ε
Φ± (u; u0,± ) :=
U±−1 (w) dw − C
(26)
u0,±
with C taken from (23). We will abrige Φε± (·; u0,± ) to Φ± (·) whenever ε and u0,± are
fixed. Now (23) can be rewritten as
εξU ′ (ξ) = f (U(ξ)) − Φ± (U(ξ)) for ξ ∈ I(0, ±∞).
(27)
As in Chapter 1.I, set Ω± := {u | Φ± (u) = f (u)} ≡ {u | ∃ξ ∈ IR± such that U(ξ) =
u, U ′ (ξ) = 0} . By the Sard lemma, Ω± have the Lebesgue measure 0 . We still obtain
that the equation in (12) is fulfilled outside of Ω± in classical sense. But on the whole of
I(a, b) = I(u0,± , u± ) , we only have (12),(13) fulfilled in the following sense:


Φ ∈ C(I(a, b)), Φ is strictly increasing and (b − a)Φ is convex on I(a, b);





(b − a)(f − Φ) ≥ 0 on I(a, b) and Φ(a) = f (a), Φ(b) = f (b);



(b − a)G(·, Φ(·), Φ̇(·)) ∈ L1loc (I(a, b)) ∩ C(I(a, b); (0, +∞]);
(28)



(b − a) Φ̈(·) − G(·, Φ(·), Φ̇(·)) ≥ 0 in the measure sense on I(a, b);





 (f (·) − Φ(·)) Φ̈(·) − G(·, Φ(·), Φ̇(·)) = 0 in the measure sense on I(a, b)
with G : (u, z, w) ∈ I(a, b) × (−∞, +∞) × (0, +∞) 7→ max{f2εw
∈ (0, +∞] . In(u)−z,0}
deed, for instance suppose u0,− < u− . Then U−−1 is a non-increasing negative function.
2
d
Hence Φ is strictly increasing and Φ̈− (u) = du
U−−1
is a non-negative measure on
(u0,− , u− ) . By (27), Φ− ≤ f and Φ− can be extended on I(a, b) by continuity, with
Φ̇−
Φ− (a) = f (a), Φ− (b) = f (b) . Since f2ε−Φ
= Φ̈− on (u0,− , u− ) \ Ω− and Ω− is of measure
−
2εΦ̇−
f −Φ−
0,
†
∈ L1loc (u0,− , u− ) . The last two properties in (28) are now evident. Besides, it follows
As above, for a, b ∈ IR , we denote by I(a, b) the interval (min{a, b}, max{a, b})
60
Riemann problem for p-Systems
from (19) that f (u0,− ) < f (u) for all u ∈ (u0,− , u− ] ; this implies, as in the proof of PropoΦ̇−
∈ C((u0,− , u− ]; (0, +∞]) .
sition 2, that Φ̇− > 0 on (u0,− , u− ] . Thus we can write f2ε−Φ
−
Conversely, a pair of functions Φ± verifying (28) on I(u0,± , u− ) , where u0,± satisfy
(19), gives rise to a bounded self-similar solution of (6), (2) through the formulae
 h
i−1
 Φ̇+ (·; u0,+ )
(ξ 2 ),
h
i−1
U(ξ) =
 Φ̇ (·; u )
(ξ 2 ),
−
0,−
V (ξ) = v− −
Z
ξ>0
(29)
ξ < 0,
ξ
ζdU(ζ)
(30)
−∞
under the following condition that generalizes (14):
v− − v+ =
Z
u+
u0,+
Z
q
ε
Φ̇+ (u; u0,+ )du +
u−
u0,−
q
Φ̇ε− (u; u0,− )du.
(31)
Indeed, (25) is obvious. As above, (28) implies that Φ̇− > 0 on (u0,− , u− ] ; further, as in
Lemma 2 in Chapter 1.I, we deduce (27) for all ξ ∈ IR . Hence (29) yields (23), (24) and the
continuity properties of Lemma 5.
We summarize Lemma 5 and the results above in
Proposition 5 Let ε, f, u± , v± be fixed. Formulae (29),(30) provide a one-to-one correspondence between the sets A and B defined by
A :=
n
f0 , Φ± (·) | f0 ∈ IR, Φ± : I(u0,± , u± ) 7→ IR and (28), (31) hold,
o
where u0,± are given by (19)
n
o
B := (U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)
Remark 3 Note that (28) is equivalent to (12),(13) under additional regularity assumptions
on f . For example, it is sufficient that f be sublinear at least on one side from each point
of IR . The argument is the same that was used in the proof of Proposition 2.
We state the existence and convergence results and the maximum principle for (28) in case
a<b:
Proposition 6 For all continuous function f , all ε > 0 and a, b ∈ IR such that f (u) >
f (a) for all u ∈ (a, b] , there exists a unique Φ that satisfy (28).
1.II.5.
Hyperbolic-elliptic case
61
Proof: Using the Maximum Principle below, as in the proof of Proposition 2 we construct a
sequence of solutions Φn to (12) penalized by troncatures. (We need the condition f > f (a)
on (a, b] in order to guarantee Φ̇n > 0 .) As n → +∞ , Φn decrease to a continuous
increasing convex function Φ on [a, b] .
Φ̇
By the Fatou Lemma, Φ̈ ≥ f2ε−Φ
in the sense of measures on (a, b) ; this also implies that
Ω := {u | Φ(u) = f (u)} is of measure 0 , and Φ ≤ f . Besides, if Φ(u) < f (u) , then
1
and Φ̇n are uniformly bounded in a neighbourhood of u , because Φn are convex
f −Φn
and Φn (b) = Φ(b) < Φ(a) for all n . Therefore we get the equality in (28) outside of
Ω . As in Proposition 2, it follows that Φ is strictly increasing. Other properties in (28)
are now obvious. Thus the existence of solution is shown; the Maximum Principle yields the
uniqueness.
⋄
Lemma 6 (Maximum Principle) Let (a, b) ⊂ IR and Φ, Ψ satisfy, in the sense of (28),
the “inequalities” Φ̈(u) ≥ G(u, Φ(u), Φ̇(u)) and Ψ̈(u) ≥ H(u, Ψ(u), Ψ̇(u)) , respectively,
with G, H : (a, b) × (−∞, +∞) × (0, +∞) 7→ (0, +∞] .
a) Assume that G(u, z, w) < H(u, ζ, w) for all u such that Φ(u) < Ψ(u) and all z, ζ, w
such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and Φ(b) ≥ Ψ(b) .
b) Assume that G(u, z, w) ≡ H(u, z, w) and increases in z (strictly) and in w ; let
Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ − Ψ) is monotone on [a, b] .
Proof: For instance, let prove a). Take c the point of minimum of Φ − Ψ ; c ∈ (a, b) .
Suppose this minimum is negative. Then Φ(c) < Ψ(c) ≤ f (c) , so that there exists δ1 > 0
such that Φ̈(·) = G(·, Φ(·), Φ̇(·)) ∈ L1loc (c, c + δ1 ) . Then there exists Φ̇(c) ; by the choice of
c , we will have Ψ̇(c+0) ≤ Φ̇(c) ≤ Ψ̇(c−0) . The convexity of Ψ implies that Ψ̇(c) = Φ̇(c) .
Therefore we have G(u, Φ(u), Φ̇(u)) < H(u, Ψ(u), Ψ̇(u)) for u = c , and consequently, for
all u ∈ [c, c + δ2 ) for some δ2 > 0 . From (28) we have Φ̈ < Ψ̈ in the sense of measures
on [c, c + δ) , where δ := min{δ1 , δ2 } . Therefore Φ − Ψ strictly decreases on [c, c + δ] ,
which contradicts to the choice of c .
⋄
Other changes, with respect to the case of monotone f , are minimal. Let only note that
we have to take into account that u0,± are not continuous as functions of f0 , in particular
in the assertion analogous to d) of Lemma 4.
As in the continuous case, we can prove the convergence of Φε± with fixed f0 ; then
establish that (31) is a bijection between f0 and v− −v+ (for instance, under the assumption
f± = ±∞ ); then prove the convergence of Φε± with fixed v− − v+ to the corresponding
convex hulls and apply Lemma 3.
The following theorem holds:
Theorem 2 Suppose lim inf f (u) = −∞, lim sup f (u) = +∞ . Then for all u± , v± ∈ IR ,
u→−∞
u→+∞
ε > 0 there exists a unique bounded self-similar distribution solution (U ε , V ε ) of the problem
(6),(2).
62
Riemann problem for p-Systems
Besides, as ε ↓ 0 (U ε , V ε )(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by
the formulae (19)-(22), so that (U, V ) is a self-similar distribution solution of the problem
(1),(2).
Remark 4 In this last section we have answered to a question posed by R.James in [Ja80],
the paper from which an extensive study of admissibility for (1),(2) in the non-monotone case
had started. James studies nonlinear elasticity and proposes to define admissible solutions as
limits of the vanishing viscosity approximations of the form
(
Ut − Vx = 0
(32)
Vt − f (U)x = εVxx ,
following the idea defended by Rayleigh ([Ray10]) for the isentropic fluid dynamics. James
calls this criterion the viscoelastic criterion. He proceeds by deducing that the energy should
decrease in time, but this condition is not sufficient for uniqueness for the Riemann problem
(which is well known in the scalar case, in absence of convexity). He proposes then to
obtain more
! restrictions on admissible shocks either by using the full strength of the viscosity
0
, either by replacing this viscosity in (32) by the artificial one used by Dafermos in
εVxx
!
!
εtUxx
0
[D74],
(or by
, which encounters less physical objections).
εtVxx
εtVxx
For the first case, there is a response in the case of flux functions with one inflexion point.
M.Shearer in [Sh82] constructs a Riemann solver which provides a unique solution to (1),(2) for
all data, using classical shocks (always admissible by the viscoelastic criterion) and additional
phase transitions propagating with zero speed. R.Pego in [Pe87] observes that only these two
kinds of shocks can be approximated by travelling waves solutions of (32).
The result of this section give a response for the second case. Evidently, in the Riemann
solver (19)-(22) the only admissible transitions between elliptic and hyperbolic regions or
between two elliptic regions are those with zero speed.
It seems that this last property is not always what one observes in physical systems. The
reason is, dissipation effects captured by vanishing viscosity can coexist with dispersion effects
provoked by capillarity. M.Slemrod in [Sl83]
! proposes a family of viscosity-capillarity criteria,
0
. The parameter A ∈ [0, 1/4] regulates phase
regularizing (1) with
εVxx − Aε2 Vxxx
transitions; an important feature is that viscosity-capillarity limits admit phase transitions of
non-zero speed whenever A > 0 . Clearly, the self-similar viscosity approach exposed above
is unable to capture this kind of effects. To pursue the study, one has to introduce additional
self-similar dissipation, as it has been done by M.Slemrod and H.Fan in [Sl89, Fan92].
Observations on the difference of diffusive and diffusive-dispersive limits for (1),(2) have
recently gave rise to a theory of non-classical Riemann solvers, in particular for nonconvex scalar
conservation laws. A survey of results on diffusive-dispersive limits and related questions can
be found in [LF98].
CHAPTER 1.III
On Viscous Limit Solutions
to the Riemann Problem
for the Equations of Isentropic Gas Dynamics
in Eulerian Coordinates†
Introduction
In this chapter we study bounded self-similar solutions to the problem
(
ρt + (ρu)x = 0
(ρu)t + (ρu2 + p(ρ))x = εtuxx
with the initial condition
(
ρ+ , x > 0
ρ(0, x) =
,
ρ− , x < 0
u(0, x) =
(1)
(
u+ ,
u− ,
x>0
x<0
(2)
and establish convergence of solutions as ε ↓ 0 .
Within the framework of isentropic gas dynamics in Eulerian coordinates, (ρ, u) : (t, x) ∈
IR × IR 7→ (ρ(t, x), u(t, x)) ∈ IR+ × IR corresponds to the density and velocity in gas,
p is the pressure law of the gas, and ε > 0 models small dissipation of the momentum.
We assume that ρ± > 0 and p(·) is continuous strictly increasing on IR+ , normalized by
p(0) = 0 .
+
Recently, the problem (1),(2) has been treated in [Kim99], following the ideas of [Tz95] (see
also [Tz96]). Under additional assumptions that prohibit vaccuum in the solutions, existence
for the problem (1),(2) was proved. The set of all solutions was shown to be compact in BV ,
and the wave-fan structure of limiting functions as ε ↓ 0 was described.
†
This chapter is being prepared upon publication [BA4]
64
Riemann Problem for Gas Dynamics in Eulerian Coordinates
Here we give a description of solutions to (1),(2) that is also valid when a solution contains
a vaccuum state. It also suggests a formula for the limiting function as ε ↓ 0 (cf. Section 3).
Let illustrate the problem with the classical example of γ -laws, i.e. p(ρ) = const · ργ ,
γ ≥ 0 . First take γ = 1 . Set f (V ) = −p(1/V ) , k(V ) = 1/V , and consider the problem
(
Vt − u y = 0
(3)
ut − f (V )y = εt(k(V )uy )y
V (0, y) =
(
1/ρ+ ,
1/ρ− ,
y>0
,
y<0
u(0, y) =
(
u+ ,
u− ,
y>0
.
y<0
(4)
This problem is (1),(2) rewritten in the Lagrangian coordinates (here y is the matherial
coordinate, and V = 1/ρ ), provided ρ > 0 (e.g., see [RoJa], [ChHs]). For the system
(3), the Riemann problem can be studied extensively (cf. Section 4 of Chapter 1.II) by the
method applied in [BA3]. It yields existence and uniqueness of a bounded self-similar solution
for all Riemann data V± > 0, u± ∈ IR , and all ε > 0 . As ε ↓ 0 , the solutions converge
to a function described by an explicit formula, based on use of convex and concave hulls of
the graph of f . In each of this solutions V is bounded, i.e. ρ = 1/V > 0 .Therefore we
can pass to the Eulerian coordinates and deduce the same results for (1),(2). The explicit
formula for the limiting function will use the images of convex/concave hulls of f (V ) under
the transformation
T : [F : V ∈ (0, +∞) 7→ F (V )] 7→ [P : ρ ∈ (0, +∞) 7→ −F (1/ρ)].
(5)
For instance, let [a, b] ⊂ (0, +∞) and F (·) be the concave hull of f (·) on [1/b, 1/a] . Let
P (·) be the function ρ ∈ (0, +∞) 7→ −F (1/ρ) . Then P (·) can be characterized by the
following properties:


(i)
P (·) ∈ C[a, b] and P ≤ p on [a, b]

−1
(6)
(ii) the function F = T P is concave on (1/b, 1/a)


(iii) for all Q(·) that satisfies (i) and (ii), one has P ≥ Q on [a, b].
In the case p(ρ) = const · ργ with γ > 1 , it is well known that, in general, one cannot
avoid the appearance of vaccuum in solutions of (1),(2) with ε = 0 (a detailed study of
this problem for γ ≥ 1 through construction of wave curves on the half-plane (ρ, u) can be
found in [ChHs]).
The same difficulty
appears for ε > 0 . This is due to the fact that the
Z 1r
Z 1/ρ r
d
dr
d
integral
p(r) =
f (v)dv converges as ρ → 0 , which impose a bound
dr
r
dv
ρ
1
on the size of Riemann data in order to have a bounded solution of (3),(4) (e.g., see Chapter
1.II and [BA3]). We will solve the problem (1),(2) independently (cf. Theorem 1) since we
cannot reduce it to (3),(4) any more† .
†
In fact, such reduction is possible, but should involve measure-valued solutions at the limit as ε ↓ 0 ; see
[Wa87] and the footnote in Section 4 of Chapter 1.II.
1.III.0.
Introduction
65
Nevertheless, it is still possible to deduce a formula for the limiting function. In fact, (6)
makes sense also for a = 0 (upon formally setting 1/a = +∞ in (ii)), which corresponds
to the presence of a vaccuum state ρ = 0 . More precisely, for all [a, b] ⊂ [0, +∞) there
exists a (unique) function P (·) that satisfies (6). Indeed, for a > 0 it is obvious; for a = 0 ,
we construct P (·) as the decreasing limit of functions Pδ (·) such that Pδ (·)|[0,δ] ≡ p(δ) ,
Pδ (·)|[δ,b] verifies (6) on [δ, b] . Proprieties (i) − (iii) in (6) follow easily from the monotony
and continuity of P (·) . This motivates the following definition.
Definition 1 Let p : IR+ 7→ IR be continuous and strictly increasing. For a ≥ 0 and
b ≥ a , the lower (-1)-hull of p(·) on [a, b] is the function P (·) that verifies (6). For b > 0
and a ≥ b , the upper (-1)-hull of p(·) on [b, a] is the function P (·) that verifies
(i)
P (·) ∈ C[b, a] and P ≥ p on [b, a]
(ii) the function F = T −1 P is convex on (1/a, 1/b)
(iii) for all Q(·) that satisfies (i) and (ii), one has P ≤ Q on [b, a].





(7)
For a′ , b′ ∈ (IR+ )2 denote the segment [min{a′ , b′ }, max{a′ , b′ }] by I(a′ , b′ ) . The (-1)-hull
of p(·) on I(a′ , b′ ) is the lower (-1)-hull on [a′ , b′ ] in case a′ ≤ b′ and the upper (-1)-hull
on [b′ , a′ ] in case a′ > b′ .
In Section 3 we consider the system
(
ρt + (ρu)x = 0
(ρu)t + (ρu2 + p(ρ))x = 0
(8)
and, using (-1)-hulls of p(·) , give a formula for the unique solution of the Riemann problem
(8),(2) that can be obtained as a limit of self-similar bounded weak solutions of (1),(2) with
ε = εn for a sequence εn ↓ 0 .
The approach by self-similar viscosity limits has been used in [Ka59, Tu64, Tu66, Tu73,
D73a, D74, DDp76] in the context of admissibility of weak solutions to the Riemann problem
for hyperbolic systems of conservation laws. In [D89], Dafermos postulated it as the wave fan
admissibility criterion. It has been successfully tested on various special systems and viscosity
matrices; see [Tz98], [Tz96] for a survey of recent results in this direction. In particular,
an
!
ρ
analysis of the problem (8),(2) regularized with the self-similar viscosity εt
is carried
ρu
out in [SlTz89], covering among others the cases where vaccuum is present. Within the same
framework, a special attention to the formation of the vaccuum state is paid in [Fan91]. In
[Tz96] the existence of an admissible solution is proved for a large class of strictly hyperbolic
systems with close Riemann data, using the identity self-similar viscosity matrix.
66
Riemann Problem for Gas Dynamics in Eulerian Coordinates
The !
main result of this chapter is that, for the case of the degenerate viscosity matrix
0 0
and under the additional assumption
0 1
(a) either p(ρ) → +∞ as ρ → +∞,
(b) or p ∈
1,1
Wloc
(R, +∞)
for some R > 0, and d
Z
+∞
R
r



d
dr
p(r) = +∞, 

dr
r
(9)
there is global existence and uniqueness of an admissible weak solution to the Riemann problem
for the nonstrictly hyperbolic system (8) in the sense of the wave fan admissibility criterion
(cf. Theorem 2). We describe the structure of this solution in Section 3.
1
Some useful properties of viscous approximations
Let fix ε > 0 . We are concerned with bounded self-similar distribution solutions to the
problem (1),(2).
Definition 2 A pair of functions (ρ, u) : IR+ × IR 7→ IR+ × IR is a solution of the problem
(1),(2) if for all k > 0 , (ρ, u)(t, x) = (ρ, u)(kt, kx) for a.a. (t, x) ∈ IR+ × IR , ρ, u ∈
L∞ (IR+ × IR) , (1) is fulfilled in D ′ (IR+ × IR) , and
ess lim kρ(t, ·) − ρ(0, ·)kL1 (−R,R) + ku(t, ·) − u(0, ·)kL1(−R,R) = 0
(10)
t↓0
for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2).
We will denote x/t by ξ and ambiguously use the same notation for a self-similar function
of the variables (t, x) and the corresponding function of ξ .
Lemma 1 A pair (ρ, u) is a solution of (1),(2) in the sense of Definition 2 if and only if the
following conditions are fulfilled:
(i) there exist continuous bounded functions ρ, u : IR 7→ IR , with u′ (·) and (· −u(·))ρ′ (·)
continuous, such that (ρ, u)(t, x) = (ρ, u)(x/t) for a.a. (t, x) ∈ (0, +∞) × IR ;
(ii) there exists a constant C ∈ IR such that one has
′
εu (ξ) = −
Z
ξ
0
(ζ − u(ζ))2ρ′ (ζ) dζ + f (ρ(ξ)) + C,
ρ(ξ)u′ (ξ) = (ξ − u(ξ))ρ′ (ξ),
lim ρ(ξ) = ρ± ,
ξ→±∞
lim u(ξ) = u± .
ξ→±∞
(11)
(12)
(13)
1.III.1.
Useful properties of viscous approximations
67
Besides, ther exists a unique ξ0 such that u(ξ0) = ξ0 . In case ρ(ξ0 ) > 0 there exist
ξ± ∈ IR , ξ− ≤ ξ0 ≤ ξ+ , such that both ρ(·) and u(·) are constant on (ξ− , ξ+ ) and
strictly monotone on (−∞, ξ− ) and (ξ+ , +∞) .
In case ρ(ξ0 ) = 0 , ξ0 is the unique vaccuum point in the solution, ξ± = ξ0 , and the same
monotony properties hold.
Moreover, u′(ξ) 6= 0 for all ξ ∈ (−∞, ξ− ) ∪ (ξ+ , +∞) , u′ (ξ0 ) = 0 in case ρ(ξ0 ) > 0 and
0 ≤ u′ (ξ0 ) < 1 in case ρ(ξ0 ) = 0 .
Remark 1 We see from Lemma 1 that in case ε > 0 , solutions of (1),(2) contain at most
one vaccuum point.
Proof: The proof consists of four steps.
I) Let (ρ, u) be solution of (1),(2). Then (ρ, u)(t, x) = (ρ, u)(x/t) and we have in D ′ (IR)
(
−ξρ′ + (ρu)′ = 0
(14)
−ξ(ρu)′ + (ρu2 + p(ρ))′ = εu′′.
It follows that εu′′ = −ξ 2 ρ′ + (ρu2 + p(ρ))′ = −(ξ 2 ρ)′ + 2ξρ + (ρu2 + p(ρ))′ in D ′ (IR) ,
′
′
′
whence u′ ∈ L∞
loc (IR) and u ∈ C(IR) . Thus ρu , (ρu)u are well defined in D (IR) , hence
ρ′ u , ρ′ u2 as well. Therefore we obtain
−(ξ − u)ρ′ + ρu′ = 0 in D ′ (IR)
(15)
and
−(ξ − u)2 ρ′ + p(ρ)′ = εu′′ in D ′ (IR).
(16)
Z ξ
It follows that −(ξ − u)ρ =
ρ(ζ) dζ + const ∈ C(IR) . Consequently, −(ξ − u)2 ρ +
0
Z ξ
p(ρ) − εu′ =
2(ζ − u(ζ))ρ(ζ) dζ + const ∈ C 1 (IR) . Thus
0
p(ρ(·)) − εu′ (·) ∈ C(IR).
(17)
II) Now consider the function ξ 7→ ξ−u(ξ) . It is continuous and tends to ±∞ as ξ → ±∞ .
Therefore there exist finite η− := min{ξ | u(ξ) = ξ} and η+ := max{ξ | u(ξ) = ξ} . One
has ξ − u(ξ) < 0 on (−∞, η− ) and ξ − u(ξ) > 0 on (η+ , +∞) . By (15), on each of
these intervals one has ρ′ ∈ L∞
loc . Therefore ρ is continuous on these intervals; by (17), so
′
′
does u ; by (15), so does ρ .
Let prove that ρ, u are monotone on (−∞, η− ) and (η+ , +∞) . Take ξ+ := sup{ξ ≥
η+ |ρ|(η+ ,ξ) ≡ const} . First show that u is strictly monotone on (ξ+ , +∞) . Indeed, suppose
the contrary. Then there exists c ∈ (ξ+ , +∞) and 0 < δ < c − ξ+ such that u′ (c) = 0
68
Riemann Problem for Gas Dynamics in Eulerian Coordinates
′
ρu
and u′ 6= 0 on (c − δ, c) . From (15) we have ρ′ = ξ−u
≥ 0 on (c − δ, c) , so that p(ρ(·))
is non-decreasing on [c − δ, c] . Hence by (15),(16) we have
Z c
Z c
′
′
εu (ξ) =
(ζ − u)ρu dζ + p(ρ(ξ)) − p(ρ(c)) ≤
(ζ − u)ρu′ dζ
ξ
ξ
pointwise on [c − δ, c] . Choosing a sequence ξn ↑ c such that u′ (ξn ) = maxξ∈[ξn,c] u′ (ξ) ,
we get εu′(ξn ) ≤ u′ (ξn ) O(c − ξn ) , where O(·) is the Landau symbol. As n → ∞ , we
deduce ε ≤ 0 , which is a contradiction.
We conclude that u is strictly monotone on (ξ+ , +∞) ; hence ρ is monotone on
(η+ , +∞) . Similarly, there exists ξ− ≤ η− such that u, ρ are strictly monotone on
(−∞, ξ− ) and ρ ≡ const on [ξ− , η− ) .
III) Let investigate the behaviour of ρ, u on the interval [η− , η+ ] . First note that there exist
finite limits ρ(η± ± 0) ; by (17), there exist finite limits u′ (η± ± 0) . We notice that, first,
u′(η+ + 0) = 0 in case ρ(η+ + 0) > 0 , and 0 ≤ u′(η+ + 0) < 1 in case ρ(η+ + 0) = 0 .
The same relation exists between u′(η− − 0) and ρ(η− − 0) .
Z η+ +1 Z η+ +1
′
u′(ζ)
dζ .
Indeed, let ρ(η+ +0) > 0 . By (15), we have
ln ρ(ζ) dζ =
ζ − u(ζ)
η+
η+
The integral on the left converge, therefore the limit u′ (η + 0) is necessarily 0 . Further, let
ρ(η+ + 0) = 0 . Then ρ is non-decreasing on (η+ , +∞) , hence u′(η+ + 0) ≥ 0 by (15). On
the other hand, the definition of η+ trivially implies that u′(η+ + 0) ≤ 1 . Besides, suppose
u′(η+ + 0) = 1 . From (16),(15) we obtain
Z ξ
′
ε(1 − u (ξ)) =
(ζ − u(ζ))ρ(ζ)u′(ζ) dζ − p(ρ(ξ))
η+
for all ξ ∈ (η+ , +∞) . Since ρu′ is continuous on this interval and tends to 0 as ξ ↓ η+ ,
there exists δ > 0 such that
Z ξ
′
ε(ξ − u(ξ)) ≤
(ζ − u(ζ)) dζ
(18)
η+
√
Z
ξ
for all ξ ∈ (η+ , η+ + δ) . Setting g(ξ) := ξ − u(ξ) and h(ξ) := εg(ξ) +
g(ζ)dζ , we
η+
√
obtain from (18) that h ∈ C 1 [η− , +∞) and h′ (ξ) ≤ 1/ ε h(ξ) . Since h(η+ ) = 0 , we get
h ≡ 0 on [η+ , η+ + δ] by the Gronwall inequality. This contradicts the definition of η+ .
Thus, definitively, 0 ≤ u′(η+ + 0) < 1 . The proof for η− − 0 in the place of η+ + 0 is
likewise.
Now we analyse separately the three cases:
a) η− = η+ =: ξ0 and one of the limits ρ(ξ0 ± 0) is non-zero. Then the two limits
coincide. Indeed, let ρ(ξ0 + 0) > 0 . First assume ρ(ξ0 − 0) > 0 ; in this case u′ (ξ0 ± 0) =
0 , so that ρ(ξ0 ± 0) coincide by (17). Further, assume ρ(ξ0 − 0) = 0 . In this case
1.III.1.
Useful properties of viscous approximations
69
p(ρ(ξ0 +0))−εu′ (ξ0 +0) = p(ρ(ξ0 +0)) > 0 , and p(ρ(ξ0 −0))−εu′ (ξ0 −0) = −εu′ (ξ0 −0) ≤ 0 ,
which contradicts (17). Hence we see that ρ, u′ are both continuous on IR . Therefore
(15),(16) can be rewritten under the form (11),(12).
We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have
ρ(ξ0 ) > 0 and u′ (ξ0 ) = 0 . Besides, with ξ± defined above, we find that ρ, u are constant
on [ξ− , ξ+ ] and that u′ , consequently ρ′ , are different from 0 on (−∞, ξ− ) ∪ (ξ+ , +∞) .
b) η− = η+ =: ξ0 and ρ(ξ0 ± 0) = 0 . Then ρ, u′ are continuous on IR , and (11),(12)
follow. Further, ξ = ξ0 is the unique vaccum point. Indeed, for instance, take ξ+ =
Z ξ+ +1 Z ξ+ +1
′
u′ (ζ)
sup{ξ | ρ(ξ) = 0} . We have
dζ , but this time the
ln ρ(ζ) dζ =
ζ − u(ζ)
ξ+
ξ+
left-hand side integral diverge. It follows that ζ − u(ζ) → 0 as ξ ↓ ξ+ , whence ξ+ = ξ0 .
We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have
ρ(ξ0 ) = 0 and 0 ≤ u′ (ξ0 ) < 1 ; besides, u′ , ρ′ are different from 0 on IR \ {ξ0 } .
c) η− < η+ Actually, we will show that Zthis case is impossible. Indeed, from (15) we have
h
iη+
η+
′
′
ρ = ((ξ − u)ρ) in D (IR) . Therefore
ρ(ζ) dζ = (ξ − u(ξ))ρ(ξ) = 0 , so that
η−
η−
ρ|(η− ,η+ ) ≡ 0 . Hence u′ |(η− ,η+ ) ≡ const by (16); taking into account that u(η± ) = η± , we
find u′(η± ∓ 0) = 1 . Besides, 0 ≤ u′(η± ± 0) < 1 in all cases; arguing as in case a), we get
a contradiction with (17).
We conclude that ρ(·), u(·) satisfy (11),(12) and have all continuity and structure properties in Lemma 1. The monotony of ρ, u trivially implies that (13) is satisfied if and only if
self-similar ρ(·, ·), u(·, ·) satisfy (10).
IV) Conversely, (11)-(13) together with the continuity of ρ(·), u(·), u′(·) and (· − u(·))ρ′ (·)
imply that (ρ, u)(t, x) := (ρ, u)(x/t) is a solution of (1),(2). Indeed, (14) is straightforward.
Besides, (11),(12) yield the monotony of ρ(·), u(·) at ±∞ . Thus (10) holds also.
⋄
Using the results of Lemma 1, set ρ0 := ρ(ξ0 ) and k := u′(ξ0 ) ; define σ := ρ0 − k .
Note that σ ∈ (−1, +∞) and
ρ0 = (σ)+ = max{σ, 0},
k = (σ)− = max{−σ, 0}.
(19)
Further, split IR into the three (may be empty) intervals (−∞, ξ− ) , (ξ− , ξ+ ) , (ξ+ , ∞) . Let
us inverse ρ(·) on (−∞, ξ− ) and (ξ+ , +∞) . We will ambiguously use the same notation
for the function ρ(·) and the independent variable ρ ∈ IR+ . The functions
ρ−1
− : I(ρ0 , ρ− ) 7→ (−∞, ξ− ),
are well defined. Set
Z
ε
Π± (ρ; σ) :=
ρ
(σ)+
ρ−1
+ : I(ρ0 , ρ+ ) 7→ (ξ+ , +∞)
h
i2
−1
ρ−1
(r)
−
u(ρ
(r))
dr − C
±
±
(20)
70
Riemann Problem for Gas Dynamics in Eulerian Coordinates
for ρ ∈ I(ρ0 , ρ± ) , where C is taken from (11). Let use the simplified notation Π± (·) for
Πε± (·; σ) whenever ε, σ are fixed; besides, let ˙ denote derivation with respect to ρ . We
can rewrite (11) under the form
εu′ (ξ) = p(ρ(ξ)) − Π± (ρ(ξ)).
(21)
Since 0 ∈
/ I(ρ0 , ρ± ) and u′(ξ) is shown to be non-zero on (−∞, ξ− )∪(ξ+ , +∞) , we deduce
that Π± ∈ C 2 (I(ρ0 , ρ± )) and that one has
ρΠ̈± + 2Π̇± =
2εΠ̇±
,
f − Π±
Π̇± > 0 and sign(p − Π± ) = sign(ρ± − ρ0 )
(22)
on I(ρ0 , ρ± ) . Further, Π± can be extended to I(ρ0 , ρ± ) by continuity. By (21) and
Lemma 1 Π± (·) admits finite limits at ρ0 , and we can assign
Π± (ρ0 ) = p(ρ0 ) − εk,
(23)
where ρ0 , k are defined by (19). Besides,
Π± (ρ± ) = p(ρ± ).
(24)
Indeed, the right-hand side in (21) admits finite limits as ρ(ξ) → ρ± , because, in case
ρ0 < ρ± , Π± (·) are increasing and bounded from above and, in case ρ0 > ρ± , Π± (·) are
concave and bounded from below. We deduce u′ (ξ) → 0 as ξ → ±∞ , since u has finite
limits at ±∞ .
In addition, in order to cover the case of ξ− = −∞ (i.e., ρ0 = ρ− ) and/or ξ+ = +∞
(i.e., ρ0 = ρ+ ), we just define Π− (·) and/or Π+ (·) by (24).
q
ρ′ (ξ)
′
for all ξ ∈ (−∞, ξ− ) and all
Finally, (20) and (12) yield u (ξ) = ± Π̇± (ρ)
ρ
ξ ∈ (ξ+ , +∞) , respectively. Since u(ξ− ) = u(ξ+ ) , (13) yields that
u+ − u− =
Z
ρ+
ρ0
Z ρ− q
q
dr
dr
Π̇+ (r)
+
Π̇− (r)
r
r
ρ0
(25)
and the integrals in the right-hand side of (25) are finite.
We have established the following result.
Proposition 1 Let (ρ, u) be a solution of (1),(2) in the sense of Definition 2. Then there
exist σ ∈ (−1, +∞) , ρ0 ∈ [0, +∞] , k ∈ [0, 1) , and functions Π± ∈ C(I(ρ0 , ρ± )) ∩
C 2 (I(ρ0 , ρ± )) such that (19) and (22)-(25) hold.
The converse assertion is also true.
1.III.1.
Useful properties of viscous approximations
71
Proposition 2 Let σ ∈ (−1, +∞) and ρ0 , k be defined by (19). Let Π± ∈ C(I(ρ0 , ρ± )) ∩
C 2 (I(ρ0 , ρ± )) satisfy (22)-(25). Then there exists a solution (ρ, u) of (1),(2) in the sense
of Definition 2, and it is given by

(

[Ξε− ]−1 (x/t), x/t < ξ−
−1

ε
[Ξ− ] (x/t), x/t < ξ+
ρ(t, x) = ρ(x/t) =
≡
ρ0 ,
ξ− < x/t < ξ+ (26)

[Ξε+ ]−1 (x/t), ξ− < x/t
 ε −1
[Ξ+ ] (x/t), ξ+ < x/t,
where

ε
ε −1

 U− ◦ [Ξ− ] (x/t),
u(t, x) = u(x/t) =
U−ε (ρ0 ) = U+ε (ρ0 ),

 ε
U+ ◦ [Ξε+ ]−1 (x/t),
U±ε (ρ)
Ξε± (ρ)
:= u± ∓
:=
Z
U±ε (ρ)
ρ
ρ±
q
Π̇± (r)
q
± Π̇± (ρ)
dr
r
x/t < ξ−
ξ− < x/t < ξ+
ξ+ < x/t,
(27)
for ρ ∈ I(ρ0 , ρ± ),
(28)
for ρ ∈ I(ρ0 , ρ± ),
(29)
and ξ± are defined by
ξ± :=
lim
ρ∈I(ρ0 ,ρ± ), ρ→ρ0
Ξε± (ρ)
if
ρ0 6= ρ± ;


ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+ . 
(30)
Proof: The cases ρ0 = ρ+ , ρ0 = ρ− are trivial; assume ρ0 6= ρ± . Let U±ε (·) , Ξε± (·)
be defined by (28),(29). Note that both integrals in (28) necessarily converge as ρ → ρ0 ,
ρ ∈ I(ρ0 , ρ± ) . Indeed, in case ρ0 = 0 both are positive and thus finite, by (25). In case
ρ0 > 0 , Π̇± are bounded as ρ → ρ0 , by monotony of ρ2 Π̇± which is evident from (22).
Therefore U±ε (ρ0 ) are well defined; by (25) they coincide. Note also that
q
ε Π̇± (ρ)
ρΠ̈± (ρ) + 2Π̇± (ρ)
q
Ξ̇ε± (ρ) = ±
=±
(31)
ρ(p(ρ) − Π± (ρ))
2ρ Π̇± (ρ)
are continuous and non-zero on I(ρ0 , ρ± ) , so that ξ± and [Ξε± ]−1 are well defined in the
graph sense. We see from (25),(28),(29) that
q
q
ξ+ − ξ− =
lim
Π̇− (ρ) +
lim
Π̇+ (ρ) ≥ 0.
(32)
ρ∈I(ρ0 ,ρ− ), ρ→ρ0
ρ∈I(ρ0 ,ρ+ ), ρ→ρ0
Clearly, ρ, u ∈ C(IR) ∩ C 1 (IR \ {ξ− , ξ+ }) ; besides, (26)-(29),(31) and
q(24) yield (13),(21)
for all ξ 6= ξ± . In fact, we could show that (24),(22) imply that
Π̇± (ρ) → +∞ as
ε
ρ → ρ± , ρ ∈ I(ρ0 , ρ± ) , so that Ξ± map I(ρ0 , ρ± ) onto (ξ+ , +∞) and (−∞, ξ− ) ,
72
Riemann Problem for Gas Dynamics in Eulerian Coordinates
respectively. Still (24) yields (21) for ξ outside the range of Ξε± even if we admit that Ξε±
can be bounded. Futhermore, by (31),(21),(29) and (26),(27) we have for all ξ 6= ξ±
(ξ − u(ξ))ρ′(ξ) = ±
ρ(p(ρ(ξ)) − Π± (ρ(ξ)))
= ρ(ξ)u′(ξ);
±ε
(33)
whence (12) follows for ξ 6= ξ± .
Now consider the two possibilities:
a) ρ0 > 0 . Then u′ (ξ± ± 0) = 0 by (21) and (24), while in case ξ− < ξ+ we have
u′(ξ± ∓ 0) = 0 by (27). Thus u ∈ C 1 (IR) and, by (33), (· − u(·))ρ′ (·) is continuous on
IR . It now follows that (11),(12) hold everywhere; besides, (13) is obvious.
b) ρ0 = 0 . Then Π̇± (ρ) → 0 as ρ → ρ0 , ρ ∈ I(ρ0 , ρ± ) . Indeed, by (22) we have for
ρ ∈ I(ρ0 , ρ± )
Z (ρ0 +ρ+ )/2 Z (ρ0 +ρ+ )/2 ·
2
ε
− 1 dr.
ln Π̇± (r) dr =
r p(r) − Π± (r)
ρ
ρ
By (23),(19) it follows that ε/(p(r) − Π± (r)) − 1 → 1/k − 1 > 0 as ρ → ρ0 . Therefore
the last integral diverge, hence ln Π̇± (ρ) → −∞ as ρ → ρ0 , ρ ∈ I(ρ0 , ρ± ) . We conclude
by (32) that ξ− = ξ+ ; by (21) and (23), u′ (ξ± ± 0) = k . Thus again u′ (·) , (· − u(·))ρ′(·)
are continuous on IR , and (11)-(13) hold.
By Lemma 1, it follows that (ρ, u) is a solution of (1),(2).
2
⋄
Existence and uniqueness of viscous approximations
In Section 1, we have reduced the problem (1),(2) to finding σ ∈ (−1, +∞) , ρ0 , k and a
pair of functions Π± ∈ C(I(ρ0 , ρ± )) ∩ C 2 (I(ρ0 , ρ± )) that satisfy (19) and (22)-(25). In this
section we prove that such σ, ρ0 , k, Π± (·) do exist (and are unique), provided p(·) satisfies
(9) (cf. Proposition 3 and Lemmae 4,5,6 below). We also prove two preliminary convergence
results (cf. Lemma 3 and Proposition 4).
Start by fixing σ ∈ [−1, +∞) and b ∈ (0, +∞) . Set a := (σ)+ , k := (σ)− and
consider the problem of finding Π ∈ C(I(a, b) ∩ C 2 (I(a, b)) satisfying

2εΠ̇(ρ)


ρΠ̈(ρ) + 2Π̇(ρ) =
with Π̇(ρ) > 0 and (b − a)(p(ρ) − Π(ρ)) > 0



p(ρ) − Π(ρ)

for all ρ ∈ I(a, b),
(34)



Π(a) = p(a) − εk,



Π(b) = p(b).
Proposition 3 There exists a unique solution to the problem (34).
Let Πε (·) denote the solution of (34) corresponding to ε , ε > 0 . Recall Definition 1.
1.III.2.
Existence and uniqueness of viscous approximations
73
Proposition 4 As ε ↓ 0 , Πε converge to the (-1)-hull of p(·) uniformly on I(a, b) .
First note that the equation in (34) is still in the scope of the maximum principle (cf.
Chapter 1.II). For the sake of completeness, we restate it here for the case a < b .
Lemma 2 [Maximum Principle] Let Π, Υ ∈ C[a, b] ∩ C 2 (a, b) and satisfy, for all ρ ∈ (a, b) ,
the equations Π̈(ρ) = G(ρ, Π(ρ), Π̇(ρ)) and Ϋ(ρ) = H(ρ, Υ(ρ), Υ̇(ρ)) , respectively, with
some G, H : (a, b) × IR × (0, +∞) 7→ (0, +∞] .
a) Assume that G(ρ, z, w) < H(ρ, ζ, w) for all ρ ∈ (a, b) such that Π(ρ) < Υ(ρ)
and all z, ζ, w such that z < ζ . Then Π ≥ Υ on [a, b] whenever Π(a) ≥ Υ(a) and
Π(b) ≥ Υ(b) .
b) Assume that G(ρ, z, w) ≡ H(ρ, z, w) , increases in w and strictly increases in z ; let
Π(a) = Υ(a) or Π(b) = Υ(b) . Then (Π − Υ) is monotone on [a, b] .
Proof: The proof is straightforward.
⋄
Secondly, note the following lemma, which will also be an ingredient of the convergence
proof in Section 3. Recall (5); as in Definition 1, we understand 1/a as +∞ in case a = 0 .
Lemma 3 Let [a, b] ⊂ IR+ and {P ε }ε≥0 ⊂ C[a, b] be a set of functions such that F ε =
T −1 P ε are concave on (1/b, 1/a) . Assume that, for all ρ ∈ [a, b] , P ε (ρ) converge to
P 0 (ρ) as ε ↓ 0 . Then the following assertions hold.
(a) This convergence is uniform on each segment [c, d] ⊂ (a, b) , and for each ε , P ε
is absolutely continuous on all segment [c, d] ⊂ (a, b) . Moreover, Ṗ ε are bounded
uniformly in ε a.e. on all segment [c, d] ⊂ [a, b) such that c > 0 .
(b) For all sequence εn ↓ 0 , Ṗ εn → Ṗ 0 a.e. on (a, b) ; the convergence take place
everywhere on (a, b) in case P εn , P 0 ∈ C 1 (a, b) .
(c) Let P ε be increasing and
Ξε± (ρ)
Z bq
q
dr
ε
:= const ∓
Ṗ (r)
± Ṗ ε (ρ).
r
ρ
(35)
Then Ξε± (·) are a.e. defined monotone functions on (a, b) , so that [Ξε± ]−1 are well
defined in the graph sense.
Z bq
Z bq
dr
dr
ε
(d) For P increasing, one has
Ṗ εn (r)
→
Ṗ 0 (r)
for all sequence εn ↓
r
r
a
a
0 , uniformly in ρ ∈ [c, d] for all [c, d] ⊂ [a, b] such that c > 0 .
(e) With the notation of (c), for all sequence εn ↓ 0 one has [Ξε+n ]−1 (ξ) → [Ξ0+ ]−1 (ξ) for
all ξ such that [Ξ0+ ]−1 is continuous at the point ξ . The same holds with [Ξε−n ]−1 ,
[Ξ0− ]−1 in the place of [Ξε+n ]−1 , [Ξ0+ ]−1 .
74
Riemann Problem for Gas Dynamics in Eulerian Coordinates
Analogous properties hold if P ε are defined on [b, a] ⊂ IR+ \ {0} and F ε = T −1 P ε are
convex on (1/a, 1/b) .
Proof: Since F ε = T −1 P ε are concave, they are differentiable a.e. Moreover, the convergence of P ε (ρ) to P 0 (ρ) implies that F ε (1/ρ) → F ε (1/ρ) as ε ↓ 0 . Therefore F ε → F 0
uniformly on all segment [1/d, 1/c] ⊂ (1/b, 1/a) , and d/dV F εn → d/dV F 0 a.e. on
(1/b, 1/a) as ε ↓ 0 . Since Ṗ ε (ρ) = 1/ρ2 d/dV F ε (V ) whenever d/dV F ε (V ) exists, (a)
and (b) are evident.
Further, d/dV F ε ≥ 0 in case (c). Substituting v = 1/r in the integral in (35), we
obtain
!
r
r
Z 1/ρ r
d ε
d ε
d ε
ε
Ξ± (ρ) = const ±
F (1/ρ) −
F (v) dv ± 1/b
F (1/ρ),
dV
dV
dV
1/b
which is monotone because d/dV F ε is monotone; hence (c). Besides, (d) follows from the
continuity and convergence of F ε (·) at V = 1/b . Indeed, one has for ρ > c > 0
r
Z bq
Z b q
Z 1/ρ r d
dr
dr
d 0 Ṗ εn (r)
Ṗ 0 (r) ≤
(36)
−
F εn (v) −
F (v) dv.
r
r
dV
dV
ρ
ρ
1/b
p
p
Take δ > 0 and integrate | d/dV F εn (V )− d/dV F 0 (V ) | separately over (1/b, 1/b+δ)
and (1/b + δ, 1/c) . For all δ > 0 , the second integral vanishes as εn ↓ 0 , by (a) and the
Lebesgue theorem. Besides, the first one can be made as small as desired by choosing δ small
enough, because one has
Z 1/b r
Z 1/b
d εn
d εn
F (v) dv ≤
(1 +
F (v)) dv = δ + (F εn (1/b + δ) − F εn (1/b)) ≤
dV
dV
1/b+δ
1/b+δ
≤ 2δ + (F 0 (1/b + δ) − F 0 (1/b))
for εn sufficiently small. Hence the left-hand side of (36) can be made as small as desired
uniformly in ρ ∈ [c, d] . Moreover, if a > 0 , we can take c = a in the reasoning above and
prove (d).
Finally, (b) and (d) imply that Ξε±n → Ξ0± a.e. on (a, b) . It is classical in the basic
probability theory that the a.e. convergence of monotone functions (interpreted as random
variables) implies the pointwise convergence of their graph inverse functions (interpreted as
their distribution functions) at the points of continuity of the limit (e.g., cf. [Sv]). Thus (e)
follows.
The case of convex F ε is similar.
⋄
Proof of Proposition 3: The case a = b is trivial; for definiteness, assume a 0 we could prove the existence directly by penalisation of the right-hand side of (34),
applying Lemmae 2,4 and following the corresponding proof in Chapter 1.II. Instead, we will
1.III.2.
Existence and uniqueness of viscous approximations
75
perform the transformation T −1 : [ρ ∈ (a, b) 7→ Π(ρ)] 7→ [V ∈ (1/b, 1/a) 7→ −Π(1/V )] . It
reduces the equation in (34) to the equation
d2
1 2ε d/dV Φ(V )
Φ(V ) =
2
dV
V f (V ) − Φ(V )
(37)
with d/dV Φ > 0 and Φ > f on (1/b, 1/a) , where Φ = T −1 Π and f = T −1 p . This
equation only differs from the one that appears in Chapter 1.II by the factor 1/V in the
right-hand side. This factor is continuous and bounded on (1/b, 1/a) since a > 0 , so
that the proof in Chapter 1.II applies without any further modification. Hence there exists
a strictly increasing concave solution Φ ∈ C[1/b, 1/a] ∩ C 2 (1/b, 1/a) to the problem (37),
Φ(1/a) = f (1/a) , Φ(1/b) = f (1/b) . Therefore Π = T Φ ∈ C[a, b] ∩ C 2 (a, b) and Π
verifies (34).
For a = 0 , let first find, for all δ ∈ (0, b) a function Πδ ∈ C[0, b] ∩ C 2 (δ, b) such that

2εΠ̇δ

ρΠ̈δ + 2Π̇δ =
with Π̇δ > 0, p − Πδ > 0 on [δ, b);
p − Πδ

Πδ (b) = p(b), Πδ |[0,δ] = −kε.
The proof of existence goes on as above. By Lemma 2(b), there exists a function Π on
[0, b] such that Πδ ↑ Π as δ ↓ 0 . Furthermore, applying one more time the same proof
from Chapter 1.II, this time to the functions T −1 Πδ , we infer that T −1 Π ∈ C 2 (1/d, 1/c)
and T −1 Π satisfies (37) on (1/d, 1/c) for all segment [1/d, 1/c] ⊂ (1/b, +∞) . Thus
Π ∈ C 2 (0, b) , and the equation in (34) holds. Besides, the continuity of Π at 0 and b
follows from Lemma 2(a) by comparison with special solutions of the equation in (34), as in
Chapter 1.II. Thus Π(0) = −kε and Π(b) = p(b) by the construction of Πδ .
⋄
Proof of Proposition 4: Let us adapt the proof from Chapters 1.I,1.II. Let a 0 and construct a barrier function
Υα ∈ C 2 (a, b) ∩ C[a, b] such that ρΫα + 2Υ̇α ≥ m(α) > 0 and α/2 ≤ P − Υα < α
on (a, b) . Then apply Lemma 2(a) to Πε and Υα . By Lemma 3(a) 1/ρ Π̇ε is uniformly
bounded on all segment [c, d] ⊂ (a, b) . It follows that P ≥ Πε ≥ Υα on [a, b] for all ε
sufficiently small.
⋄
Finally, note the following property of solutions of (34).
Lemma 4 Let Π(·; σ) verify (34) with a = 0 , k = −σ ∈ [0, 1) . Then Π̇(ρ; σ) → 0 as
Z bq
dr
ρ → 0 , and the integral S(σ) =
Π̇(r; σ)
is finite. In addition, S(σ) ↑ +∞ as
r
0
σ ↓ −1 .
Proof: Clearly, there is no problem in convergence of the integral on the upper limit. Let
prove that S(σ) converge at the lower limit. Set κ = 1 − (1 − k)/2 ∈ [1/2, 1) . Since Π, p
are continuous on [0, b] and p(0) − Π(0; σ) = kε , there exists δ > 0 such that p − Π ≤ κε
76
Riemann Problem for Gas Dynamics in Eulerian Coordinates
Z δ
2
ε
on (0, δ) . By (34), one has on (0, δ) Π̇(δ; σ) ≥ Π̇(ρ; σ)+
− 1 Π̇(r; σ) dr .
κε
r
ρ
−2(1/κ−1)
Hence Π̇(ρ; σ) ≥ const ρ
by the Gronwall inequality, so that the first two assertions
are evident.
Now consider the function Π̇(·; −1) . If S(−1) diverge, then limσ↓−1 S(σ) = +∞ ,
by Lemma 2(b) and the Levi theorem. Assume that S(−1) < +∞ ; we will arrive to a
contradiction. Indeed, there are two cases.
a) lim supρ↓0 Π̇(ρ; σ) = 0 . Proceeding as in Proposition 2, we can therefore define ξ+ :=
q
Z ξ+ q
dr
lim
Π̇+ (r; −1)
+
Π̇+ (ρ; −1) = S(−1) . Introduce the C 1 functions r, u :
ρ↓0
r
ρ
ξ ∈ [ξ+ , +∞) 7→ ρ(ξ), u(ξ) by formulae (26),(27) and rewrite (34) for ξ, ρ, u . This yields
ρ(ξ+ ) = 0 , ξ+ − u(ξ+ ) = 0 , u′(ξ+ + 0) = 1 and
Z ξ
′
′
ε(u (ξ) − u (ξ+ + 0)) = −
(ζ − u(ζ))ρ(ζ)u′(ζ) dζ + p(ρ(ξ)).
ξ+
for all ξ > ξ+ . As it is shown in part III of the proof of Lemma 1, these properties are
incompartible.
b) lim supρ↓0 Π̇(ρ; σ) = l2 > 0 . Take κ ∈ (0, 1) . There exists δ0 ∈ (0, b) and σ0 ∈
(−1/2, −1) such that p(ρ) − Π(ρ; σ0 ) ≥ κε for all ρ ∈ [0, δ0 ] . Take δ1 ∈ (0, δ0 ] such
that Π̇(δ1 ; −1) ≥ l2 /2 . Note that, by Lemma 2(a), Π(·; σ) ↓ Π(·; −1) as σ ↓ −1 .
By Lemma 3(b), we also have Π̇(δ1 ; σ) → Π̇(δ1 ; −1) as σ ↓ −1 . Therefore there exists
σ1 ∈ [σ0 , −1) such that Π̇(δ1 ; σ1 ) ≥ l2 /4 ; in addition, p(ρ) − Π(ρ; σ1 ) ≥ κε for all
Z δ1 2
ε
− 1 Π̇(r; σ1 ) dr for all ρ ∈
ρ ∈ [0; δ1 ] . Now (34) yields Π̇(δ1 ; σ1 ) ≤ Π̇(ρ; σ1 ) +
κε
r
ρ
[0, δ1 ] . Applying the Gronwall inequality and performing easy calculations, we get S(σ1 ) ≥
Z δ1 q
dr
l
δ 1−1/κ
l κ
Π̇(r; σ1 )
≥ δ 1/κ−1
=
. Letting κ ↓ 1 , we conclude that S(r
2
1/κ − 1
21−κ
0
1) = limσ↓−1 S(σ) = +∞ , which contradicts our assumption.
⋄
Z ρ± q
dr
For σ ∈ (−1, +∞) denote by S±ε (σ) the integrals
Π̇ε± (r; σ)
, where Πε± (·; σ)
r
+
(σ)
denotes the unique solution of (22)-(24), according to Proposition 3. By Lemma 4, S±ε (σ)
are finite for σ ∈ (−1, 0] ; evidently, it is also true in case σ ∈ (0, +∞) .
The next step consists in varying σ in order to satisfy the condition (25), i.e.,
u+ − u− = S+ε (σ) + S−ε (σ),
which is now shown to be meaningful. We will prove that, for ρ± and ε fixed, (25) establishes
a bijection between σ ∈ (−1, +∞) and u+ − u− ∈ IR , provided p(·) satisfies (9). For
ρ0 = (σ)+ ∈ [0, +∞) , denote by P± (·; ρ0 ) the (-1)-hulls of p(·) on I(ρ0 , ρ± ) , respectively.
It will be convenient to extend Πε± (·; σ) , P± (·; ρ0 ) to continuous functions on IR+ , by
setting them constant on each component of IR+ \ I(ρ0 , ρ± ) . As in Chapter 1.II, we have
the following two results.
1.III.2.
Existence and uniqueness of viscous approximations
77
Lemma 5 With the notation above and σ ∈ (−1, +∞), ρ0 = (σ)+ , the following holds.
(a) For all ρ ∈ IR+ and ε > 0 , σ 7→ Πε± (ρ; σ) do not decrease; nor do ρ0 7→ P± (ρ; ρ0 ) .
(b) For all ρ ∈ IR+ and ε > 0 , σ 7→ sign(ρ± − ρ0 )Π̇ε± (ρ; σ) do not increase; nor do
σ 7→ sign(ρ± −ρ0 )Ṗ± (ρ; ρ0 ) .
(c) For all ε > 0 the maps σ 7→ Πε± (·; σ) are continuous for the L∞ (IR+ ) topology; so
do σ 7→ P± (·; ρ0 ) .
ε
(d) For all ε > 0 , the functions
Z ρ± q σ 7→ S± (σ) are continuous and strictly decreasing; so do
dr
the functions ρ0 7→
.
Ṗ± (r; ρ0 )
r
ρ0
Proof: The properties (a)-(c) for ε > 0 follow from Lemma 2. Hence (a)-(c) for P±
follow by Proposition 4 and Lemma 3(b). Now (b) and the Levi theorem yield (d).
⋄
Lemma 6 Let ρ± , u± be fixed.
(a) Assume (9)(a) holds. Then for all ε > 0 one has S±ε (σ) → −∞ as σ → +∞ .
(b) Assume (9)(b) holds. Then for all L > 0 there exists ε0 (L) > 0 such that for all
0 < ε < ε0 (L) one has limσ→+∞ S±ε (σ) < −L .
Z ρ± q
dr
(c) In both cases, one has
Ṗ± (r; ρ0 )
→ −∞ as ρ0 → +∞ .
r
ρ0
Proof: The proof is much the same as the one of Proposition 4 in Chapter 1.II.
(a) Assume, for instance, that S + (σ) is bounded from below by some constant −M ∈
IR− . Set V0 := 1/(σ)+ , V+ := 1/ρ+ > V0 , and perform the transformation T −1 p = f ,
T −1 Πε+ (·; σ) = Φ(·; V0 ) . We have Φ(·; V0 ) which is convex on [V0 , V+ ] , Φ(V0 ; V0 ) =
f (V0 ) , Φ(V+ ; V0 ) = f (V+ ) , and Φ(·; V0 ) satisfies the equation (37) on (V0 , V+ ) . Note that
f (V0 ) → −∞ as V0 → −∞ , by (9)(a). Therefore the convexity of Φ(·; V0 ) implies that
p
Φ(V+ /2; V0 ) → −∞ as V0 → 0 ; on the other hand, it yields (V+ /2) d/dV Φ(V+ /2; V0) ≤
−S+ε (σ) ≤ M independently of V0 ∈ (0, V+ /2) . Hence we have a family of functions
{Φ(·; V0 )}V0 ∈(0,V+ /2) such that Φ(V+ /2; V0 ) ↓ −∞ as V0 ↓ 0 , Φ̇(V+ /2; V0 ) ≤ (2M/V+ )2
uniformly in V0 , Φ(V+ ; V0 ) = f (V+ ) , and Φ(·; V0 ) satisfy (37) on (V+ /2, V+ ) . Let us
compare Φ(·; V0 ) to the solution Ψ of the Cauchy problem Ψ(V+ /2) = Φ(V+ /2; V0) ,
Ψ̇(V+ /2) = (2M/V+ )2 + 1 for (37). By the maximum principle Lemma 2(b), which is valid
for the equation (37), we arrive to the conclusion that Φ(V+ ; V0 ) < Ψ(V+ ) , while this last
value can be made well-defined and strictly less than f (V+ ) for V0 small enough. This
contradiction proves (a).
For the proof of (c) in this case it is sufficient to pass to F± (·; V0 ) = T −1 P± (·; ρ0 ) and
Z ρ± q
Z V± p
dr
note that
Ṗ± (r; ρ0 )
=−
d/dV F± (v; V0 ) dv . The last integrals diverge as
r
ρ0
V0
78
Riemann Problem for Gas Dynamics in Eulerian Coordinates
V ↓ 0 ; it is easy to see because F± (·; V0 ) are convex, F± (V0 ; V0 ) = f (V0 ) → −∞ and
p0
d/dV F± (V ; V0 ) ≥ min{1, d/dV F± (V ; V0 )} .
(b) First assume that f = T −1 p has a finite number of points of inflexion. By Definition 1,
the function F+ (·; V0) = T −1 P+ (·; ρ0 ) is the convex hull of f (·) on [V0 , V+ ] = [1/ρ0 , 1/ρ+ ] .
Its graph consists of a finite number of fragments of the graph of f (·) and of chords of this
graph. It follows easily by the Cauchy-Schwarz inequality that
Z ρ+ q
Z V+ p
dr
Ṗ+ (r; ρ0 )
d/dV F+ (v; V0 ) dv ≤
=−
r
ρ0
V0 Z
Z ρ+ p
V+ p
dρ
≤−
d/dV f (v) dv =
ṗ(r)
→ −∞
ρ
V0
ρ0
Z ρ+ q
dr
→ −∞
as ρ0 → +∞ , by (9)(b). It follows by a density argument that
Ṗ+ (r; ρ0 )
r
ρ0
as ρ0 → +∞ for a general flux function p(·) , which ends the proof of (c). Besides, the
claim of (b) follows now by Proposition 4 and Lemma 3(d).
⋄
Combining Propositions 1,2 and 3 and Lemmae 4,5,6, one easily deduces the main result
of this section.
Theorem 1 Let p(·) be continuous and strictly increasing on IR+ .
(a) Assume (9)(a) holds. Then for all ρ± > 0 , u± ∈ IR , ε > 0 there exists a unique
solution to the problem (1),(2) in the sense of Definition 2.
(b) Assume (9)(b) holds. Then for all ρ± > 0 , u± ∈ IR there exists ε0 = ε0 (ρ± , u+ −u− )
such that for all 0 < ε < ε0 there exists a unique solution to the problem (1),(2) in
the sense of Definition 2.
Remark 2 At the present stage, it is not clear to the author whether the condition (9)(b)
suffices to have the existence of solutions to (1),(2) for all ε > 0 . On the other hand, it is
easy to see that if p(·) is locally absolutely continuous but fails to satisfy (9)(b), there is no
existence of a bounded self-similar solution to (1),(2) for arbitrary Riemann data. The exact
bound on u+ − u− in order to have the existence for ε in some non-empty neigbourhood of
0 is
Z ρ0 q
Z ρ0 q
dr
dr
−(u+ − u− ) < lim
Ṗ+ (r; ρ0 )
+ lim
Ṗ− (r; ρ0 ) ,
(38)
ρ0 →+∞ ρ
ρ
→+∞
r
r
0
ρ−
+
where P± (·; ρ0 ) are the (-1)-hulls of p(·) on [ρ+ , ρ0 ] and [ρ− , ρ0 ] , respectively. This
assertion follows from the extremal property of the upper (-1)-hull P (·) on [a, b] ⊂ (0, +∞) ,
Z bq
dr
which in fact maximizes the integral
Q̇(r)
with respect to all Q(·) that satisfy the
r
a
assumptions (7)(i),(ii) in Definition 1.
1.III.3.
3
Admissible solution of the problem (8),(2)
79
The admissible solution of the problem (8),(2)
In this section we study the global (with respect to data ρ± , u± ) solvability for the system
(8),(2) subject to the wave fan admissibility criterion (cf. [D74, D89, Tz96]).
Definition 3 Let (ρ, u) be a pair of functions such that ρ ∈ L∞ (IR+ × IR; IR+ ) and
u ∈ L∞ ({ρ > 0}; IR) , where {ρ > 0} := {(t, x) ∈ IR+ × IR | ρ(t, x) > 0} , ρ being an a.e.
defined representative of ρ . Then (ρ, u) is a wave-fan admissible solution of the problem
(8),(2), if
(i) the equations ρt + qx = 0 , qt + (e + p(ρ))x = 0 are fulfilled in D ′ (IR+ × IR) , where
q = ρu, e = ρu2 on the set {ρ > 0} and q = 0, e = 0 on its complementary;
(ii) one has
ess lim
t↓0
kρ(t, ·) − ρ(0, ·)kL1(−R,R) + ku(t, ·) − u(0, ·)kL1(−R,R) = 0
for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2);
(iii) in addition, there exists a sequence {εn }n∈IN ⊂ (0, +∞) , εn → 0 as n → ∞ , such
that (ρεn , uεn ) tends to (ρ, u) as n → ∞ , where (ρεn , uεn ) is a solution of (1),(2)
in the sense of Definition 2; more exactly, this means that ρεn → ρ , ρεn uεn → q and
ρεn (uεn )2 → e a.e. on IR+ × IR .
Note that, according to (i) of Definition 3, u remains undefined within the vaccuum state.
While (i),(ii) define a weak solution, (iii) presents an additional selection criterion which is
stronger than (i) unless p(·) degenerates to p(ρ) = const/ρ on some interval of IR+
depending on ρ± , u± . If such a degeneration does not occur, an infinity of weak solutions
that are not admissible can be observed (cf. [KrSt93]).
Due to (iii), a wave-fan admissible solution is actually self-similar. According to formulae
(40)-(45) and Theorem 2 below, it is unique and has the usual wave-fan structure. The
solution contains at most three “main” constant states: (ρ− , u− ) at −∞ ; the vaccuum
state ρ ≡ 0 or a constant state in a neighbourhood of the unique point ξ0 such that there
exists limξ→ξ0 u(ξ) = ξ0 (in case there is no vaccuum state); and (ρ+ , u+ ) at +∞ . These
states are separated by two wave fans of the first and the second family, respectively. Each
wave fan is a sequence of shocks, rarefactions, contact discontinuities and (in case p(·) is
not smooth) “rarefaction type” constant states. There is no vaccuum state in the solution
unless the intermediate “main” state is the one. In this case, u(ξ) − ξ → 0 as ξ enters
the vaccuum state from any side (cf. (41),(43) and (48) below). The necessary and sufficient
condition for vaccuum state to appear is that
u+ − u− ≥
Z
ρ+
0
Z ρ− q
q
dr
dr
Ṗ+ (r; 0)
+
Ṗ− (r; 0) ,
r
r
0
(39)
80
Riemann Problem for Gas Dynamics in Eulerian Coordinates
where P± (·; 0) are the (-1)-hulls defined below. All these properties can be observed from
the following formulae for the wave-fan admissible solution, suggested by Propositions 2 and
4:

(

[Ξ− ]−1 (x/t), x/t < ξ−
−1

[Ξ− ] (x/t), x/t < ξ+
≡
ρ(t, x) = ρ(x/t) =
ρ0 ,
ξ− < x/t < ξ+ (40)

[Ξ+ ]−1 (x/t), ξ− < x/t

−1
[Ξ+ ] (x/t), ξ+ < x/t,
where

U− ◦ [Ξ− ]−1 (x/t),



 U (ρ ) = U (ρ ),
− 0
+ 0
u(t, x) = u(x/t) =




U+ ◦ [Ξ+ ]−1 (x/t),
U± (ρ) := u± ∓
Z
ρ±
ρ
Ξ± (ρ) := U± (ρ) ±
q
q
Ṗ± (r; ρ0 )
Ṗ± (ρ; ρ0 )
dr
r
x/t < ξ−
ξ− < x/t < ξ+
(in case ρ0 > 0)
ξ+ < x/t,
(41)
for ρ ∈ I(ρ0 , ρ± ),
(42)
for ρ ∈ I(ρ0 , ρ± ),
(43)
ξ± are defined by
ξ± :=
lim
ρ∈I(ρ0 ,ρ± ), ρ→ρ0
Ξ± (ρ)
if
ρ0 6= ρ± ;


ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+ , 
(44)
ρ0 = 0 in case (39) holds, and ρ0 ∈ (0, +∞) is the unique value that satisfies the relation
Z ρ+ q
Z ρ− q
dr
dr
u+ − u− =
Ṗ+ (r; ρ0 )
+
Ṗ− (r; ρ0 )
(45)
r
r
ρ0
ρ0
in case (39) fails; finally, P± (·; ρ0 ) are the (-1)-hulls of the graph of p(·) on I(ρ0 , ρ± ) ,
respectively, as defined in Definition 1.
Let state the main result of this chapter.
Theorem 2 Assume that continuous strictly increasing function p(·) satisfies (9). Let ρ± >
0 , u± ∈ IR . Then the solution (ρε , uε ) to the problem (8),(2) (which exists and is unique, at
least for ε sufficiently small) tends as ε ↓ 0 , in the sense of Definition 3(iii), to (ρ, u) given
by the formulae (40)-(45). The pair (ρ, u) is the unique solution of the Riemann problem
(8),(2) in the sense of Definition 3.
Proof: According to Theorem 1, there exists a unique solution to (1),(2), at least for
ε sufficiently small; let denote this solution by (ρε , uε ) . By Propositions 1 and 2 (ρε , uε )
corresponds to some σ ε ∈ (−1, +∞) and Πε± (·; σ ε ) such that (22)-(25) and (26)-(30)
hold. Consider the set {(σ ε )+ } ⊂ IR+ ; it has an accumulation point ρ0 in IR+ . Choose
1.III.3.
Admissible solution of the problem (8),(2)
81
a sequence εn ↓ 0 such that σ εn → σ 0 as n → ∞ ; let omit the subscript n , because
we will prove later that σ 0 does not depend on any particular sequence of ε ↓ 0 . Consider
separately the three possibilities: ρ0 ∈ (0, +∞) , ρ0 = +∞ and ρ0 = 0 .
a) ρ0 ∈ (0, +∞) . As in Chapter 1.II, from Proposition 4 and Lemma 5(a),(c) it follows
that
Πε± (·; σ ε ) → P± (·; ρ0 ) in L∞ (IR+ ) as ε → 0,
(46)
where Πε± , P± are extended to IR+ as in Lemma 5. Similarly, extend U±ε , Uε in (28)
and (42) by constants to continuous functions on IR+ . By Lemma 3(d), U±ε (·) converge to
U± (·) uniformly on I(ρ0 , ρ± ) , respectively. Since (25) and (45) write also as U−ε ((σ ε )+ ) =
U+ε ((σ ε )+ ) and U− (ρ0 ) = U+ (ρ0 ) , one deduces (45). It follows by Lemma 5 that ρ0 is
uniquely determined by p(·) , ρ± and u± in case a).
Further, by Lemma 3(c) and (43),(44) one has Ξ± (ρ1 ) ≤ ξ− ≤ U− (ρ0 ) = U+ (ρ0 ) ≤
ξ+ ≤ Ξ+ (ρ2 ) for all ρ1 ∈ I(ρ0 , ρ− ) , ρ2 ∈ I(ρ0 , ρ+ ) . Considering separately the cases
ρ0 > ρ+ , ρ0 = ρ+ , ρ0 < ρ+ , one deduces by Lemma 3(e) from (46),(26) and (40) that
ρε (·) → ρ(·) a.e. on (ξ− , +∞) . Similarly, one gets ρε (·) → ρ(·) a.e. on (−∞, ξ+ ) , so
that the convergence actually takes place for a.a. ξ ∈ IR . Consequently, uε (·) → uε (·) a.e.
on IR .
b) ρ0 = +∞ . Actually, this case is impossible. Indeed, for all L > 0 we have
Z ρ+ q
Z ρ− q
dr
dr
ε
u+ − u− =
Π̇+ (r; σ ε )
+
Π̇ε− (r; σ ε )
≥
r
r
ε
σεZ ρ q
σZ
q
ρ−
+
dr
dr
≥
Π̇ε+ (r; L)
+
Π̇ε− (r; L)
r
r
L
L
for ε small enough, by Lemma 5(d). Passing to the limit as ε → 0 , we obtain
Z ρ− q
Z ρ+ q
dr
dr
Ṗ+ (r; L)
+
Ṗ− (r; L)
u+ − u− ≥
r
r
L
L
by Proposition 4 and Lemma 3(d). By Lemma 6(c), this last quantity tends to −∞ as
L → +∞ , provided (9) holds. Thus u+ − u− = −∞ , which is contradictory.
c) ρ0 = 0 . As in case a), one has (46). Therefore Π̇ε± (·; σ ε ) → Ṗ± (·; 0) a.e. on (0, ρ± ) ,
respectively, by Lemma 3(b). It follows by the Fatou Lemma that
Z ρ− q
Z ρ+ q
dr
dr ε
ε
u+ − u− = lim inf
Π̇+ (r; σ )
+
Π̇ε− (r; σ ε )
≥
ε→0
r
r
0 Z ρ q
0 Z ρ q
(47)
+
−
dr
dr
ε
ε
≥
Π̇+ (r; 0)
+
Π̇− (r; 0) .
r
r
0
0
Note that, since c) is excluded and since we have seen that in case a) this last quantity is
necessarily greater than u+ − u− , the accumulation point ρ0 is always unique and finite.
Let prove that ρε → ρ , ρε uε → ρu and ρε (uε )2 → ρu2 on the set {ξ | ρ(ξ) > 0} , and
that ρε → 0 , ρε uε → 0 and ρε (uε )2 → 0 on its complementary.
82
Riemann Problem for Gas Dynamics in Eulerian Coordinates
First assume (47) holds with equality. Then we still have U− (0) = U+ (0) . Moreover,
Ṗ (ρ; 0) → 0 as ρ → 0
(48)
whenever vaccuum
Indeed,
Z ρ±appears.
q
q one can show as in the proof of Lemma 3(c) that
dr
Ξ± (ρ) = u± ∓
Ṗ± (r; 0) ± Ṗ± (ρ; 0)
are monotone on (0, ρ± ) ; on the other
r
Z ρ± q ρ
dr
hand,
Ṗ± (r; 0)
are also monotone and converge as ρ ↓ 0 . Therefore there exist
r
ρ
q
limits as ρ ↓ 0 of
Ṗ± (ρ; 0) , which are necessarily zero. Thus Ξ− (ρ1 ) ≤ ξ− = U− (0) =
U+ (0) = ξ+ ≤ Ξ+ (ρ2 ) for all ρ1 ∈ [0, ρ− ] , ρ2 ∈ [0, ρ+ ] ; moreover, ρ(ξ) > 0 for all
ξ 6= ξ− = ξ+ . The convergence of ρε , uε to ρ, u , respectively, a.e. on IR follows as in case
a).
Secondly, assume that the inequality in (47) is strict. By (48) we have Ξ− (ρ1 ) ≤ ξ− =
U− (0) < U+ (0) = ξ+ ≤ Ξ+ (ρ2 ) , with ρ1 , ρ2 as above. One more time, we deduce
the convergence of ρε , uε to ρ, u , respectively, a.e. on (−∞, ξ− ) ∪ (ξ+ , +∞) . Note
that Ξ± (ρ) are strictly monotone at ρ = 0 , since P± (·; 0) are strictly increasing and
Ṗ± (+0; 0) = 0 . It follows that ρ(ξ) → 0 as ξ ↑ ξ− or ξ ↓ ξ+ . For ε sufficiently small, ρε
has no points of maximum on IR , so that for all δ > 0 we infer maxξ∈[ξ− −δ,ξ+ +δ] ρε (ξ) =
max{ρε (ξ− − δ), ρε (ξ+ + δ)} . By the last observation, it follows that ρε → 0 uniformly
on [ξ− , ξ+ ] . Besides, uε is nondecreasing on IR for ε sufficiently small, hence uniformly
bounded, so that ρε uε → 0 and ρε (uε )2 → 0 on [ξ− , ξ+ ] .
Thus we have shown that (ρ, u)(·, ·) satisfies Definition 3(i),(iii). Besides, (ρ, u)(±∞) =
(ρ± , u± ) by (40)-(42). Since, in addition, ρ(·) , u(·) are both monotone at ±∞ , Definition 3(ii) holds as well.
⋄
Remark 3 In the equations of isentropic gas dynamics u represents the velocity in gas and
has no physical meaning inside the vaccuum state, while the specific impulse q and specific
energy e are, naturally, both zero within vaccuum. Nevertheless, one could ask for a limit
of uε even inside the vaccuum state. In case p ∈ W 1,1 (0, R) for some R > 0 and
limρ↓0 ṗ(ρ) = 0 , we are able to prove that the unique limit of uε inside the vaccuum state
is the identity function
u(ξ) = ξ . This is due to the formula (43) and the uniform in ε
q
convergence of
Π̇ε± (ρ; 0) to 0 as ρ ↓ 0 . In turn, this last property results from (48),
Proposition 4, Lemma 3(b) and the following kind of maximum principle:
max Π̇ε± (ρ; 0) = max{Π̇ε± (δ; 0), sup ṗ(ρ)} for δ
ρ∈[0,δ]
small enough,
ρ∈(0,δ)
which can be easily deduced from the equation (22). In general, we ignore whether the
convergence always take place to the same limit u(ξ) ≡ ξ within the vaccuum state; but we
still observe that u(ξ) coincides with ξ on its boundary.
CHAPTER 1.IV
L1 -Theory of Scalar Conservation Law
with Continuous Flux Function†
Introduction
We consider the Cauchy problem

 ∂u
+ divx φ(u) = g on Q = {(t, x); t ∈ (0, T ), x ∈ IRN }
∂t
 u(0, ·) = f
on IRN
(CP )
where φ : IR 7→ IRN is only assumed to be continuous and (f, g) satisfy

 f = f0 + c with c ∈ IR, f0 ∈ L1 (IRN ) ∩ L∞ (IRN ),
Z T
(1)
 g ∈ L1 (Q), g(t, ·) ∈ L∞ (IRN ) for a.a. t ∈ (0, T ) and
kg(t, ·)k∞dt < ∞.
0
A solution of (CP) will be understood in the sense of the generalized entropy solution
(g.e.s.) as introduced by S.N.Kruzhkov (cf. [K69a, K69b, K70a]). In the case of a locally
Lipschitz continuous flux function φ , there exists a unique bounded g.e.s.; this is actually
true for any (f, g) satisfying

 f ∈ L∞ (IRN ), g ∈ L1loc (Q),
Z T
(2)
N
∞
 g(t, ·) ∈ L (IR ) for a.a. t ∈ (0, T ) and
kg(t, ·)k∞ dt < ∞.
0
For the general continuous flux function
situation
is more delicate. Let consider the
α−1φ the
|u|
u |u|β−1 u
particular case N = 2 , φ(u) =
, β
. It has been shown in [KP90] that if
α
α 6= β , α + β < 1 , then for some f ∈ L∞ (IR2 ) the problem (CP), with g = 0 , has
a one-parameter family of different bounded g.e.s.. On the other hand, it has been shown
†
This chapter will be published in [BABK].
84
Scalar Conservation Law with Continuous Flux Function
in [BK96] that if (f, g) satisfy (1), then for any α, β > 0 there exists a unique bounded
g.e.s. of (CP). In this chapter we shall improve this last result, showing (cf. Theorem 3) that
(CP) has a unique bounded g.e.s. for any (f, g) satisfying (1) according to whether the flux
function φ satisfies
(
There exist orthonormal vectors ξ1 , . . . , ξN −1 such that
(3)
r ∈ IR 7→ ξi · φ(r) ∈ IR is nondecreasing , i = 1, . . . , N − 1.
Actually, while we shall prove another uniqueness result (cf. Theorem 4), we still do not
know whether there is or is not uniqueness of bounded g.e.s. for (f, g) satisfying (1) with
any continuous flux function φ ; but we shall prove that there always exist a maximum and
a minimum bounded g.e.s. of (CP). More precisely, for any continuous flux function φ and
(f, g) satisfying

N
N
∞
N
†
 f = f0 + c with c ∈ IR, f0 ∈ L∞
0 (IR ) = h ∈ L (IR ); λ {|h| > δ} < ∞ ∀δ > 0
Z T
(4)
N
 g ∈ L1loc (Q), g(t, ·) ∈ L∞
(I
R
)
for
a.a.
t
∈
(0,
T
)
and
kg(t,
·)k
dt
<
∞,
∞
0
0
for all c ∈ IR there exist a maximum and minimum bounded g.e.s. of (CP), which coincide
except for a countable set of values of c depending on φ , f0 and g (cf. Theorem 1 and
Proposition 1).
As pointed out in [C72] and [B72], solutions of (CP) for (f, g) satisfying (1) can be
constructed through the nonlinear semigroup theory from the solutions of the equation
u + divx φ(u) = f
on IRN .
(E)
As was done in [BK96], we shall derive for the equation (E) the same properties as for the
Cauchy problem (CP); actually we shall prove (cf. Corollary 1), for φ and c given, that
there is uniqueness of a bounded g.e.s. of (CP) for all (f, g) satisfying (1) if and only if there
is uniqueness of a bounded g.e.s. of (E) for all f = f0 + c with f0 ∈ L1 (IRN ) ∩ L∞ (IRN ) .
Existence of maximum and minimum generalized
entropy solutions
1
Throughout this chapter φ : IR 7→ IRN is a continuous function and we consider the Cauchy
problem (CP) as well as the equation (E). Recall the following definition:
Definition 1 Let f ∈ L1loc (IRN ) . A sub-g.e.s. (generalized entropy subsolution) (respecN
tively, super-g.e.s.) of (E) is a function u ∈ L∞
loc (IR ) satisfying
in D ′ (IRN ) for any k ∈ IR,
α · (u − k) + divx α · (φ(u) − φ(k)) ≤ α · (f − k)
λN denote the N-dimensional Lebesgue measure; {|h| > δ} stands for {x ∈ IRN ; |h(x)| > δ} and
so on.
†
1.IV.1.
Existence of maximum and minimum g.e.s.
85
where α = sign+ (u − k) ‡ (respectively, sign− (u − k) ). A function u is a generalized
entropy solution (g.e.s.) of (E) if it is both sub- and super-g.e.s.
Definition 2 Let f ∈ L1loc (IRN ) and g ∈ L1loc (Q) . A sub-g.e.s. (respectively, super-g.e.s.)
of (CP) is a function u ∈ L∞
loc (Q) satisfying
∂
α · (u − k) + divx α · (φ(u) − φ(k)) ≤ α · g
in D ′ (Q) for any k ∈ IR,
∂t
where α = sign+ (u −k) (respectively, sign− (u −k) ), and (u(t, ·) −f )+ → 0 (respectively,
(u(t, ·) − f )− → 0 ) in L1loc (IRN ) as t → 0 essentially. A function u is a g.e.s. of (CP) if
it is both sub- and super-g.e.s.
The main result is the following theorem.
Theorem 1 Let (f, g) satisfy (4). Then there exist a maximum and a minimum bounded
g.e.s. of (E) and of (CP).
N
More precisely, considering the equation (E) and f = f0 +c with c ∈ IR , f0 ∈ L∞
0 (IR ) ,
we shall prove that there exists a (unique) g.e.s. u ∈ L∞ (IRN ) such that u ≥ u a.e. on
IRN for any sub-g.e.s. u ∈ L∞ (IRN ) of (E).
This g.e.s. u will be obtained as the a.e. pointwise limit of a nonincreasing sequence
{un } , where un is any bounded g.e.s. of (E) corresponding to f = f0 + cn with a sequence
{cn } in IR decreasing to c .
The same corresponding results are valid for (CP) and minimum solutions.
The main new ingredient in the proof of Theorem 1 is the following lemma.
Lemma 1 a) Let u and û be bounded sub- and super-g.e.s., respectively, of (E), corresponding to f and fˆ ∈ L1loc (IRN ) , respectively. Assume that
N
N
λ
x ∈ IR ; u(x) > û(x) < ∞,
(5)
then
Z
+
(u − û) +
Z
{u>û}
(f − fˆ)− ≤
Z
{u≥û}
(f − fˆ)+ ,
(6)
and in particular, if f ≤ fˆ a.e. on {u ≥ û} , then u ≤ û a.e. on IRN .
b) Let u and û be bounded sub- and super-g.e.s., respectively, of (CP), corresponding
to (f, g) and (fˆ, ĝ) ∈ L1loc (IRN ) × L1loc (Q) , respectively. Assume that
N +1
λ
(t, x) ∈ Q; u(t, x) > û(t, x) < ∞,
(7)
We use the notation sign+ for the Heaviside function, i.e. the characteristic function of (0, +∞) , and
sign− (r) = −sign+ (−r) .
‡
86
Scalar Conservation Law with Continuous Flux Function
then for a.a. t ∈ (0, T )
Z
Z Z
t
+
(u(t, ·) − û(t, ·)) +
0 {u>û}
−
(g − ĝ) ≤
Z
(f − fˆ) +
+
Z Z
t
0 {u≥û}
(g − ĝ)+ ,
(8)
and in particular, if f ≤ fˆ a.e. on IRN and g ≤ ĝ a.e. on {u ≥ û} , then u ≤ û a.e. on
Q.
Proof: Applying Lemma 3.1a) in [BK96] (cf. also [K70a, B72, C72, Ba88]), we have in the
case a)
Z
Z
Z
Z
+
−
+
(u − û) ζ +
(f − fˆ) ζ ≤
(f − fˆ) ζ + |φ(u) − φ(û)|χ{u>û} |Dζ|
{u>û}
{u≥û}
for all ζ ≥ 0 , ζ ∈ D(IRN ) . By the assumption (5), |φ(u) − φ(û)|χ{u>û} ∈ L1 (IRN ) , so
that we may let ζ tend to 1 to obtain (6) at the limit and prove a). The proof of b) is
identical using Lemma 3.1b) in [BK96] and (7).
⋄
We also need the following general existence result, partially contained in [B72] and [KH74],
for which we give a complete proof in the Appendix.
Lemma 2 Let f ∈ L∞ (IRN ) (resp., and g ∈ L1loc (Q) satisfying g(t, ·) ∈ L∞ (IRN ) for a.a.
Z T
t ∈ (0, T ) and
kg(t, ·)k∞dt < ∞ ). Then there exists a bounded g.e.s. of (E) (resp.
(CP)).
0
Proof of Theorem 1: Let {cn } be a sequence in IR decreasing to c and, for n ∈ IN ,
un be a bounded g.e.s of (E) corresponding to fn = f0 + cn . Such a g.e.s. exists by Lemma
2.
m
n
n
Fix n > m . Set h = cn +c
and take 0 < δ < cm −c
; δ = cm −c
− α for some α > 0 .
2
2
2
Using Theorem 2.2a) in [BK96], we have
Z
Z
Z
Z +
+
+
+
(un −um +2δ) ≤ (un −h+δ) + (h+δ−um ) =
(un −cn )−α +
Z Z
Z
− Z
+
−
+
(um −cm )+α ≤ (f0 −α) + (f0 +α) = (|f0 | − α)+ < ∞;
it follows that |{un > um }| < ∞ and thus we deduce from Lemma 1a) that un ≤ um .
Define u = lim un ; this is, clearly, a bounded g.e.s. of (E) corresponding to f = f0 + c .
n→∞
Let now u be a bounded sub-g.e.s. of (E); with the same argument as above, u ≤ un a.e.
for all n and thus u ≤ u a.e. In other words, u is the maximum bounded g.e.s. of (E).
The proof of existence of the maximum bounded g.e.s. of (CP) is similar using Lemma
1b); considering a bounded g.e.s. un of (CP) corresponding to (fn , g) , we only need to
show that
Z + Z − sup
(un (t)−cn )−α +
(um (t)−cm )+α
< ∞.
(9)
t∈[0,T ]
L1 semigroup approach
1.IV.2.
87
Z
+
To prove (9), for M > 0 set κM (t) = inf κ ; (|g(t)| − κ) ≤ M . We have
κM (t) ≤ kg(t)k∞ and thus κM ∈ L1 (0, T ) ; on the other hand, for a.a. t ∈ (0, T ) ,
N
since g(t) ∈ L∞
) , κM (t) decrease to 0 as M increase to ∞ . Thus there exists
0 (IR
Z T
M > 0 such that
κM (t)dt ≤ α/2 . Fix M such that this is satisfied, and set kM (t) =
0
Z t
α/2 +
κM (s)ds for t ∈ [0, T ] . Applying Theorem 2.2b) from [BK96], we get for all
0
t ∈ [0, T ]
Z + Z − Z +
(un (t)−cn )−α +
(um (t)−cm )+α ≤
(un (t)−cn )−kM (t) +
Z − Z + Z tZ +
+
(um (t)−cm )+kM (t) ≤
|f0 |−α/2 +
|g(s)|−κM (s) ds ≤
Z 0
+
≤
|f0 |−α/2 +MT.
The proof of existence of the minimal bounded g.e.s. for (E) and (CP) is identical.
⋄
We actually do not know whether there is in general uniqueness of a bounded g.e.s. of (E)
or (CP). However, we can prove the following result.
N
N
1
∞
Proposition 1 Let f0 ∈ L∞
0 (IR ) (resp., and g ∈ Lloc (Q) , g(t, ·) ∈ L0 (IR ) for a.a.
Z T
t ∈ (0, T ) and
kg(t, ·)k∞dt < ∞ ). Then there exists an at most countable set N in
0
IR such that for all c ∈ IR \ N the equation (E) (resp., the problem (CP)) with f = f0 + c
has a unique bounded g.e.s..
Proof: For c ∈ IR , denote by u(c) (resp., u(c) ) the maximum (resp. minimum) bounded
g.e.s. By the proof above, we know that c 7→ u(c) and c 7→ u(c) are nondecreasing from
IR into L∞ continuous from the right and the left, respectively, for the L1loc topology in
L∞ ; moreover, for c1 < c2 , u(c1 ) ≤ u(c1 ) ≤ u(c2 ) . Thus it follows that u(c) = u(c) a.e.
for any c except an at most countable set in IR .
⋄
2
The L1 semigroup approach
In this section, using the nonlinear semigroup theory in L1 , we make the relation between
the equation (E) and the problem (CP) under the assumption (1) on the data (f, g) clearer.
For simplicity we shall assume c = 0 .
For λ > 0 and f ∈ L1 (IRN ) ∩ L∞ (IRN ) , the equation
u + divx λφ(u) = f
in IRN
(10)
has a maximum bounded g.e.s. that we shall denote by Jλ+ f ; by Corollary 2.1 in [BK96]
Jλ+ f ∈ L1 (IRN ) . In other words, Jλ+ maps L1 (IRN ) ∩ L∞ (IRN ) into itself. Let us start
with the following results.
88
Scalar Conservation Law with Continuous Flux Function
Proposition 2 With the notation above, the following properties hold:
1. for any λ > 0 , Jλ+ is a T-contraction for the L1 -norm, i.e.
Z
(Jλ+ f
−
Jλ+ fˆ)+
≤
Z
(f − fˆ)+
2. {Jλ+ }λ>0 is a resolvent family, i.e.
Jλ+ f
=
Jµ+
µ
λ−µ +
f+
Jλ f
λ
λ
∀f, fˆ ∈ L1 (IRN ) ∩ L∞ (IRN );
∀λ, µ > 0, f ∈ L1 (IRN ) ∩ L∞ (IRN );
3. the range R(Jλ+ ) , independent of λ by 2) , is dense in L1 (IRN ) .
Proof: For Part 1), let f, fˆ ∈ L1 (IRN ) ∩ L∞ (IRN ) and, for δ > 0 , denote by uδ , ûδ
bounded g.e.s. of (E) corresponding to f + δ, fˆ + δ respectively.
As in the Zproof of Theorem
Z
1, we have λN {uδ > û2δ } < ∞ and then, by Lemma 1,
(uδ − û2δ )+ ≤ (f − fˆ − δ)+ ≤
Z
(f − fˆ)+ . At the limit as δ → 0 , uδ → Jλ+ f and û2δ → Jλ+ fˆ a.e., so that we get
Part 1) by the Fatou Lemma.
For Part 2), let f ∈ L1 (IRN ) ∩ L∞ (IRN ) and assume first that λ > µ > 0 . Set
u = Jλ+ f ; it is a bounded g.e.s. of
v + divx µφ(v) =
µ
λ−µ
u
f+
λ
λ
and so u ≤ v = Jµ+ µλ f + λ−µ
u
. Then µλ f + λ−µ
u ≤ µλ f + λ−µ
v and v is a bounded
λ
λ
λ
sub-g.e.s. of u + divx λφ(u) = f . We deduce v ≤ u and thus v = u . To complete the
proof of Part 2), we apply the abstract Lemma 3 below.
For the proof of Part 3), let f ∈ L1 (IRN ) ∩ L∞ (IRN ) and, for λ > 0 , set uλ = Jλ+ f .
We have uλ ∈ R(Jλ+ ) and this set is clearly, by Part 2) , independent of λ . Since kuλk∞ ≤
kf k∞ (see Corollary 2.1 in [BK96]), it follows immediately that uλ → f in D ′ (IRN ) as
λ → 0 ; indeed, being a g.e.s., uλ is also Za solution of (10) in the sense
Z of distributions. Now
using translation invariance and Part 1),
|uλ (x + h) − uλ (x)|dx ≤
|f (x + h) − f (x)|dx ,
so that the set {uλ }λ>0 is relatively compact in L1loc (IRN ) and uλ → f in L1loc (IRN ) . At
last kuλ k1 ≤ kf k1 (cf. Corollary 2.1 in [BK96]) so that uλ → f in L1 (IRN ) ; indeed, for
any compact set K in IRN we have
Z
Z
Z
Z
lim supkuλ − f k1 ≤ lim sup
|uλ − f | + |uλ | −
|uλ| +
|f | ≤
λ→0
K
IRN \K
Z
Z λ→0 Z K
Z
≤ |f | −
|f | +
|f | = 2
|f |,
K
IRN \K
which can be made as small as we want.
IRN \K
⋄
1.IV.2.
L1 semigroup approach
89
Lemma 3 Let X0 be a linear subspace of a Banach space X and {Jλ }λ>0 be a family of
Jλ
non-expansive mappings from X0 into X0 . If the resolvent identity Jλ = Jµ µλ I + λ−µ
λ
holds for all 0 < µ < λ , then it still holds for any λ, µ > 0 .
Proof: Following [BCP], Exercise E8.2, denote by Aλ the multivalued operator from X0
into itself defined by
v ∈ Aλ u
⇔
u, v ∈ X0 , u = Jλ (u + λv);
the graph of this operator is (Jλ f, f −Jλ λ f ); f ∈ X0 and one has (I + λAλ )−1 = Jλ .
Jλ is equivalent to the inclusion
For given λ, µ > 0 , the equality Jλ = Jµ µλ I + λ−µ
λ
Jλ ⊂ Jµ µλ I + λ−µ
J
since these two maps are everywhere defined in the linear space X0 ;
λ
λ
so it is also equivalent to the inclusion Aλ ⊂ Aµ .
By assumption Aλ ⊂ Aµ for 0 < µ < λ . We deduce that for any λ > 0 , Aλ is
an accretive operator; indeed, for µ > 0 small enough ( µ < λ ), (I + µAλ )−1 is a nonexpansive mapping since it is contained in Jµ . Thus, for 0 < λ < µ , (I + µAλ )−1 is a
single-valued operator in X0 containing (I + µAµ )−1 = Jµ , which is everywhere defined in
X0 ; so (I + µAλ )−1 = (I + µAµ )−1 and Aλ = Aµ .
⋄
As we have seen in the proof above, there exists a multivalued operator A+ in L1 (IRN ) ∩
L∞ (IRN ) such that Jλ+ = (I + λA+ )−1 for any λ > 0 . This operator is accretive densely
defined in L1 (IRN ) and R(I + λA+ ) = D(Jλ+ ) = L1 (IRN ) ∩ L∞ (IRN ) is dense in L1 (IRN )
for any λ > 0 . This operator A+ is exactly defined by
v ∈ A+ u ⇔ ∃f ∈ L1 (IRN ) ∩ L∞ (IRN ) such that
u is the maximum bounded g.e.s. of (E) and v = f − u;
it follows that A+ is actually single-valued since for v ∈ A+ u one has v = divx φ(u) in
D ′ (IRN ) . By the Crandall-Liggett theorem (cf. [CL71, B72, C76, BCP]) for any (f, g) ∈
L1 (IRN ) × L1 (Q) there exists a unique mild (or integral) solution u ∈ C([0, T ]; L1 (IRN )) of
du
+ A+ u = g
dt
on (0, T ),
u(0) = f.
(11)
Theorem 2 With the notations above, for (f, g) satisfying (1) with c = 0 , the mild solution
of (11) is the maximum bounded g.e.s. of (CP).
Proof: With the same argument as in [C72] and [B72], it is clear that, under the assumptions
(1), the mild solution u ∈ C([0, T ]; L1 (IRN )) of (11) is in L∞ (Q) and a g.e.s. of (CP).
Therefore u ≤ u a.e. on Q .
Now we prove that u satisfies
Z
Z
Z th
i
+
+
(u(t) − w) ≤ (f − w) +
u(τ ) − w, g(τ ) − A+ w dτ
0
+
(12)
90
Scalar Conservation Law with Continuous Flux Function
for a.a. t ∈ (0, T ) and for all w ∈ D(A+ ) , where for u, f ∈ L1 (IRN )
Z
Z
+
Z
Z
(u + µf ) − u+
h
i
u, f =
f+
f + ≡ inf
.
µ>0
+
µ
{u>0}
{u=0}
Z
h
i
d
Using translation invariance in time, we shall have dt (u(t)−w)+ ≤ u(t)−w, g(t)−A+ w
+
in D ′ ((0, T )) . Applying the results of [BaB92], we shall conclude that u ≤ u a.e. on Q
and this will end the proof.
Let w ∈ D(A+ ) , δ > 0 . By definition w = J1+ h , A+ w = h − J1+ h with some
h ∈ L1 (IRN )∩L∞ (IRN ) . Consider w δ a bounded g.e.s. of w +divx φ(w) = h+δ . Take w δ
as a stationary bounded g.e.s. of the corresponding (CP); since w δ − δ ∈ L∞ (0, T ; L1 (IRN ))
and u ∈ L∞ (0, T ; L1(IRN )) (cf. Corollary 2.1 in [BK96]), we have λN +1 u > w δ < ∞
with the same argument as in the proof of Theorem 1. Thus Lemma 1b) yields
Z +
u(t) − w δ
≤
Z Z + Z t h
i
+
δ
δ
δ
≤
f −w
+
u(τ ) − w , g(τ ) − (h+δ−w ) dτ ≤
f − wδ +
(13)
+
0
Z
Z t Z +
+
1
u(τ ) − w δ + µ(g(τ )−A+ w+w δ −w−δ) −
u(τ ) − w δ
dτ
+
0 µ
δ
for
decreases to w ; moreover for 0 < µ ≤ 1 ,
any µ > 0 . As δ decreases to 0 , w
+
+
u(τ ) − w δ + µ(g(τ )−A+ w+w δ −w−δ)
increases to u(τ ) − w + µ(g(τ ) − A+ w) .
So we may pass to the limit in (13) and obtain
Z +
u(t) − w ≤
Z + Z t 1 Z + Z + +
≤
f −w +
u(τ ) − w + µ(g(τ ) − A w) −
u(τ ) − w
dτ
0 µ
for any 0 < µ ≤ 1 . Letting µ → 0 yields (12).
⋄
Remark 1 Of course, one may consider minimum bounded g.e.s. of (10), define the corresponding operator A− , and prove the following result analogous to Theorem 2:
The mild solution of (11) with A− in place of A+ is exactly the minimum bounded
g.e.s. of (CP).
Corollary 1 For a given continuous flux function φ and c ∈ IR , the following assertions are
equivalent:
(i) for all f = f0 + c with f0 ∈ L1 (IRN ) ∩ L∞ (IRN ) there exists a unique bounded
g.e.s. of (E);
(ii) for all (f, g) satisfying (1) there exists a unique bounded g.e.s. of (CP).
1.IV.3.
Uniqueness results in L1 (IRN ) ∩ L∞ (IRN )
91
Proof: Replacing φ(r) by φ(r + c) , we may assume c = 0 .
If (i) holds, the operators A+ and A− coincide and then by Theorem 2, (see also
Remark 1), for any (f, g) satisfying (1) the maximum and minimum bounded g.e.s. of (CP)
coincide, so that (ii) holds.
Conversely, assume that (ii) holds and for f ∈ L1 (IRN ) ∩ L∞ (IRN ) let u, û be two
bounded g.e.s. of (E). Then u(t) ≡ u is a bounded g.e.s. of (CP) corresponding to (u, g(t) ≡
f −u) and so, by uniqueness, it is the maximum bounded g.e.s. and then, by Theorem 2,
it is the unique mild solution of the corresponding evolution problem (11). In the same way
û(t) ≡ û is the unique mild solution of (11) corresponding to (û, g(t) ≡ f −û) . Then by the
integral inequality (see [B72, BCP, BW94])
Z
Z
h
i
d
− |u − û| = u(t) − û(t), (f − u) − (f − û) ≥
|u(t) − û(t)| = 0,
dt
where [·, ·] stands for the Zbracket associatedZ with the standard norm in L1 , i.e., for all
h
i
N
1
u, f ∈ L (IR ) , u, f =
f sign u +
|f | . It follows that u = û a.e. in IRN
{u6=0}
{u=0}
so that (i) holds.
⋄
Remark 2 For f = f0 + c with f0 ∈ L1 (IRN ) ∩ L∞ (IRN ) , any bounded g.e.s. of (E) is
in c + L1 (IRN ) (cf.
Z Corollary 2.1 in [BK96]); so there is uniqueness of a bounded g.e.s. to
(E) if and only if
u(f ) − u(f ) = 0 , where u(f ) and u(f ) are the maximum and the
minimum bounded g.e.s. of (E), respectively.
By Part 1) of Proposition 2, for given c ∈ IR the map f0 7→ u(f0Z+c)−c is a contraction
1
for the L -norm; the same holds for f0 7→ u(f0 +c)−c so that f0 7→
u(f0 +c)−u(f0 +c)
is continuous for the L1 -topology. It follows that (i) of Corollary 1 is equivalent to the
uniqueness of a bounded g.e.s. of (E) for all f0 in some L1 -dense subset of L1 (IRN ) ∩
L∞ (IRN ) .
Consequently, since the L1 -topology in L1 (IRN ) ∩ L∞ (IRN ) is separable, Proposition 1
can be improved as follows.
Proposition 3 There exists an at most countable set N in IR such that, for all c ∈ IR\N ,
the two properties (i) and (ii) of Corollary 1 hold.
3
Some uniqueness results in L1 (IRN ) ∩ L∞ (IRN )
As noted in the introduction, we still do not know if, for any continuous flux function φ , there
is uniqueness of a bounded g.e.s. to (CP) under assumption (1) or to (E) for all f = f0 + c ,
f0 ∈ L1 (IRN ) ∩ L∞ (IRN ) , c ∈ IR . In this section we shall improve some uniqueness results
shown in [BK96].
92
Scalar Conservation Law with Continuous Flux Function
Theorem 3 Assume there exist orthonormal vectors ξ1 , . . . , ξN −1 and C : IR → [0, +∞)
continuous such that
d
ξi · φ(r) ≤ C(r)
dr
in D ′ (IR) for i = 1, . . . , N − 1.
(14)
Then for any c ∈ IR the two properties of Corollary 1 hold.
We shall need the following lemma.
Lemma 4 Let ξ ∈ IRN , ξ 6= 0 , such that r ∈ IR 7→ ξ · φ(r) is nondecreasing.
Let α ∈ IR
o
N
N
∞
and f ∈ L (IR ) with support contained in H = {x ∈ IR ; ξ · x ≥ α . Assume that one
of the following conditions holds:
a) there exists a unique bounded g.e.s. of (E);
N
b) f ∈ L∞
0 (IR ) .
Then for all bounded g.e.s. u of (E) the support of u is also contained in H .
Proof: a) This is clearly true if φ is locally Lipschitz continuous. Indeed, by the definition
of g.e.s.,
Z
Z
α−ξ·x
α−ξ·x
|u(x)|ρ
ζ(x)dx ≤ |f (x)|ρ
ζ(x)dx+
ε
ε
Z
ξ ′ α−ξ·x
α−ξ·x
+ sign u(x)(φ(u(x)) − φ(0)) − ρ
ζ(x) + ρ
Dζ(x) dx
ε
ε
ε
for ζ ∈ D(IRN ) , ζ ≥ 0 , ρ ∈ C ∞ (IR) with ρ′ ≥ 0 , ρ = 0 on (−∞, 0] , ρ = 1 on
[1, +∞) , and ε ≥ 0 . Since sign u(x)(φ(u(x)) − φ(0)) · ξ ≥ 0 and f (x)ρ α−ξ·x
≡ 0,
ε
using the Lipschitz continuity of φ we get
Z
Z
α−ξ·x
α−ξ·x
|u(x)|ρ
ζ(x)dx ≤ C |u(x)|ρ
|Dζ(x)|dx.
ε
ε
Let ε → 0 and ζ → 1 ; it follows that u = 0 a.e. in IRN \ H (see, for instance, Lemma
1.1 in [BK96]).
For the general case, let φn = φ ∗ ρn , where {ρn } is a sequence of mollifiers, and let un
be the bounded g.e.s. of (E) corresponding to the flux φn . Using the contraction property
and translation invariance, we see that the sequence {un } is relatively compact in L1loc (IRN ) ;
clearly any limit point is a bounded g.e.s. of (E) and then by the uniqueness assumption u is
the limit in L1loc (IRN ) of the sequence {un } . Note that r ∈ IR 7→ ξ · φn (r) is nondecreasing
for all n ; thus by the argument above supp un ⊂ H and the same is true at the limit.
N
b) Let f ∈ L∞
0 (IR ) . By Theorem 1 the equation (E) has a maximum bounded g.e.s.
u , which is the limit of any sequence {un } of bounded g.e.s. of (E) corresponding to
fn = f + cn with cn ↓ 0 . Moreover, by Proposition 1 we may choose cn so that there
is uniqueness of bounded g.e.s. of (E) corresponding to fn . By the first part of Lemma 4,
1.IV.3.
Uniqueness results in L1 (IRN ) ∩ L∞ (IRN )
93
supp (un − cn ) ⊂ H , therefore supp u ⊂ H . Using the same argument for the minimum
bounded g.e.s. u of (E), we see that the conclusion of Lemma 4 still holds.
⋄
Proof of Theorem 3: Replacing φ(r) by φ(r + c) − φ(c) , we may assume c = 0 and
φ(0) = 0 .
Since we are working with bounded solutions, we may also assume that C(r) is constant.
By replacing ξi by −ξi , it is then equivalent to assume instead of (14) that
d
ξi · φ(r) + C ≥ 0
dr
in D ′ (IRN ) for i = 1, . . . , N − 1.
Now we notice that for η ∈ IRN , u is a bounded g.e.s. of (CP) corresponding to
φ, f, g if and only if ũ(t, x) = u(t, x − tη) is a bounded g.e.s. of (CP) corresponding to
φ̃(r) = φ(r) + rη , f˜(x) = f (x) , and g̃(t, x) = g(t, x − tη) . Then, according to Corollary 1,
the conclusion of Theorem 3 holds for φ if and only if it holds for the flux function φ(r)+rη .
Choosing η ∈ IRN such that η · ξi > C for i = 1, . . . , N − 1 , which is always possible since
the vectors are linearly independent, we may assume that
r ∈ IR 7→ ξi · φ(r) ∈ IR
is an increasing homeomorphism for i = 1, . . . , N − 1, (15)
a slightly strengthened version of (3).
Under the assumption (15), we prove the result by induction in the dimension N . The
result is true for N = 1 (see [B72]). Assuming that it is true for N −1 , we prove it for
N ≥ 2 . Changing coordinates, we may assume from (15) that φ(r) = φ1 (r), . . . , φN (r)
with φi (·) increasing homeomorphism from IR to IR for i = 1, . . . , N − 1 . We shall
prove that the equation (E) has a unique bounded g.e.s. for any f ∈ L1 (IRN ) ∩ L∞ (IRN ) .
According to Corollary 1, this will end the proof of the theorem.
So let f ∈ L1 (IRN ) ∩ L∞ (IRN ) and u be a bounded g.e.s. of (E); one has u ∈ L1 (IRN )
′
′
′
(see Corollary 2.1 in [BK96]). Set
x = (x1 , x ) with x = (x2 , . . . , xN ) , w(x1 , x ) =
φ1 (u(x1 , x′ )) , β = φ−1
1 , ψ(r) = φ2 (β(r)), . . . , φN (β(r)) . Suppose that w(x1 + t, ·) →
w(x1 , ·) in L1loc (IRN −1 ) as t → 0 for some x1 ∈ IR ; then for every T > 0 the function
v : (t, x′ ) ∈ Q′ = (0, T ) × IRN −1 7→ w(x1 + t, x′ ) is a bounded g.e.s. of the Cauchy problem
∂v
+ divx′ ψ(v) = g
∂t
on Q′ ,
v(0, ·) = v0 (·) on IRN −1 ,
(16)
where v0 (x′ ) = w(x1 , x′ ) and g(t, x′ ) = f (x1 +t, x′ )−β(w(x1 +t, x′ )) ; g ∈ L1 (Q′ )∩L∞ (Q′ )
since f and β(w) = u are in L1 (IRN ) ∩ L∞ (IRN ) .
According to Remark 2, it suffices to prove the uniqueness of a bounded g.e.s. of (E)
corresponding to compactly supported f . So assume supp f ⊂ H = {x1 ≥ α0 } and suppose
there exist u, û two bounded g.e.s. of (E). By Lemma 4, supp u ⊂ H , supp û ⊂ H . Take
x1 = α < α0 ; consider v(t, x′ ) = φ1 (u(t+α, x′ )) , v̂(t, x′ ) = φ1 (û(t+α, x′ )) . The functions
94
Scalar Conservation Law with Continuous Flux Function
v, v̂ are bounded g.e.s. of (16) corresponding to v0 (·) ≡ 0 , g(t, ·) = f (t+ α, ·) −β(v(t, ·))
and v̂0 (·) ≡ 0 , ĝ(t, ·) = f (t + α, ·) − β(v̂(t, ·)) , respectively.
By the inductive assumption the Cauchy problem (16) has a unique bounded g.e.s., which
is in L1 (IRN −1 ) for a.a. t ∈ (0, T ) ; as in the proof of Corollary 1, it follows that the integral
inequality holds:
Z
Z
h
i
d
|v(t) − v̂(t)| ≤ v(t) − v̂(t), g(t) − ĝ(t) = − |β(v(t)) − β(v̂(t))| ≤ 0.
dt
Z
Z
Hence
|v(t) − v̂(t)| ≤ |v0 − v̂0 | = 0 , so that v = v̂ a.e. in Q′ . Thus u = û a.e. in
⋄
H , which proves the Theorem.
In [B72] it has been proved that for any f ∈ L1 (IRN ) ∩ L∞ (IRN ) there is uniqueness of
u ∈ L1 (IRN ) ∩ L∞ (IRN ) g.e.s. of (E) under the isotropic assumption lim rkφ(r)k
1−1/N = 0 . In the
r→0
next theorem we shall prove the uniqueness under the anisotropic assumption introduced in
[KP90] and [BK96].
Theorem 4 Let c ∈ IR and ω1 , . . . , ωN be moduli of continuity, i.e., increasing sub-additive
continuous functions from [0, δ] into [0, +∞) , δ > 0 , with ωi (0) = 0 , i = 1, . . . , N ;
assume that
lim inf
r→0
1
r N −1
N
Y
i=1
ωi (r) < ∞.
(17)
Assume that there exist orthonormal vectors ξ1 , . . . , ξN such that |ξi ·φ(c+r)−ξi ·φ(c)| ≤
ωi (|r|) for all r ∈ [−δ, δ] , i = 1, . . . , N . Then the two assertions of Corollary 1 hold† .
Proof: We may assume that c = 0 , φ(0) = 0 and φ = (φ1 , . . . , φN ) with |φi (r)| ≤
ωi (|r|) for r ∈ [−δ, δ] , i = 1, . . . , N . Recalling Remark 2 and Corollary
Z
Z1, we only need
to prove for f, u ∈ L1 (IRN ) ∩ L∞ (IRN ) with u g.e.s. of (E) that
†
u=
f . Replacing
In spite of the fact that this result underlines one more time (cf. [B72]) that Hölder continuity of the flux
function at zero simplifies the issue of uniqueness in L1 (IRN ) ∩ L∞ (IRN ) , Proposition 3 hereabove suggests
that the non-uniqueness, if there is any, can have no intrinsic relation with regularity properties of the flux
function. As we see, the appropriate Hölder continuity of the flux function permits to pass to the limit directly
in the estimations of Lemma 3.1 in [BK96], and thus obtain the contraction property. But even in the case
one seems unable to deduce this property, there is uniqueness for a.a. translations φ(· − c) − φ(·) of the flux
function φ(·) , with c ∈ IR . This should be compared to the fact that one can construct functions that have
equally bad Hölder continuity properties at all points of their domain; for exemple, one can use realisations of
a Wiener process. Thus we can suggest that, if the uniqueness in L1 (IRN ) ∩ L∞ (IRN ) is determined by some
regularity property of the flux function, this must be a property that holds on the domain of any continuous
function φ(·) everywhere, except an at most countable set of points. The author ignores what could be the
nature of such property.
On the other hand, note that the assumption of Theorem 3 makes appeal to a global property of the flux
function, which also differs from the “pointwise regularity” point of view suggested by Theorem 4.
1.IV.3.
Uniqueness results in L1 (IRN ) ∩ L∞ (IRN )
95
ωi (r) , φi , f , and u by ωi (Mr)/M , φ(Mr)/M , f /M , and u/M , respectively, we
may assume that kuk∞ ≤ δ .
Clearly, it suffices to show that for all µ > 0, R > 0 there exists a function ζ such that
Z
N
N
0 ≤ ζ ≤ 1 on IR , ζ(x) = 1 for all x ∈ [−R, R] , and (u − f )ζ < µ; (18)
for this we follow the proof of Lemma 1.1 in [BK96].
For r > 0 set λi (r) = ωi (r)/r . If all λi are bounded, then φ is Lipschitz continuous
and the result is well known (see the Introduction). Without loss of generality we may assume
that lim λi (r) = +∞ for i = 1, . . . , l and λi (r) ≤ λ for i = l + 1, . . . , N with some
r→0
l ∈ {1, . . . , N} . Since ωi are sub-additive and positive for r > 0 , ωi (r) ≥ λ0 r for some
λ0 > 0 , so that it is equivalent to assume instead of (17) that lim inf C(r) = C < ∞ , where
r→0
C(r) = rλN −l
l
Q
λi (r) . Note that if l = 1 , then clearly C = 0 .
i=1
For all u bounded g.e.s. of (E), for all ζ ∈ D(IRN ) we have
Z
Z
Z
uζ = φ(u) · Dζ + f ζ.
(19)
Moreover, since f and u are bounded, (19) is also valid for ζ given by
+ |xi |
ζ(x1 , . . . , xN ) =
exp −
−1
R
i
i=1
N
Y
with arbitrary positive Ri . Take ζ corresponding to Ri = λi (ε)/η for i = 1, . . . , l and
Ri = λ/α for i = l + 1, . . . , N ; positive numbers α, η, ε will be chosen later. We note
Z
l
N
Q
22N N −l Y
λi (ε) , and |Di ζ| =
that 0 ≤ ζ ≤ 1 , ζ(x) ≡ 1 on
[−Ri , Ri ] ,
ζ = l N −l λ
ηα
i=1
i=1
ηζ
αζ
χ{|xi |>Ri } for i = 1, . . . , l , |Di ζ| =
χ{|xi |>Ri } for i = l+1, . . . , N .
λi (ε)
λ
Z
N Z
X
From (19) we get (u − f )ζ ≤
ωi (|u|)|Di ζ| .
i=1
Now, by the sub-additivity of ωi , for i = 1, . . . , l we have ωi (r) ≤ rωi (ε)/ε + ωi (ε) =
rλi (ε)+ελi (ε) for all ε > 0 ; for i = l+1, . . . , N we have ωiZ(r) ≤ rλ . Hence by substituting
into the last estimate the expressions above for |Di ζ| and
ζ , we get
Z
Z
Z
Z
l
l
N
X
X
X
(u − f )ζ ≤
η
|u|ζ +
εη
ζ+
α
|u|ζ ≤
{|x
|>R
}
{|x
|>R
}
{|x
|>R
}
i
i
i
i
i
i
i=1
i=1
i=l+1
Z
l Z
l
2N
X
Y
l2
N −l
≤η
|u| + l−1 N −l · ελ
λi (ε) + α(N − l) |u|.
η α
i=1 {|xi |>Ri }
i=1
96
Scalar Conservation Law with Continuous Flux Function
Take µ > 0 , R > 0 . Choose α0 > 0 such that λ/α0 > R and α0 ·(N −l)kuk1 < µ/3 .
1
l22N
Choose η0 > 0 such that l−1 · N −l · 2C < µ/6 ; note that if l = 1 then C = 0
η0
α0
and whatever η0 is good. Finally, since λi (ε) → ∞ as ε → 0 for i = 1, . . . , l and
l
Y
u ∈ L1 (IRN ) , by definition of C there exists ε0 > 0 satisfying ε0 λN −l
λi (ε0 ) < 2C +
α0N −l η0l−1
l22N
µ
·
such that Ri = λi (ε0 )/η0 > R ,
6
holds for ζ constructed with α0 , η0 , ε0 .
l Z
X
i=1
{|xi |>Ri }
i=1
|u| <
µ
. It follows that (18)
3η0
⋄
Remark 3 Introducing ξ1 , . . . , ξN in the condition (17) is not superfluous. Indeed, take
2/3
N = 2 and let φ = u , u/|u|
in some orthonormal basis ξ1 , ξ2 ; here (17) holds.
Changing coordinates by rotation by any angle θ such that θ 6= πk/2, k ∈ ZZ , we see that
condition (17) fails in the new basis.
Appendix: proof of Lemma 2
We give here the complete proof of Lemma 2. More precisely, we shall prove the following
result:
Theorem 5 Let φ : IRN 7→ IR be a continuous function.
a) There exists a map G : L∞ (IRN ) 7→ L∞ (IRN ) satisfying:
(i) for any f ∈ L∞ (IRN ), u = Gf is a g.e.s. of (E);
(ii) G is a T-contraction for the L1 -norm, i.e., for any f, fˆ ∈ L∞ (IRN )
Z + Z +
ˆ
ˆ
Gf − Gf
≤
f −f .
n
o
b) Set X = (f, g) ; (f, g) satisfies (2) ; there exists a map U : X 7→ L∞ (Q)∩
C([0, T ]; L1loc (IRN )) satisfying:
(i) for any (f, g) ∈ X, u = U(f, g) is a g.e.s. of (CP);
(ii) for any (f, g), (fˆ, ĝ) ∈ X, the T-contraction property holds:
Z + Z + Z Z +
sup
U(f, g)(t) − U(fˆ, ĝ)(t) ≤
f − fˆ +
g − ĝ .
t∈[0,T ]
IRN
IRN
Q
Proof of a): First, let f ∈ L1 (IRN ) ∩ L∞ (IRN ) . For ε > 0 , take φε : IR → IRN
Lipschitz continuous functions such that φε converge to φ uniformly on compact sets in
IR , as ε → 0 . It is well-known that there exists a unique solution uε to the equation
uε + divx φε (uε ) = ε∆x uε + f
on IRN ;
moreover, the map Gε : f ∈ L1 (IRN ) ∩ L∞ (IRN ) 7→ uε ∈ L1 (IRN ) ∩ L∞ (IRN ) is a Tcontraction for the L1 -norm, the maximum principle ( kuε k∞ ≤ kf k∞ ) holds and there is
1.IV.4.
Appendix
97
translation invariance in x . Thus the family {Gε f }ε>0 is relatively compact in L1loc (IRN ) .
Take a countable L1 -dense set M in L1 (IRN ) ∩ L∞ (IRN ) ; by the diagonal process, there
exist εn → 0 such that Gεn f → u =: G0 f in L1loc (IRN ) for all f ∈ M . It is clear
that u is a g.e.s. of (E) and G0 : M 7→ L1 (IRN ) ∩ L∞ (IRN ) is a T-contraction for the
L1 -norm. Thus G0 can be extended to the whole of L1 (IRN ) ∩ L∞ (IRN ) so that G0 is a
T-contraction for the L1 -norm, G0 f is a g.e.s. of (E) and the maximum principle holds.
Now for the general case f ∈ L∞ (IRN ) , set fn,m = f + χ{|x|≤n} −f − χ{|x|≤m} ∈ L1 (IRN )∩
L∞ (IRN ) . As n → ∞ , G0 fn,m ↑ um ∈ L∞ (IRN ) ; further, as m → ∞ , um ↓ u =:
Gf
that u is a bounded
g.e.s. of (E); by the Fatou Lemma, it follows that
Z . It is clear
Z +
+
fn,m − fˆn,m
for f, fˆ ∈ L∞ (IRN ) . It is easy to check
Gf − Gfˆ ≤ lim inf
n→∞,m→∞
Z + Z +
ˆ
that lim lim
fn,m − fn,m
=
f − fˆ ∈ [0, +∞] , so that (ii) also holds.
⋄
m→∞ n→∞
h
i h
i
N
N
1
∞
1
∞
Proof of b): First, set X0 = L (IR ) ∩ L (IR ) × L (Q) ∩ L (Q) and let (f, g) ∈
X0 . For ε > 0 , take φε as in the proof of a); there exists a unique solution uε to the
Cauchy problem

ε
 ∂u
+ divx φε (uε ) = ε∆x uε + g on Q
∂t
 uε (0, ·) = f
on IRN ;
moreover, the map Uε : (f, g) ∈ X0 7→ uε ∈ L1 (Q) ∩ L∞ (Q) ∩ C([0, T ]; L1loc (IRN )) satisfies
Z T
ε
the maximum principle ( ku k∞ ≤ kf k∞ +
kg(τ, ·)k∞dτ ), the T-contraction property
0
holds , and there is translation invariance in x . Hence there exists a modulus of continuity
ωf,g such that
Z Uε (f, g)(t, x+∆x) − Uε (f, g)(t, x)dx ≤ ωf,g (∆x)
uniformly in ε > 0 and t ∈ [0, T ] . By Theorem 2 in [K69a], it follows that for any compact
set K ⊂ IRN
Z Uε (f, g)(t+∆t, x) − Uε (f, g)(t, x)dx ≤ ωf,g,K (∆t)
K
uniformly in ε > 0 and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Take a
countable set M dense in X0 for the L1 (IRN ) × L1 (Q) -topology. By the diagonal process,
there exist εn → 0 such that Uεn (f, g) → u =: U0 (f, g) in L1loc (Q) for all (f, g) ∈ M ;
u is a g.e.s. of (CP), and the maximum principle and the T-contraction property hold for
U0 : M 7→ L1 (Q) ∩ L∞ (Q) ∩ C([0, T ]; L1loc (IRN )) . Thus U0 can be extended to the whole
of X0 , so that U0 (f, g) is a g.e.s. of (CP), the T-contraction property holds and there is
translation invariance in x and the maximum principle holds.
Now for the general case (f, g) ∈ X , set fn,m = f + χ{|x|≤n} − f − χ{|x|≤m} and gn,m =
min{n, g + }χ{|x|≤n} −min{m, g − }χ{|x|≤m} , so that we have (fn,m , gn,m) ∈ X0 . As n → ∞ ,
98
Scalar Conservation Law with Continuous Flux Function
U0 (fn,m , gn,m ) ↑ um ∈ L∞ (Q) ; further, as m → ∞ , um ↓ u =: U(f, g) ∈ L∞ (Q) . By the
Fatou Lemma, it follows that (ii) holds.
We now show that u = U(f, g) is in C([0, T ]; L1loc (IRN )) and a g.e.s. of (CP). Indeed,
there exists an increasing sequence {(ni , mi )}i∈IN in IN2 such that ui = U0 (fi , gi ) → u
in L1loc (Q) as i → ∞ , where fi = fni ,mi , gi = gni ,mi . By Lemma 3.1 in [BK96] and
translation invariance, for ζ ∈ D(IRN )
Z Z sup
ui (x+∆x, t)−ui (x, t) ζ(x) dx ≤ fi (x+∆x)−fi (x) ζ(x) dx+
t∈[0,T ]
Z TZ +
gi (s, x+∆x)−gi (s, x) ζ(x) dxds +
0
Z TZ φ(ui (s, x+∆x))−φ(ui (s, x)) Dζ(x) dxds.
0
Z TZ The last term tends to
φ(u(s, x+∆x))−φ(u(s, x)) Dζ(x) dxds as i → ∞ , therefore
0
for any compact set K ⊂ IRN
Z ui (t, x+∆x) − ui (t, x)dx ≤ ωf,g,K (∆x)
K
uniformly in i ∈ IN and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Hence, again by
Theorem 2 in [K69a], the family {ui (t, ·)}i∈IN is equicontinuous from [0, T ] to L1loc (IRN ) .
Thus u ∈ C([0, T ]; L1loc (IRN )) and u(0, ·) = lim fi = f , so that u is a bounded g.e.s. of
i→∞
(CP).
⋄
Part 2
Weak Solutions
for Elliptic-Parabolic Systems
CHAPTER 2.I
Elliptic-Parabolic Problems: Existence
and Continuity with Respect to the Data
of Weak Solutions†
Introduction
Let Ω be a bounded domain of IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let
Γ0,i be a closed set in ∂Ω and Γ1,i = ∂Ω\ Γ0,i . We consider initial boundary value problems
for elliptic-parabolic systems of the form

b(·, v)t = div a(·, v, Dv) + f (·, v) on Q = (0, T ) × Ω



 a (·, v, Dv) · ν = g (·, v) on Σ = (0, T ) × Γ for i = 1, . . . , N
i
i
1,i
1,i

vi = hi on Σ0,i = (0, T ) × Γ0,i for i = 1, . . . , N



b(·, v)(0, ·) = u0 on Ω,
(P)
where b : Q × IRN 7→ IRN , a = (a1 , . . . , aN ) : Q × IRN × (IRd )N 7→ (IRd )N , f : Q × IRN 7→
IRN , g = (g1 , . . . , gN ) : Σ × IRN 7→ IRN , h = (h1 , . . . , hN ) : Σ 7→ IRN , u0 : Ω 7→ IRN ,
and ν is the external unit normal vector to Σ = (0, T ) × ∂Ω . We denote by “ · ” the scalar
product in IRN , and by “ : ” the scalar product in (IRd )N . Except in Section 3.3, we assume
Q
that |Γ0 | = N
i=1 |Γ0,i | > 0 .
The assumptions on the data will be made precise in Section 1. In the particular case
b(t, x, z) = b(z) , a(t, x, z, ξ) = a(z, ξ) , f ≡ 0 , g ≡ 0 , h ≡ 0 , the assumptions reduce to
the three following conditions on b, a, u0 :
b = ∂Φ, with Φ : IRN 7→ IR convex differentiable , Φ(0) = 0;
†
This chapter is being prepared upon publication [BAB]
(1)
102
Continuous Dependence for Elliptic-Parabolic Problems
Z 1
as usual, we set B(z) =
b(z) − b(σz) · z dσ = b(z) · z − Φ(z) ≥ 0 for any z ∈ IRN ;
0


a is continuous, monotone in ξ, and satisfies Leray-Lions type conditions :





there exist c > 0, C ≥ 0 and a “sublinear” function L : IR+ 7→ IR+



(i.e., a function with lim L(r)/r = 0) such that
r→∞


p
p

a(z,
ξ)
:
ξ
≥
c|ξ|
−
L(|z|
)
−
C
1
+
B(z)
,




′

 |a(z, ξ)|p ≤ L B(z) + |z|p + C 1 + |ξ|p ,
(2)
with 1 < p < ∞ , p′ = p/(p − 1) ;
u0 ∈ L1 (Ω; IRN ) with Ψ(u0 ) ∈ L1 (Ω),
(3)
where Ψ is the Legendre conjugate of Φ defined by Ψ(z ∗ ) = supz∈ IR N z · z ∗ − Φ(z) ∈
[0, +∞] for any z ∗ ∈ IRN . Note that one has B(z) = Ψ(b(z)) for all z ∈ IRN . In this
chapter, adding the structure condition
(
a(z, ξ) = e
a(b(z), ξ) for all z ∈ IRN , ξ ∈ (IRd )N ,
(4)
where e
a : R(b(·)) × (IRd )N 7→ (IRd )N is Caratheodory,
we prove existence of a weak solution (in the variational sense) to (P ) .
Such result has already been proved by Alt-Luckhaus in [AL83], while under more restrictive
assumptions. Since then, many similar results have been obtained (e.g., cf. [Kac90, DiDT94,
FKac95, BW96, Bou97]), in particular in the case of time-space dependent elliptic part and
space dependent parabolic part. Equations with the parabolic part of the form c(t, x, v) vt
have been studied in [Pl98, Pl00] by a different approach; existence results for
Z some of equaz
tions of this form can be derived from the ours, upon introducing b(t, x, z) =
c(t, x, ζ) dζ .
0
The aim and the interest of this work is, first, in proving the continuous dependence of
weak solutions on the data in (P ) ; existence is an easy corollary of this result. Secondly,
we use arguments that permit to make less restrictive assumptions (in particular, all data are
(t, x) - dependent) and, at the same time, clarify the essence of the proof.
The continuity theorem is presented in Section 1 (Theorem 1); in the reduced case considered above it reads as follows:
Theorem 0 Let bn , an , u0,n be a sequence of data satisfying the assumptions (1)-(3) with
c, C, L(·) independent of n ∈ IN and corresponding Bn , Ψn . Assume that bn (·) → b(·) in
C(IRN ; IRN ) † , for all ξ ∈ (IRd )N an (·, ξ) → a(·, ξ) in C(IRN ; (IRd )N ) , and u0,n → u0 a.e
on Ω with Ψn (u0,n ) → Ψ(u0 ) in L1 (Ω) . For n = 1, 2, . . . let vn be a weak solution of
the corresponding problem (Pn ) with f ≡ 0 , g ≡ 0 , h ≡ 0 , and where |Γ0 | > 0 . Then
†
I.e., for all compact subset K of IRN there is uniform convergence for z ∈ K
2.I.1.
Assumptions and results
103
(i) The sequence {vn } is bounded in Lp (0, T ; W 1,p (Ω; IRN )) and the sequence {bn (vn )}
is relatively compact in L1 (Q; IRN ) .
(ii) Any weak limit point of {vn } in Lp (0, T ; W 1,p(Ω; IRN )) is a weak solution of (P ) ,
provided (P ) satisfies (4).
A precise definition of a weak solution is given in Section 1. Note that, while not assuming
the strict monotonicity of b in z , we have to impose the structure condition (4) , which
is trivial for b strictly monotone. However, this condition is not intrinsically related to the
existence of weak solution to (P ) (cf. [BW99] and Section 3.2).
The proof of the continuity theorem in general setting is given in Section 2 and includes
three essential arguments. First, we establish a time-dependent version of the chain rule
Lemma 1.5 from [AL83] and apply it to get a priori estimates which imply compactness in
x of {vn } . Secondly, using a general compactness lemma due to S.N.Kruzhkov ([K69a]) we
deduce the compactness of {bn (vn )} in (t, x) . Finally, under additional structure conditions,
the Minty-Browder argument is used for passage to the limit in the equation (Pn ) . The proofs
of the appropriate versions of this three results (Lemma 1, Lemma 6 and Lemma 7) are given
in the Appendix.
In Section 3 we give some remarks and further existence results. More specifically, we
treat the case of inhomogeneous Dirichlet data, discuss structure conditions of type (4), give
one direct extension to the case |Γ0 | = 0 , and indicate possible applications to proving
convergence of approximate methods.
1
Assumptions and results
In this section and Section 2 we state the results for the problem (P ) with h ≡ 0 and
|Γ0 | > 0 . See Section 3 for some comments on the general case.
Let Ω be a bounded set in IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let
Γ0,i be a closed
set in ∂Ω and Γ1,i = ∂Ω \ Γ0,i . Let 1 0 , and set
Q = (0, T ) × Ω , Σ = (0, T ) × ∂Ω .
Our assumptions include a collection of hypotheses that impose restrictions on the growth
of the coefficients ( (H2), (H3), (H4), (H6), (H7), (H8), (H13) below). We will say that
104
Continuous Dependence for Elliptic-Parabolic Problems
this collection of hypothesis is satisfied if there exist


c = const > 0,





C = const ≥ 0,



+
+


such that
 K0 : [0, T ] × Ω × IR 7→ IR
1
K0 (·, r) ∈ L (Q) and K0 (0, ·, r) ∈ L1 (Ω) for all r ∈ IR+ ,



K1 : (0, T ) 7→ IR+ , K1 ∈ L1 (0, T ),





K2 : Q 7→ IR+ , K2 ∈ L1 (Q),



 K : Σ 7→ IR+ , K ∈ L1 (Σ),
3
3
(5)
for all ε > 0 there exist K ε , K1ε , K2ε and K3ε such that
 ε
K = const ≥ 0,



 K ε : (0, T ) 7→ IR+ , K ε ∈ L1 (0, T ),
1
1
+
ε
ε
 K2 : Q 7→ IR , K2 ∈ L1 (Q),


 ε
K3 : Σ 7→ IR+ , K3ε ∈ L1 (Σ),
(6)
and if the corresponding inequalities holds.
Let b : [0, T ] × Ω × IRN 7→ IRN satisfy
(
b(t, x, z) = ∂Φ(t, x, ·)(z), where Φ : Q × IRN 7→ IR
is convex differentiable in z and Φ(t, x, 0) = 0 for all t ∈ [0, T ], x ∈ Ω;
(H1)
|b(t, x, z)| ≤ K0 (t, x, |z|) for all z ∈ IRN , t ∈ [0, T ] and a.a. x ∈ Ω.
(H2)
Z 1
Set B(t, x, z) =
b(t, x, z) − b(t, x, σz) · z dσ , and Ψ(t, x, z ∗ ) = Φ(t, x, ·)∗ (z ∗ ) =
0
sup z · z ∗ − Φ(t, x, z) for any z ∗ ∈ IRN . One has B(t, x, z) = b(t, x, z) · z − Φ(t, x, z) =
z∈IR N
Ψ t, x, b(t, x, z) .
Note that one has (e.g., cf. Remark 1.2 in [AL83]) for all δ > 0 ,
|b(t, x, z)| ≤ δB(t, x, z) + sup |b(t, x, δ)| ≤ δB(t, x, z) + K0 (t, x, 1/δ).
(7)
|ζ|≤1/δ
In addition to (H1), (H2) we assume that Φ is absolutely continuous in t on [0, T ]
for all z ∈ IRN and a.a. x ∈ Ω , and that there exists Φt : Q × IRN 7→ IR Caratheodory
d
such that
Φ(·, x, z)(t) = Φt (t, x, z) for all z ∈ IRN and a.a. (t, x) ∈ Q . Moreover, we
dt
require that for a.a. x ∈ Ω , a.a. t, s ∈ (0, T ) and all z ∈ IRN the function Φt satisfy
p
|Φt (t, x, z)| ≤ ε K1 (t)B(s, x, z) + |z| + K2ε (t, x).
(H3)
2.I.1.
Assumptions and results
105
It follows that Φ, b, B are measurable in x for all t ∈ [0, T ] and z ∈ IRN , and continuous
in (t, z) for a.a. x ∈ Ω .
Let f : Q × IRN 7→ IRN be Caratheodory and assume for a.a. (t, x) ∈ Q
′
|f (t, x, z)|p ≤ ε K1 (t) B(t, x, z) + |z|p + K2ε (t, x).
(H4)
Let a : Q × (IRN × (IRd )N ) 7→ (IRd )N be Caratheodory and assume for a.a. (t, x) ∈ Q
ˆ : (ξ − ξ)
ˆ ≥ 0 for all z ∈ IRN , ξ, ξ̂ ∈ (IRd )N ,
a(t, x, z, ξ) − a(t, x, z, ξ)
(H5)
a(t, x, z, ξ) : ξ − f (t, x, z) · z + Φt (t, x, z) ≥ c|ξ|p − ε |z|p −
−K1ε (t) B(t, x, z) − K2ε (t, x),
p′
p
|a(t, x, z, ξ)| ≤ ε K1 (t) B(t, x, z) + |z| + K2ε (t, x) + C|ξ|p.
(H6)
(H7)
Let gi : (0, T ) × Γ1,i × IRN 7→ IRN be Caratheodory, i = 1, . . . , N . For convenience, for
all z ∈ IRN extend gi (·, z) by zero on (0, T ) × Γ0,i , and assume for a.a. (t, x) ∈ Σ
(
′
|g(t, x, z)|p ≤ C|z|p + K3 (t, x)
(H8)
g(t, x, z) · z ≤ ε |z|p + K3ε (t, x).
Remark 1 The growth restrictions in (H3), (H4), (H6)−(H8) are in fact of the same type
as those in (2). It is convenient to pass to the form chosen above, rather than majorate
′
|Φt (t, x, z)| , |f (t, x, z)|p , etc., by terms of the form L(t, x, B(t, x, z)) and L(t, x, |z|p ) ,
with “sublinear” functions L(t, x, ·) subject to additional restrictions on their dependence on
(t, x) .
A typical situation where, for instance, the hypothesis (H4) is satisfied, is when one has
′
|f (t, x, z)|p ≤ M(t, x) (B(t, x, z))1/κ + |z|p/κ + K2 (t, x)
′
with κ > 1 , M ∈ Lκ (Q) and K2 ∈ L1 (Q) . An assumption of slightly different kind that
is also covered by (H4) is that
′
|f (t, x, z)|p ≤ N (t) L(B(t, x, z)) + L(|z|p ) + K2 (t, x)
with N ∈ L1 (0, T ) and L(·) “sublinear”, independent of (t, x) . Other growth hypotheses
can be simplified in similar ways.
To shorten the notation, for all v : Q 7→ IRN we will denote the function b(t, x, v(t, x))
by b(v) , the function a(t, x, v(t, x), Dw(t, x)) by a(v, Dw) , the function Ψ(τ, x, w(x))
by Ψ(τ, w) and so on.
†
It seems that a more natural condition would be (H3) written only for s = t ; see Remark 6 in the
Appendix for a discussion of this issue
106
Continuous Dependence for Elliptic-Parabolic Problems
Definition 1 For u0 ∈ L1 (Ω; IRN ) and h ∈ Lp (0, T ; W 1,p(Ω; IRN )) , a weak solution of
(P ) is a function v : Q 7→ IRN satisfying
(i) v ∈ h + Lp (0, T ; V ) , B(v) ∈ L∞ (0, T ; L1 (Ω)) (whence b(v) ∈ L1 (Q) by (7));
(ii) for all ζ ∈ Lp (0, T ; V ) with ζt ∈ L∞ (Q) and ζ(T ) = 0 ,
ZZ
Q
b(v) · ζt +
Z
Ω
u0 (·) · ζ(0, ·) =
ZZ
Q
a(v, Dv) : Dζ −
ZZ
Q
f (v) · ζ −
ZZ
Σ
g(v) · ζ.
Let denote by V ′ the dual space of V , and by < ·, · > the duality pairing between V ′ and
V . We remark that, according to (i) and the conditions (H4), (H7), (H8) , the condition
(ii) in the Definition 1 makes sense and can be rewritten, as in [AL83], under the equivalent
form:
′
there exists χ ∈ Lp (0, T ; V ′ ) such that
 ZZ
Z
Z T


b(v) · ζt + u0 (·) · ζ(0, ·) =
< χ, ζ >
(ii1 )
Q
Ω
0


for all ζ ∈ Lp (0, T ; V ) with ζt ∈ L∞ (Q) and ζ(T ) = 0,
and
(ii2 )
 Z


T
< χ, ζ >=
0
ZZ
Q
a(v, Dv) : Dζ −

 for all ζ ∈ Lp (0, T ; V ).
ZZ
Q
f (v) · ζ −
ZZ
Σ
g(v) · ζ
According to (ii1 ) , the distribution derivative of b(v) with respect to t , b(v)t = −χ , is in
′
Lp (0, T ; V ′ )) .
Following [AL83], in addition to u0 ∈ L1 (Ω; IRN ) we assume
Ψ(0, u0) ∈ L1 (Ω).
(H9)
One has the following result.
Lemma 1 Let b satisfy (H1), (H2), (H3) with corresponding functions
B, Φ . Assume
N
1
1
p
that u0 ∈ L (Ω; IR ) with Ψ(0, u0) ∈ L (Ω) and v ∈ L 0, T ; V
with B(v) ∈
∞
1
1
L (0, T ; L (Ω)) (whence b(v) ∈ L (Q) ) be given such that (ii1 ) holds with some χ ∈
′
Lp (0, T ; V ′ ) . Then one has Φt (v) ∈ L1 (Q) , and for a.a. t ∈ (0, T )
Z
Z
Z t
Z tZ
B(v)(t) =
Ψ(0, u0) −
< χ(τ ), v(τ ) > dτ −
Φt (v)(τ ) dτ.
(8)
Ω
Ω
0
0
Ω
This is a time-dependent version of the chain rule lemma (cf. [Bam77, AL83, Ot96, CaW99]),
which is crucial in this framework. We give a proof of it in the Appendix.
2.I.1.
Assumptions and results
107
Remark 2 Under the hypothesis we take, we avoid much unnecessary technicalities by requiring into Definition 1 and Lemma 1 that B(v) ∈ L∞ (0, T ; L1(Ω)) (which was one of the
claims in the corresponding lemma in [AL83]). Still this property, together with (ii1 ) and
(ii2 ) , could be deduced from the assumption that b(v) ∈ L1 (Q) and Definition 1 (ii) . We
also take h ≡ 0 in the statement of Lemma 1, hoping to get the general case by translation
by h (cf. Section 3.1).
In order to state the main result of this
chapter, we consider a sequence of problems (Pn ) ,
n ∈ IN , with data bn , an , fn , gn , u0,n satisfying
 ε
ε


 bn , an , fn , gn verify (H1) − (H8) with c, K0 , K1 , K , K2 , K3
independent of n and corresponding functions Bn , Φn,t , Ψn ;


 {Ψ (0, u )} is bounded in L1 (Ω).
n
(H)
0,n
We
that a sequence of problems (Pn ) converge to the problem (P ) with data
will say b, a, f, g, u0 , if:
 
bn (t, x, ·), Φn,t (t, x, ·), an (t, x, ·, ξ), fn (t, x, ·) −→






−→
b(t,
x,
·),
Φ
(t,
x,
·),
a(t,
x,
·,
ξ),
f
(t,
x,
·)
t


N
N
in C IR ; IR × (IRd )N × IRN for a.a. (t, x) ∈ Q, for all ξ ∈ (IRd )N ; (9)





gn (t, x, ·) → g(t, x, ·) in C(IRN ; IRN ) for a.a. (t, x) ∈ Σ;



 u → u a.e. on Ω with Ψ (0, u ) → Ψ(0, u ) in L1 (Ω)
0,n
0
n
0,n
0
Theorem 1 Let |Γ0 | > 0 and the Dirichlet part of the boundary data be homogeneous, i.e.,
h ≡ 0,
(H10)
and bn , an , fn , gn , u0,n be a sequence of data satisfying (H) . Assume that (Pn ) converge
to (P) in the sense (9). For n = 1, 2, . . . let vn be a weak solution of the corresponding
problem (Pn ) . Then
(i) The sequence {vn } is bounded in Lp (0, T ; W 1,p (Ω; IRN )) and the sequence {bn (vn )}
is relatively compact in L1 (Q; IRN ) .
(ii) Any weak limit point of {vn } in Lp (0, T ; W 1,p(Ω; IRN )) is a weak solution of (P ) ,
provided (P ) satisfies the structure conditions (H11) − (H13) † :
(
f (t, x, z) = fe(t, x, b(t, x, z)) for all z ∈ IRN and a.a. (t, x) ∈ Q,
(H11)
where fe : {(t, x, β) ∈ Q × IRN | β ∈ R(b(t, x, ·))} 7→ IRN is Caratheodory;
†
By R(b(t, x, ·)) we denote the image of IRN by b(t, x, ·)
108
Continuous Dependence for Elliptic-Parabolic Problems


a(t, x, b(t, x, z), ξ) for all z ∈ IRN , ξ ∈ (IRd )N and a.a. (t, x) ∈ Q,
 a(t, x, z, ξ) = e
(H12)
where e
a : {(t, x, β, ξ) ∈ Q × (IRN × (IRd )N ) | β ∈ R(b(t, x, ·))} 7→ (IRd )N


is Caratheodory;

e x, b(t, x, z)) + z · ϕ(t,
Φt (t, x, z) = φ(t,
e x, b(t, x, z)) for all z ∈ IRN and a.a. (t, x) ∈ Q,



 where (φ,
e ϕ)
e : {(t, x, β) ∈ Q × IRN | β ∈ R(b(t, x, ·))} 7→ IR × IRN
′

is Caratheodory and‡ |ϕ(t,
e x, b(t, x, z))|p ≤ K1 (t)Ψ(t, x, b(t, x, z)) + C|z|p + K2 (t, x)



for a.a. (t, x) ∈ Q and all z ∈ IRN .
(H13)
Remark 3 In the convergence part, we will manage with the dependence of coefficients on
(t, x) by considering them as mappings from Q to C(IRN ; IRN ) or C(IRN×(IRd )N ; (IRd )N )
and repeatedly applying the Egorov and Lusin theorems. It is possible, since a Caratheodory
function Y1 × Y2 7→ Y3 , where Yi ⊂ IRDi , i = 1, 2, 3 , is measurable considered as mapping
from Y1 to C(Y2 ; Y3 ) . Indeed, it is weakly measurable since it is measurable in y1 ∈ Y1 for
all fixed y2 ∈ Y2 ; hence it is strongly measurable (e.g., cf. [BbkiXIII],Chap.IV,S5,Prop.10).
Corollary 1 Let the data in (P ) satisfy (H1) − (H13) , and Ψ(u0 ) ∈ L1 (Q) . Then there
exists a weak solution to (P ) with h = 0 .
Proof of Corollary 1: Let approximate (P ) in the sense (H) ,(9) by a sequence of
problems (Pn ) with bn bilipschitz in z , and with data and coefficients regular in (t, x) .
Note that we have to assure that Ψn (0, u0,n ) → Ψ(0, u0) in L1 (Ω) . This is done by
choosing for all n ∈ IN some m ∈ IN and a measurable function u0,n such that u0,n
is a value within the image of [−m, m]N by b(0, x, ·) , and kΨn (0, u0,n ) − Ψ(0, u0)k ≤
1/n . In turn, this is possible since Bn (0, x, ·) are dominated on [−m, m]N by the function
√
√
nm K0 (0, x, nm) ∈ L1 (Ω) uniformly in n , since the assumption (H2) holds uniformly
in n .
To prove the existence for (Pn ) itself, we use the Galerkin approximations in the way it is
done in [LJLL65, JLL] and [ALpr]. The global in time existence of Galerkin approximations for
(Pn ) follows from an a priori L∞ (0, T ; L2(Ω)) estimate, obtained as in [ALpr]. It is at this
level that we also find that B(v) ∈ L∞ (0, T ; L1(Ω)) , which we have required in Definition 1.
The convergence of Galerkin approximations for (Pn ) follows as in Theorem 1 and yields
existence of a weak solution vn .
Applying Theorem 1 to the constructed sequence {vn } , we get existence for (P ) .
‡
⋄
This hypothesis actually seems amount to nothing more than this inequality. Moreover, a natural candidate
for ϕ
e is bt , and this inequality, with (H3) and (H4) taken into account, means that it could be considered
as a part of the second member in (P ) . See Lemma 3 for a discussion of the condition (H13) in the scalar
case.
2.I.2.
Proof of the continuity theorem
109
Proof of the continuity theorem
2
In order to prove Theorem 1, we start with the following lemma:
Lemma 2 Assume (P ) satisfy (H1), (H2), (H3), (H6), (H8) and Ψ(0, u0 ) ∈ L1 (Ω) ; let
v is a weak solution of (P ) . Then
Z
(i)
sup
B(v)(τ, ·) + kvkLp (0,T ;W 1,p (Ω; IR N )) ≤ M
τ ∈[0,T ]
(ii)
Ω
there exists a function ω ∈ C(IR+ ; IR+ ) , ω(0) = 0 such that for all E ⊂ Q
ZZ
E
|b(v)| ≤ ω(|E|);
here M and ω(·) are determined solely by kΨ(0, u0)kL1 (Ω) , c, K0 , K1 , K2 in (5) and the
dependence of K ε , K2ε in (6) on ε .
Proof of Lemma 2: Take vχ[0,t)×Ω for the test function in (ii2 ) of Definition 1. Applying
Lemma 1, on account of (H3), (H6), (H8) we get for a.a. t ∈ (0, T )
Z
Z
Z tZ
Z tZ
Z tZ
p
p
p
B(v)(t) −
Ψ(0, u0) + c
|Dv| ≤ ε
|v| +
|v| +
Ω
Ω
0 Ω
0 Ω
Z t
Z
Z0 tZ∂Ω
Z tZ
ε
ε
+
K1 (τ ) B(v)(τ ) dτ +
K2 +
K3ε
0
Ω
0
Ω
0
∂Ω
with ε > 0 and the corresponding K1ε , K2ε , K3ε . Hence, using the imbedding of W 1,p (Ω)
into Lp (∂Ω) and the Poincaré inequality, we get for all ε sufficiently small
Z
c
B(v)(t) + kvkLp (0,T ;W 1,p (Ω;IRN )) ≤ kΨ(u0)kL1 (Ω) +
2
Ω
Z t
Z
ε
+
K1 (τ ) B(v)(τ ) dτ + kK2ε kL1 (Q) + kK3ε kL1 (Σ) .
0
Ω
Thus (i) follows from the Gronwall inequality.
Consequently, (ii) follows readily from (7); it suffices to take
ZZ
ω(r) = min Mδ + sup
K0 (·, 1/δ) .
δ>0
E⊂Q,|E|≤r
E
⋄
Now let prove the compactness part of the continuity theorem.
Proof of (i) in Theorem 1: By Lemma 2, kvn kLp (0,T ;W 1,p (Ω; IR N )) are bounded;
moreover, un = bn (vn ) are equiintegrable on Q .
Let prove the “compactness in x ” of {un } in L1 (Q) (i.e. the property (10) below); its
compactness in (x, t) in L1 (Q) will follow from Lemma 6 (see the Appendix and [K69a]).
Indeed, un = bn (vn ) satisfies the evolution equation ∂/∂t un = Fn in D ′ (Q) , with Fn
110
Continuous Dependence for Elliptic-Parabolic Problems
bounded in L1 (0, T ; W −1,1(Ω)) by virtue of (H4), (H7), (H8) and (i) of Lemma 2. Moreover, {un } is bounded in L1 (Q) by (ii) of Lemma 2. So we only need to show that for all
compact set K ⊂ Ω , for all h ∈ [0, dist(K, ∂Ω)/2] one has
In = sup
|∆x|≤h
Z TZ
0
K
|un (t, x + ∆x) − un (t, x) dxdt ≤ ωK (h)
(10)
with some function ωK ∈ C(IR+ ; IR+ ) with ωK (0) = 0 , ωK independent of n .
Fix α > 0 . First, by Remark 3 and the Lusin and Egorov theorems there exists an
open set Qα ⊂ Q , |Qα | < α such that bn → b in C(Q \ Qα , C(IRN ; IRN )) = C((Q \
Qα ) × IRN ; IRN ) . Thus it follows by the Arzela-Ascoli theorem that there exists a function
′
′
ω α,M ∈ C((IR+ )3 ; IR+ ) with ω α,M (0, 0, 0) = 0 such that for all
(t, x), (t , x ) ∈ Q \ Qα , all
z, z ′ ∈ [−M, M]N one has |bn (t, x, z) − bn (t′ , x′ , z ′ )| ≤ ω α,M |t − t′ |, |x − x′ |, |z − z ′ | .
Second, take M = M(α) = 1/α supn kvn kL1 (Q) ; by the Chebyshev inequality, for each
n ∈ IN there exists another set Qα,n ⊂ Q , |Qα,n | < α such that |vn | ≤ M(α) a.e. on
Q \ Qα,n .
Now we can estimate In in (10) by integrating separately over the set Qα,n (∆x) =
{(t, x) ∈ (0, T ) × K : (t, x) ∈ Qα ∪ Qα,n or (t, x + ∆x) ∈ Qα ∪ Qα,n } , with |Qα,n (∆x)| <
4α , and
set Q′ α,n (∆x) = ((0, T ) × K) \ Qα,n (∆x) . By the concavity of
the complementary
ω α,M 0, |∆x|, ·
, we get
In ≤ ω(4α) +
ZZ
|bn (t, x + ∆x, vn (t, x + ∆x)) − bn (t, x, vn (t, x))| dxdt ≤
α,M
≤ ω(4α) + +
ω
0, |∆x|, |vn (t, x + ∆x) − vn (t, x)| ≤ ω(4α) +
Q
ZZ
1
α,M
+ |Q| ω
0, |∆x|,
|vn (t, x + ∆x) − vn (t, x)| ≤
|Q| Q kDvn kL1 (Q)
≤ ω(4α) + |Q| ω α,M 0, h, sup
h
.
|Q|
Q′ α,n (∆x)
ZZ
n
Minimizing the right-hand side in α > 0 , we get a function ωK with the desired properties.
⋄
Finally, let prove the convergence part of the continuity theorem.
Proof of (ii) in Theorem 1: By compactness, choose a subsequence (which we still
denote by vn ) such that vn ⇀ v in Lp (0, T ; V ) , vn → v in Lp (Σ; IRN ) and a.e., and
bn (vn ) → u in L1 (Q; IRN ) and a.e. on Q .
It follows that u = b(v) , by the argument introduced in [BrSt73]. More precisely, by
Remark 3, the Egorov theorem and the Chebyshev inequality, for all α > 0 there exists an
open set Qα ⊂ Q such that |Qα | < α , bn → b in C(Q \ Qα ; C(IRN )) , bn (vn ) → u in
L∞ (Q \ Qα ) , and v is bounded on Q \ Qα . Therefore for all η, ζ ∈ L∞ (Q \ Qα ) we have
bn (η) → b(η) in L∞ (Q \ Qα ) ; in addition, b(v + λζ) → b(v) in L∞ (Q \ Qα ) as λ → 0 .
2.I.2.
Proof of the continuity theorem
111
Thus, by the monotonicity of bn ,
ZZ
ZZ
u(v − η) = lim
bn (vn )(vn − η) ≥
Q\Qα
Q\Qα
ZZ
ZZ
≥ lim
bn (η)(vn − η) =
b(η)(v − η).
Q\Qα
Q\Qα
Taking η = v + λζ with arbitrary ζ ∈ L∞ (Q \ Qα ) , letting λ decrease to 0 and then
increase to 0 , we conclude that u = b(v) in L1 (Q \ Qα ) . Letting α go to 0 , we get
u = b(v) a.e. on Q .
′
It follows that bn (vn )t → b(v)t in Lp (0, T ; V ′ ) ; indeed, note that bn (vn )t are uniformly bounded in this latter space by (ii2 ) of Definition 1, Lemma 2 and hypotheses
(H4), (H7), (H8) .
The initial condition (ii1 ) is therefore satisfied at the limit; indeed, (9) implies that
u0,n → u0 in L1 (Ω; IRN ) . Note also that, by the Fatou lemma, B(v) ∈ L∞ (0, T ; L1 (Ω)) ,
which will permit to apply Lemma 1 to the function v .
Let prove that (ii2 ) holds as well.
′
Start by showing that one has gn (vn ) → g(v) and fn (vn ) → f (v) in Lp (Σ; IRN ) and
′
Lp (Q; IRN ) , respectively. The former follows readily from (H8) and the Lebesgue dominated
convergence theorem. We need the structure condition (H11) in order to prove the latter.
Let us fix ǫ > 0 and show that, for all n sufficiently large, one has
ZZ
′
|fn (vn ) − f (v)|p < ǫ.
(11)
Q
′
First, it follows from (H4) and Lemma 2 that |fn (vn )|p are equiintegrable on Q .
Indeed, take α > 0 . For all E ⊂ Q , |E| < α , we have for ε > 0 and the corresponding
K2ε
ZZ
ZZ
Z TZ
ZZ
p′
ε
p
|fn (vn )| ≤ ε
K2 (t)Bn (vn )(t)dt +
|vn | +
K2 ≤
E
0 Ω
Q
E
ZZ
p
≤ ε supn kK1 kL1 (0,T ) kBn (vn )kL∞ (0,T ;L1 (Ω)) + kvn kLp (0,T ;V ) +
K2ε ,
E
which is independent of n and can ZbeZ made as small as desired by a choice of ε and α
′
small enough. Fix α > 0 such that
|fn (vn ) − f (v)|p < ǫ/3 whenever |E| < α .
E
Further, by the Chebyshev inequality, there exists M > 0 such that for all n ∈ IN one
can choose an open set Qα,n ⊂ Q with |Qα,n | < α so that |vn | ≤ M , |v| ≤ M on
t,x
Q \ Qα,n . Fix M ; for all (t, x) ∈ Q define the set KM
⊂ IRN as the image of [−M, M]N
by b(t, x, ·) . Let TM (t, x, ·) be the projection of IRN on this set, i.e.,
(
t,x
TM (t, x, ·) : z ∈ IRN 7→ TM (t, x, z) = ẑ ∈ KM
,
(12)
where dist(z, ẑ) = minζ∈Kt,x dist(z, ζ).
M
112
Continuous Dependence for Elliptic-Parabolic Problems
t,x
The projection is well defined since, clearly, KM
is compact and convex. Note that TM :
N
N
Q × IR 7→ IR is Caratheodory. Indeed, it is a contraction in IRN for all (t, x) ∈ Q ; for
all z ∈ IRN it is measurable in (t, x) , because b is measurable in (t, x) and TM depends
continuously on b with respect to the norm k · kC([−M,M ]N ) .
Now for all n ∈ N , for a.a. (t, x) ∈ Q , TM ◦ bn = TM (t, x, bn (t, x, ·)) is well defined.
By (H11) , one has for a.a. (t, x) ∈ Q
p′
′
|fn (vn ) − f (v)|p ≤ const fn (vn ) − f (vn ) +
p′
p′
e
e
e
e
+ (f ◦ b)(vn ) − (f ◦ TM ◦ b)(vn ) + (f ◦ TM ◦ b)(vn ) − (f ◦ TM ◦ bn )(vn ) + (13)
p′
p′ e
e
e
e
+ (f ◦ TM )(bn (vn )) − (f ◦ TM )(b(v)) + (f ◦ TM ◦ b)(v) − (f ◦ b)(v) .
By Remark 3, we can apply the Lusin and Egorov theorems to fn , f and fe◦ TM ◦ bn , fe◦
TM ◦ b . Indeed, by (9) fn → f and fe◦ TM ◦ bn → fe◦ TM ◦ b in C([−M, M]; IRN ) for a.a.
(t, x) ∈ Q . Besides, one also have bn (vn ) → b(v) a.e. on Q . It follows that there exists
an open set Qα ⊂ Q with |Qα | such that one has


f → f and fe ◦ TM ◦ bn → fe ◦ TM ◦ b in C((Q \ Qα ) × [−M, M]N ; IRN );

 n
bn (vn ) → b(v) in C(Q \ Qα ; IRN );
(14)


 fe ◦ T
is uniformly continuous on (Q \ Qα ) × [−M, M]N .
M
In addition, for a.a. (t, x) ∈ Q , TM ◦ b ≡ b on [−M, M]N .
For each n , we can integrate in (11) separately over Qα ∪ Qα,n and Q \ (Qα ∪ Qα,n ) .
The second integral vanishes as n → ∞ , due to (13) and (14), and the first one is estimated
by 2ǫ/3 . Hence (11) holds for n small enough.
It remains to justify the passage to the limit in the elliptic term. This can be made through
a usual Minty-Browder argument summarized in Lemma 7 in Appendix. Let introduce the
′
operators An : η ∈ Lp (0, T ; V ) 7→ Lp (0, T ; V ′ ) by defining the duality product of An η with
ϕ ∈ Lp (0, T ; V ) :
Z T
ZZ
< An η, ϕ >=
an (vn , Dη) : Dϕ.
0
Q
By (H7) and Lemma 2, this last integral makes sense. Likewise, the operator A : η ∈
Z T
ZZ
p
p′
′
L (0, T ; V ) 7→ Aη ∈ L (0, T ; V ) is defined by assigning
< Aη, ϕ >=
a(v, Dη) :
0
Q
∗
Dϕ. . Since vn is a weak solution of (Pn ) , it follows from the analysis above that An vn ⇀ χ
′
in Lp (0, T ; V ′ ) , where
Z T
Z T
ZZ
ZZ
< χ, ϕ >=
< −b(v)t , ϕ > +
f (v) · ϕ +
g(v) · ϕ
0
for ϕ ∈ Lp (0, T ; V ) .
0
Q
Σ
2.I.3.
Comments and further results
113
Let verify the other assumptions of Lemma 7. We have vn ⇀ v in Lp (0, T ; V ) ; all
An are monotone; besides, A is hemicontinuous by (H7) and the Lebesgue dominated
′
convergence theorem. Further, under the assumption (H12) , An η → Aη in Lp (0, T ; V ′ )
for all fixed η ∈ Lp (0, T ; V ) ; the arguments are the same as used for the proof of convergence
of fn (vn ) above.
Furthermore, let prove that one also has
lim inf
Z
T
< An vn , vn >≤
0
Z
T
< χ, v > .
(15)
0
First, under the structure condition (H13) we can show that
ZZ
Φn,t (vn ) →
ZZ
Φt (v) as
e
n → ∞ (the last integral makes sense, by Lemma 1). Indeed, by (H13) Φt (v) = φ(b(v))
+
′
vZ Z· ϕ(b(v))
e
and ϕ(b(v))
e
∈ Lp (Q) . Since vn ⇀ v in Lp (Q) , it suffices to show that
e
|Φn,t (vn ) − φ(b(v))
− vn · ϕ(b(v))|
e
vanishes as n → ∞ . This can be done by the same
Q
arguments as used for the proof of convergence of fn (vn ) . With TM (t, x, ·) defined by (12),
we have to replace the key estimate (13) by the lengthy, but trivial estimate
e
e
e
e
≤ Φn,t (vn ) − Φt (vn ) + φ(b(vn )) − φ(b(v)) +
Φn,t (vn ) − φ(b(v)) − vn · ϕ(b(v))
e
e
(
φ
◦
b)(v
)
−
(
φ
◦
T
◦
b)(v
)
+
Φ
(v
)
−
Φ
(v
)
≤
e
))
−
ϕ(b(v))
e
+ vn ϕ(b(v
n
M
n +
n,t n
t n
n
+ (φe ◦ TM ◦ b)(vn ) − (φe ◦ TM ◦ bn )(vn ) + (φe ◦ TM )(bn (vn )) − (φe ◦ TM )(b(v)) +
p′
p
e
e ◦ b)(vn ) − (ϕ
e ◦ TM ◦ b)(vn ) +
+ (φ ◦ TM ◦ b)(v) − (φe ◦ b)(v) + ǫvn + C(ǫ) (ϕ
p′
p′ e ◦ TM )(bn (vn )) − (ϕ
e ◦ TM )(b(v)) +
e ◦ TM ◦ b)(vn ) − (ϕ
e ◦ TM ◦ bn )(vn ) + (ϕ
+ (ϕ
p′ ,
e ◦ TM ◦ b)(v) − (ϕ
e ◦ b)(v)
+ (ϕ
′
where we have used the inequality |a · b| ≤ ǫ|a|p + C(ǫ)|b|p for a, b ∈ IRN . The required
convergence will follow from the convergences of bn , Φn,t to b, Φt , respectively, given by (9),
and the a.e. convergence of bn (vn ) to b(v) . Secondly, without loss of generality we can
apply Lemma 1 with t = T to v and all functions vn , n ∈ IN . Using the convergence of
Ψn (0, u0,n) in (9), we have
Z
Z
Z
lim sup
< −bn (vn )t , vn >= lim sup − Ψn (bn (vn ))(T ) +
Ψn (0, u0,n ) −
n→∞
n→∞ Z
Ω
Ω
ZZ 0
Z
ZZ
Z T
−
Φn,t (vn ) ≤ − Ψ(b(v))(T ) + Ψ(u0) −
Φt (v) =
< −b(v)t , v >;
Q
T
Ω
Ω
Q
0
the inequality here is due to the Fatou Lemma. Together with the strong convergence of
′
gn (vn ) and fn (vn ) , this yields (15). By Lemma 7, χ = Av in Lp (0, T ; V ′ ) , which implies
(ii2 ) .
⋄
114
3
Continuous Dependence for Elliptic-Parabolic Problems
Comments and further results
3.1. On the inhomogeneous Dirichlet boundary conditions
One can deduce from Corollary 1 some existence results for non-zero Dirichlet boundary conditions h ∈ Lp (0, T ; W 1,p(Ω; IRN )) . Indeed, Definition 1 permits to perform the translation
Th : v 7→ v − h ; writing down the restrictions induced by (H2)−(H13) and Th , we obtain
hypotheses that guarantee the existence of weak solutions. Within certain classes of h , the
hypotheses of Section 1 remain invariant. Let give two examples. For simplicity, assume that
K1 , K1ε ∈ L∞ (0, T ) for the first case.
Corollary 2 Let (H1), (H4)−(H9), (H11)−(H13) hold, with K1 = const in (H4), (H7) ,
(H13) and K1ε = constε in (H6) for each ε > 0 . Assume that for all z ∈ IRN , all
t ∈ [0, T ] and a.a. x ∈ Ω

 |b(t, x, z)| ≤ K
e 0 (t, x) 1 + |z|p/κ , where κ ∈ [1, p), κ′ = κ/(κ − 1)
(H2’)
 and K
e ∈ Lν (Q) with K
e (0, ·) ∈ Lν (Ω) for some ν ∈ [κ′ , +∞];
0
0
assume that for all ε > 0 there exists K2ε ∈ L1 (Q) such that for all z ∈ IRN and a.a.
(t, x) ∈ Q
|Φt (t, x, z)| ≤ ε|z|p + K2ε (t, x).
(H3’)
Let 1/σ + 1/ν + 1/κ = 1 . Then for all Dirichlet data h ∈ Lp (0, T ; W 1,p(IRN )) such that

N
1,σ
σ


 h ∈ W (0, T ; L (Ω; IR )),
(H10’)
h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω


 with h(0, ·) ∈ Lp (Ω; IRN ) and h(0, ·) · u (·) ∈ L1 (Ω),
0
there exists a weak solution to (P ) .
Proof of Corollary 2: It is equivalent to consider instead of (P ) the problem (P )
with the zero Dirichlet data and b(t, x, z) = b(t, x, z + h(t, x)) , a(t, x, z, ξ) = a(t, x, z +
h(t, x, ), ξ + Dh(t, x)) , f (t, x, z) = f (t, x, z + h(t, x)) and g(t, x, z) = g(t, x, z + h(t, x)) .
Note that (H1) holds with Φ(t, x, z) = Φ(t, x, z + h(t, x)) − Φ(t, x, h(t, x)) , so that
Ψ(t, x, z ∗ ) = Ψ(t, x, z ∗ ) − h(t, x) · z ∗ + Φ(t, x, h(t, x)) and
B(t, x, z) = B(t, x, z + h(t, x)) − h(t, x) · b(t, x, z + h(t, x)) + Φ(t, x, h(t, x)).
Clearly, the initial condition is unchanged: u0 = u0 on Ω .
Since h(0, ·) ∈ Lσ (Q; IRN ) by the Fatou lemma, and because
e 0 (0, x) |h(0, x)| (1 + |h(0, x)|p/κ) ≤
|Φ(0, x, h(0, x))| ≤ K
ν
σ
p
e
≤ const (K0 (0, x)) + |h(0, x)| + 1 + |h(0, x)| ,
(16)
2.I.3.
Comments and further results
115
it follows from (H2′ ) and (H10′) that Ψ(u0 ) ∈ L1 (Ω) . Hence (H9) holds. The hye 0 |h|p/κ ∈ L1 (Q) . Further, (H5), (H11), (H12) are
pothesis (H2) is satisfied because K
obvious. Since h, Dh ∈ Lp (Q) , the invariance of (H8) follows by the Hölder inequality;
moreover, checking (H4), (H6), (H7) amounts to showing that
B(t, x, z + h(t, x)) ≤ B(t, x, z) + C|z|p + K2 (t, x)
(17)
with some function K2 (t, x) ∈ L1 (Q) . Since Φ is non-negative, one has by (H2′) and
(16)
B(t, x, z + h(t, x)) ≤ B(t, x, z) + h(t, x) · b(t, x, z + h(t, x)) ≤
e 0(t, x)| (1 + |z + h(t, x)|p/κ ),
≤ B(t, x, z) + |h(t, x)| |K
(18)
whence (17) follows.
It remains to show that (H3) and (H13) hold for (P ) as well. One has
Φt (t, x, z) = ∂/∂t h(t, x) · b(t, x, z + h(t, x)) − b(t, x, h(t, x) +
+Φt (t, x, z + h(t, x)) − Φt (t, x, h(t, x)).
(19)
f x, b(t, x, z)) + z · (ϕ)(t,
f x, b(t, x, z)) with (ϕ)
f ≡ ϕ
Hence Φt (t, x, z) = (φ)(t,
e . As in the
estimate (18), using in addition the inequality c|a · b| ≤ C(ǫ)cν + C(ǫ)|a|σ + ǫ|b|κ valid for all
c ∈ IR+ , a, b ∈ IRN , one sees that (H3′ ) and the inequality in (H13) are invariant under
the translation by h , whenever (H2′ ), (H10′) hold.
Therefore (P ) is in the scope of Corollary 1.
⋄
Remark 4 Clearly, under the assumptions (H1), (H2′), (H3′), (H4)−(H9), (H11)−(H13)
on (P ) , for the class of Dirichlet data h verifying (H10′) there is also a continuous dependence on h of weak solutions of (P ) in the sense of Theorem 1.
In case without dependence of coefficients on (t, x) , the assumptions on h of Corollary 2
meet exactly the assumptions in the remark in [AL83] that follows Lemma 1.5. Nevertheless,
if we could modify the condition (H3) by excluding s 6= t (cf. Remark 6 in the Appendix),
there would be no need to strengthen (H3) to (H3′ ) . The problem here is that, in what
concerns s 6= t , (H3) is utterly non-invariant under translations Th with natural assumptions on h . For instance, with the hypothesis (H3) we cannot attain the assumptions on h
made in the statement of Lemma 1.5 in [AL83] unless requiring (H2′′ ) below, which imposes
a growth restriction in z † . We state the corresponding result in case where b is independent
of t , so that (H3) does not seem pecular.
†
This hypothesis is verified up to exponential growthes of b(x, z) in z
116
Continuous Dependence for Elliptic-Parabolic Problems
Corollary 3 Let b be independent of t , and (H1), (H3)−(H9), (H11), (H12) hold. Assume

+
1
e

 for all r ∈ IR there exist Λ(r) < ∞ and K0 (·, r) ∈ L (Ω)
(H2”)
such that for all λ, z ∈ IRN , and a.a. x ∈ Ω one has


e 0 (x, |λ|) and |b(x, 0)| ≤ K
e 0 (x, 0).
|b(x, z + λ)| ≤ Λ(|λ|)|b(x, z)| + K
Then there exists a weak solution of (P ) for all Dirichlet data h ∈ Lp (0, T ; W 1,p(IRN )) such
that
(
h ∈ L∞ (Q; IRN ), ∂/∂t h ∈ L1 (0, T ; L∞ (Ω; IRN ))
(H10”)
and h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω.
Proof of Corollary 3: We proceed as in the proof above. First note that it follows
e 0 (x, 0)+ K
e 0 (x, |z|) . Let M = khkL∞ (Q) .
from (H2′′ ) that |b(x, z)| ≤ K0 (x, |z|) = Λ(|z|)K
Clearly, |b(t, x, z)| ≤ K0 (x, |z| + M) , so that (H2), (H9) hold for (P ) . Besides, one has
to show that
|Φt (t, x, z)| ≤ εK1 (t)B(s, x, z) + K2ε (t, x)
(20)
for all z ∈ IRN , a.a. t, s ∈ (0, T ) and a.a. x ∈ Ω , where K1 ∈ L1 (0, T ) and K2ε (t, x) ∈
L1 (Q) , and that
B(x, z + h(t, x)) ≤ B(t, x, z) + K2 (t, x)
(21)
for all z ∈ IRN and a.a. (t, x) ∈ Q , where K2 ∈ L∞ (0, T ; L1(Ω)) .
First, by (H2′′ ) and (7) with δ = ε/Λ(2M) one has
e 0 (x, 2M) ≤ Λ(2M)|b(s, x, z)|+
|b(x, z + h(t, x))| ≤ Λ(2M)|b(x, z + h(s, x))| + K
e 0 (x, 2M) ≤ εB(s, x, z) + K0 (x, ε/Λ(2M) + M) + K
e 0 (x, 2M).
+K
Hence it follows from (19), with
Φt ≡ 0 , that (20) holds with K1 (t) = k∂/∂t h(t,
·)kL∞ (Ω)
ε
e 0 (x, 2M) .
and K2 (t, x) = |∂/∂t h(t, x)| K0 (x, M) + K0 (x, ε/Λ(2M) + M) + K
Further, (21) holds, because by (16) and (7) with δ = 1 one has
e 0 (x, M + 1). ⋄
B(x, z + h(t, x)| ≤ B(t, x, z) + |h(t, x)| |b(t, x, z)| ≤ (1 + M)B(t, x, z) + K
3.2. On the structure conditions
Let us turn now to the relevancy of the hypotheses (H11) −(H13) . First note one direct
generalization of Theorem 1 and Corollaries 1,2,3.
2.I.3.
Comments and further results
117
Remark 5 An additional term of the form fˆ(·, v) can be considered in the right-hand side
of (P ) , provided fˆ : Q × IRN 7→ IRN is Caratheodory and satisfy
(
(fˆ(t, x, z) − fˆ(t, x, ẑ)) · (z − ẑ) ≤ 0,
(H14)
′
|fˆ(t, x, z)|p ≤ C |z|p + K1 (t) B(t, x, z) + K2 (t, x)
for all z, ẑ ∈ IRN , for a.a. (t, x) ∈ Q , with some K2 ∈ L1 (Q) and C = const ≥ 0 . For
(P ) modified in such a way, (H) and (9) modified correspondingly, and (H14) admitted,
Theorem 1 and Corollary 1 still hold. Indeed, in this case we can include the terms −fˆn (t, x, ·)
into the operators An , and An remain monotone. Therefore we do not need a structure
condition of the kind (H11) on fˆ .
Besides this remark, it is easy to see that in case the dependence on z of f and a is
weak, (H11) and (H12) can be superfluous while proving the existence of a weak solution.
Indeed, for p = 2 consider the model problem
b(v)t = ∆v + div F (v) on Q = (0, T ) × Ω
(MP)
with F : IRN 7→ IRN which is globally Lipschitz continuous, and with the homogeneous
Dirichlet boundary condition (so that V = W01,p (Ω; IRN ) ). The operator A : η ∈ L2 (0, T ; V )
7→ L2 (0, T ; V ′ ) defined by
Z T
ZZ
< Aη, ϕ >=
{Dη : Dϕ + F (η) · Dϕ}
0
Q
for all ϕ ∈ L2 (0, T ; V ) is monotone provided the diameter of Ω is small, due to the Poincaré
inequality. Therefore we can use Lemma 7 with An ≡ A while passing to the limit in a
sequence of approximating problems (MPn ) with bn bilipschitz, and with solutions vn such
that vn ⇀ v in L2 (0, T ; V ) . This yields that Avn ⇀ Av in L2 (0, T ; V ′ ) , but also that
vn → v strongly in L2 (0, T ; V ) ; so one can pass to the limit in each term in (MP ) and
infer the existence.
For scalar problems of type (MP ) , an existence result for arbitrary locally Lipschitz
continuous F has been shown in [BW99]. The methods used in this paper are those of the
nonlinear semigroup theory; unfortunately, this gives no information on the type of convergence
of approximating solutions to the weak solution of (MP ) .
Practically, (H11) , (H12) often hold in problems motivated by physics. For instance, in
the Richards equation (cf. [Bear]) v and u represent the liquid pressure and the saturation of
the medium, respectively; the nonlinearity in the right-hand side of the corresponding equation
is the permeability of the medium which is a function of u . On the other hand, the condition
(H13) is proper to the function b(·) and seems to impose restrictions on its dependence on
t . While the author was not able to prove that the representation
(
e x, b(t, x, z)) + z · ϕ(t,
Φt (t, x, z) = φ(t,
e x, b(t, x, z))
(22)
N
for all z ∈ IR and a.a. (t, x) ∈ Q
118
Continuous Dependence for Elliptic-Parabolic Problems
eϕ
with φ,
e Caratheodory generically holds, the following result fixes the situation in the scalar
case.
Lemma 3 Let N = 1 . Assume that bt is Caratheodory. Then (22) holds, and one can take
ϕ
e such that for all z ∈ IRN and a.a. (t, x) ∈ Q one has ϕ(t,
e x, b(t, x, z)) = bt (t, x, z) .
Proof: The dependence on x is immaterial here, and we neglect it in the notation.
Let first prove that, for a.a. t0 ∈ (0, T ) , if Φ(t0 , ·) is affine in an interval [z, z] (i.e.,
b(t0 , ·) degenerates) it follows that its derivative with respect to t is also affine on [z, z] .
Indeed, assume that this property fails on some subset B of (0, T ) with |B| > 0 . For each
t0 ∈ B there exist two rational points z̃, ẑ ∈ [z, z] such that the function
∆(t0 , z̃, ẑ; ·) = (z̃ − ẑ)(Φ(t0 , ·) − Φ(t0 , ẑ)) − (· − ẑ)(Φ(t0 , z̃) − Φ(t0 , ẑ))
is identically zero on [z, z] , while the continuous function
∇(t0 , z̃, ẑ; ·) = (z̃ − ẑ)(Φt (t0 , ·) − Φt (t0 , ẑ)) − (· − ẑ)(Φt (t0 , z̃) − Φt (t0 , ẑ))
is non-zero at some rational point z0 ∈ (z, z) . Still ∇(t0 , z̃, ẑ; z0 ) = ∆t (t0 , z̃, ẑ; z0 ) . Hence
S
B ⊂ Bz̃,ẑ,z0 , where Bz̃,ẑ,z0 = {t0 ∈ (0, T ) | ∆(t0 , z̃, ẑ; z0 ) = 0, but ∆t (t0 , z̃, ẑ; z0 ) 6= 0} ,
and the union is taken over all z̃, ẑ, z0 ∈ Q
I . Since ∆(·, z̃, ẑ; z0 ) is absolutely continuous on
[0, T ] , |Bz̃,ẑ,z0 | = 0 for all z̃, ẑ, z0 ∈ Q
I , which is a contradiction.
Note that for t0 ∈
/ B , one has bt (t0 , ·) = (Φt (t0 , z̃) − Φt (t0 , ẑ))/(z̃ − ẑ) = const on
[z, z] . Hence we have shown that for a.a. (t, x) ∈ Q , all z, z ∈ IRN one has


 bt (t, z) = bt (t, z) and
b(t, z) = b(t, z) ⇒
(23)
Φt (t, z) − Φt (t, z) = (z − z)bt (t, z)


for all z between z and z.
This is equivalent to the statement of the lemma. Indeed, (23) means that bt (t, ·) is in
fact a function of b(t, ·) : bt (t, z) = ϕ(t,
e b(t, z)) . The function ϕ(t,
e ·) is continuous. Indeed,
let βn , β ∈ R(b(t, ·)) and βn → β as n → ∞ . Due to the monotony of b(t, ·) , there
exist zn ∈ b(t, ·)−1 (βn ) , z ∈ b(t, ·)−1 (β) such that zn → z . Hence ϕ(t,
e βn ) = bt (t, zn ) →
bt (t, z) = ϕ(t,
e β) as n → ∞ . Furthermore, by (23), Φt (t, ·) − z ϕ(t,
e b(t, ·)) stays constant
e b(t, ·)) ,
as b(t, ·) degenerates. By the same argument, it can be written under the form φ(t,
e ·) continuous.
with φ(t,
⋄
3.3. On pure Neumann boundary conditions for one or more components
The restriction |Γ0 | > 0 , imposed in Sections 1,2, can be relaxed in two ways. The first
one consists in imposing hypotheses that provide a priori estimates on kDvkLp (Q) and
kB(v)kL∞ (0,T ;L1 (Q)) , and another hypothesis that permits to control |v| by |B(v)| . This
requires some modification of our arguments, and we do not pursue this line here. On the
2.I.4.
Appendix
119
other hand, some cases can be included by requiring directly the coercivity of the elliptic part
in the space V for all t ∈ (0, T ) ; more precisely, it is sufficient (in case h ≡ 0 ) that, instead
of (H6) and the second part of (H8) , the following assumption be fulfilled:
Z n
o
a(t, ·, w(·), Dw(·)) : Dw(·) − f (t, ·, w(·)) · w(·) + Φt (t, ·, w(·)) −
Ω
N Z
X
−
gi (t, ·, w(·)) wi(·) ≥ ckwkV − K1 (t) B(t, ·, w(·)) − K1 (t)
i=1
Γ1,i
for all w ∈ V and a.a. t ∈ (0, T ) , where K1 ∈ L1 (0, T ) and c > 0 . In this case, Lemma 2
is immediate, and other arguments run without any modification.
3.4. On the convergence of approximate methods
Note that Theorem 1 can be used to prove the convergence while solving (P ) by various
approximate methods. One can almost directly apply the theorem to Galerkin approximations.
Indeed, the equation for Galerkin approximations is (P ) itself with an error term that tends
′
to zero weakly in Lp (0, T ; V ′ ) , which still permits to apply the Minty-Browder argument
for convergence. For finite volumes methods, an example is given in [BAGuW] and Chapter
2.II, where the convergence for a class of schemes is proved for a simple model ellipticparabolic system. The steps of the proof of Theorem 1 are the same, provided the system that
determines the discret solution is rewritten under the form of equation in D ′ similar to (P ) .
Approximations by time implicit discretization (at least, with uniform step) can be treated in
a similar way; in this case writing the system that determines the solution under the form of
equation in D ′ is trivial.
Appendix
Here we present the appropriate versions of the three essential arguments involved in the proof
of Theorem 1.
A1. The chain rule argument
Let use the shortened notation w(t) for w(t, x) , b(w)(t) for b(t, x, w(t, x)) , Φt (w)(t) for
Φt (t, x, w(t, x)) , Φt (τ, w(t)) for Φt (τ, x, w(t, x)) and so on. Let start with two auxiliary
results.
Lemma 4 For k > 0 , let Tk : z ∈ IRN 7→ min{|z|, k} z/|z| ∈ IRN . Assume that b satisfy
(H1) and Φ is absolutely continuous on [0, T ] for all z ∈ IRN and a.a. x ∈ Ω . Then one
has (omitting the dependence in x )
Z t
B(t, Tk (z)) ≤ B(s, ẑ) + (b(t, z) − b(s, ẑ)) · Tk (z) −
Φt (τ, Tk (z)) dτ
s
N
for all z, ẑ ∈ IR , t, s ∈ [0, T ] , x ∈ Ω .
120
Continuous Dependence for Elliptic-Parabolic Problems
Proof: By the convexity of Φ in z , one has
B(t, Tk (z)) − B(s, ẑ) = B(s, Tk (z)) − B(s, ẑ) + B(t, Tk (z)) − B(s, Tk (z)) ≤
≤ (b(s, Tk (z)) − b(s, ẑ)) · Tk (z) + b(t, Tk (z)) · Tk (z) − b(s, ẑ) · Tk (z)−
Z t
−Φ(t, Tk (z)) + Φ(s, Tk (z)) = (b(t, Tk (z)) − b(s, ẑ)) · Tk (z) −
Φt (τ, Tk (z)) dτ.
s
Since b is monotone and z −Tk (z) = (|z|/k −1)+ Tk (z) , it follows that b(t, Tk (z)) · Tk (z) ≤
b(t, z) · Tk (z) .
⋄
Lemma 5 Let φ : Q × IRN 7→ IR be Caratheodory. Assume that for all z ∈ IRN , for a.a.
(t, x) ∈ Q one has
|φ(t, x, z)| ≤ F (t, x, |z|) with F (·, x, r) ∈ L1 (0, T ) for all r > 0.
(24)
Z t
h
For h 6= 0 , set φ (t, x, z) = 1/h
φ(τ, x, z) dτ , where φ(·, z) is extended by 0 outside
t−h
Q . Assume that v : Q 7→ IRN is measurable and for all h one has
|φh (v)(·)| ≤ F (·) for all h, where F ∈ L1 (Q).
(25)
Then φh are Caratheodory and φh (v) → φ(v) in L1 (Q) .
Proof: By (24), for all h > 0 , for all bounded set K in IRN , |φh (·, z)| are dominated
by a fixed function in L1 (Q) for all z ∈ K . Therefore φh are Caratheodory.
By Remark 3 it follows that φ, φh are strongly measurable from Q to C(IRN ) . Moreover,
φh → φ a.e. on Q in C(IRN ) . Therefore for all sequence hk → 0 one can apply the Lusin
and Egorov theorems to φhk . Thus for all α > 0 there exists an open subset Qα of Q
with |Qα | < α such that φhk → φ in C(Q \ Qα ; C(IRN )) = C(Q \ Qα × IRN ) .
By (25), φhk (v) are equiintegrable on Q , so that it is sufficient to prove the lemma in
case v(·) is bounded on Q by some constant M and φhk , φ are equicontinuous in (t, x, z)
on Q × [−M, M]N . Let approximate v a.e. on Q by a sequence of functions vm each
taking a finite number of values in [−M, M]N . By the diagonal process, one can extract a
subsequence hkn such that φhkn (vm ) → φ(vm ) a.e. on Q for all m ∈ IN . Take (t, x) in
the set where this convergence takes place. For each m fixed, one has
|φhkn (v) − φ(v)| ≤ |φhkn (v) − φhkn (vm )| + |φhkn (vm ) − φ(vm )| + |φ(vm ) − φ(v)|,
which can be made as small as desired by a choice of m and n large enough. Since the
sequence hk → 0 is arbitrary, the assertion of the lemma follows by the Lebesgue theorem.
⋄
Proof of Lemma 1: Take v0 measurable such that b(v0 ) = u0 , and extend v(t) by v0
to t < 0 . One has b(v)t = −χ ≡ 0 for t < 0 .
2.I.4.
Appendix
121
First, take t ∈ (0, T ) , h > t − T and apply Lemma 4 to s = t − h , z = v(t) ,
ẑ = v(t − h) . Denote Tk (v(·)) by vk (·) ; note that kvk (t)kV ≤ kv(t)kV . After integration
in x , one obtains
Z
Z
B(vk )(t) ≤ B(v)(t − h)+
Z
Z t Z
(26)
+ (b(v)(t) − b(v)(t − h)) · vk (t) −
Φt (τ, vk (t)) dτ.
t−h
By (ii1 ) of Definition 1 and (H3) we deduce
Z T Z
Z
Z Tn Z t
1
B(v)(t) − B(v)(t − h) dt ≤ const |h|
kχ(τ )kV ′ dτ kv(t)kV +
0
0
Z t
Z
Z
Zh t t−h
Z
o
1
1
p
ε
+
K1 (τ ) dτ
B(v)(t) +
|v(t)| +
K2 (τ ) dτ dt ≤
h t−h
h t−h
n
≤ const |h| kχkLp′ (0,T ;V ′ ) kvkLp (0,T ;V ) + kK1 kL1 (0,T ) ×
o
×kB(v)kL∞ (0,T ;L1 (Ω)) + kvkpLp (Q) + kK2ε kL1 (Q) ≤ const |h|.
Z
Z
1,1
Therefore
B(v) ∈ W (0, T ) . In particular, there exists l = lim B(v)(h) . We have
h↓0
Z
to show that l =
B(v0 ) . Take t = 0 , z = v0 , s = h , ẑ = v(h) in Lemma 4. By
Definition 1 (ii1 ) and (H3) one has
Z
Z
Z h
Z hZ
B(v)(h) ≥ B(Tk (v0 )) −
(u0 − b(v)(h)) · Tk (v0 ) −
Φt (τ, Tk (v0 )) dτ ≥
0
0
Z
Z h
Z h
≥ B(Tk (v0 )) −
kχ(τ )kV ′ dτ kTk (v0 )kV −
K1 (τ ) dτ ×
0
Z 0h
×kB(0, Tk (v0 ))kL1 (Ω) − kTk (v0 )kpLp (Q) −
kK2ε (τ )kL1 (Ω) dτ.
As h ↓ 0 and k → +∞ , one gets l ≥
Z
0
B(v0 ) by the Fatou lemma. The inverse inequality
can be obtained in a similar way from (26) with t = h .
To prove (8), it remains to show that
Z
Z
d
B(v) = − < χ(t), v(t) > − Φt (v)(t).
dt Ω
Ω
By (26), it suffices to show that there exists a sequence hn ↓ 0 such that
Z
1
(b(v)(t) − b(v)(t ∓ hn )) · v(t) =
hn
1
< b(v)(t) − b(v)(t ∓ hn ), v(t) >→ ± < χ(t), v(t) >
hn
in L1 (Q) , and
Z t Z
Z
1
Φt (τ, v(t)) dτ → ± Φt (v)(t)
hn t∓hn
122
Continuous Dependence for Elliptic-Parabolic Problems
′
in L1 (0, T ) . Since (b(v)(·) − b(v)(· ∓ hn ))/hn → ±χ(·) in Lp (0, T ; V ′ ) , the former
convergence is clear. Besides, the latter one can be proved by applying Lemma 5 to φ = Φt .
Indeed, (25) follows from (H3) ; (24) follows from (H3) and the continuity of B in (t, z) .
This ends the proof.
⋄
Remark 6 Formally, the formula (8) would make sense under the weaker version of (H3) :
|Φt (t, x, z)| ≤ ε K1 (t)B(t, x, z) + |z|p + K2ε (t, x).
(H3? )
for a.a. (t, x) ∈ Q and all z ∈ IRN . On the other hand, except in the proof above, we
actually need this weaker form whenever (H3) is used. At the present stage, it is not clear to
the author whether the analogue of Lemma 1 does hold when (H3) is replaced by (H3? ) .
A2. The compactness argument
In his paper [K69a] Kruzhkov has proved that, for a bounded weak solution of a general
evolution equation, the L1 -modulus of continuity in t can be estimated via the L1 -modulus
of continuity in x and an a priori bound on the right-hand side in some Sobolev space
L1 0, T ; W −m,1 . Here we present this lemma, with somewhat less restrictive assumptions,
as a kind of compactness result.
Lemma 6 Let Ω be an open subset of IRN , E be a set of indexes, and uε ∈ L1 ((0, T )×Ω ,
ε ∈ E , satisfy
X
∂ ε
u (t, x) =
(−1)α D α Aεα (t, x)
∂t
in D ′((0, T ) × Ω).
|α|≤m
Assume that for all compact set K ⊂ Ω the following estimates hold uniformly in ε :
(i) kuε kL1 ((0,T )×K) ≤ CK ;
(ii)
P
|α|≤m
kAεα kL1 ((0,T )×K) ≤ CK ;
(iii) for all h > 0 small enough
sup
|∆x|≤h
with limh→0 ωK (h) = 0 .
Z TZ
0
K
|uε(t, x + ∆x) − uε (t, x)|dxdt ≤ ωK (h) ,
Then the family {uε }ε∈E is relatively compact for the L1loc ((0, T ) × Ω) -topology.
Proof of Lemma 6: We only need to find a function ω
eK on IR+ such that lim ω
eK (τ ) = 0
τ →0
Z T −τZ
and I :=
|uε(t + ∆t, x) − uε (t, x)|dxdt ≤ ω
eK (τ ) for all ∆t ∈ [0, τ ] and ε ∈ E .
0
K
Indeed, the result will follow from (i),(iii) and the Kolmogorov theorem.
2.I.4.
Appendix
123
Let revise the proof from [K69a]. Fix ε ∈ E , ∆t ∈ [0, τ ] . For t ∈ [0, T − τ ] set
w (·) := uε (t + ∆t, ·) − uε (t, ·) . For all ϕ ∈ D(K) we have
Z
Z t+∆tZ X
t
w (x)ϕ(x)dx =
Aεα (θ, x)D α ϕ(x)dxdθ.
(27)
t
K
|α|≤m
K
t
For each t ∈ [0, T −τ ] take for the test function ϕ(·) a regularisation of sign w t (·) . More
precisely, let δ := dist(K, ∂Ω) and h ∈ (0, δ/2) . Set K−h = {x ∈ K : dist(x, ∂K) ≥ h}
and denote by χK−h (·) its characteristic function. Choose ρ ∈ C0∞ ([−1, 1]N ) , ρ ≥ 0 such
Z
Z x − y
t
−N
that
ρ(σ)dσ = 1 , and take in (27) ϕ (x) = h
ρ
sign w t (y)χK−h (y)dy . It
h
is clear that for |α| ≤ m , kD α ϕt kL1 ([0,T ]×K) ≤ const · h−m uniformly in t and ε .
Z T −τZ
Z T −τZ t
t
Note that I = I1 +I2 , where I1 =
w (x)ϕ (x)dxdt and I2 =
|w t (x)|−
0
K
0
K
P
t
t
−m
ε
w (x)ϕ (x) dxdt . Using (27) and (ii), we get |I1 | ≤ const· τ h
|α|≤m kAα kL1 ([0,T ]×K) ≤
nZ T
o
−m
CK τ /h . Besides, by (i),(iii) and the Kolmogorov theorem the family
|uε (t, ·)|dt
0
1
ε∈E
is relatively compact in L (K) . Therefore these functions are equiintegrable on K , so that
Z TZ
|uε (t, x)|dxdt ≤ ω̂K (h) with limh→0 ω̂K (h) = 0 uniformly in ε . Hence
0
K\K−2h
Z
T −τZ
|I2 | ≤ 2
|w t(x)|dxdt +
0
K\K−2h
Z T −τZ
Z
x−y
t
t
−N
+
sign w t (y)dydxdt ≤ 4ω̂K (h) +
|w (x)| − w (x) h ρ
h
0
K−2h
Z T −τ
Z
Z
x − y t
−N
+
h ρ
|w (x)| − w t (x) sign w t (y)dydxdt.
h
0
K−2h
Since for all a, b ∈ IR we have ||a| − a sign b| ≤ 2|a − b| , it follows that
Z T −τZ
Z
x−y
−N
|I2 | ≤ 4ω̂K (h) + 2
h ρ
|w t(x) − w t (y)|dydxdt ≤
h
0
K
Z
Z −2h
TZ
≤ 4ω̂K (h) + 4 ρ(σ)
|uε (t, x) − uε (t, x − hσ)|dxdtdσ ≤ 4ω̂K (h) + 4ωK (h).
0
K−2h
The function ω
eK (τ ) = min CK
0<h≤δ/2
τ → 0 , which ends the proof.
τ
hm
+ ω̂K (h) + ωK (h)
majorates I and tends to 0 as
⋄
A3. The Minty-Browder argument
Lemma 7 Let E be a Banach space, E ′ its dual and (·, ·) denote the duality product of
elements of E ′ and E . Take a sequence vn in E such that vn ⇀ v . Take a sequence
of monotone operators An : E 7→ E ′ such that An converge pointwise to some operator
∗
A : E 7→ E ′ and An vn ⇀ χ in E ′ .
Then χ = Av whenever A is hemicontinuous (i.e., continuous in the weak-∗ topology
of E ′ along each direction) and lim inf n→∞ (An vn , vn ) ≤ (χ, v) .
124
Continuous Dependence for Elliptic-Parabolic Problems
Proof of Lemma 7: The proof is standard (e.g., see [JLL]); we give it here for the sake
of completeness. Under the assumptions of Lemma 7, for all η ∈ E one has
(χ, v − η) ≥ lim inf (An vn , vn − η) ≥ lim inf (An η, vn − η) = (Aη, v − η).
n→∞
n→∞
Taking η = v + λζ with λ ∈ IR , ζ ∈ E and letting λ increase to zero, one gets (χ, ζ) ≥
(Av, ζ) . As λ decreases to zero, the inverse inequality follows, so that (χ, ζ) = (Av, ζ) for
all ζ ∈ E .
⋄
CHAPTER 2.II
Convergence of Finite Volumes Approximations
for a Nonlinear Elliptic-Parabolic Problem:
a Variational Approach†
Introduction
Let Ω be an open bounded polygonal domain in Rd , d ≥ 1 and T > 0 . We consider the
initial boundary value-problem for a system of nonlinear elliptic-parabolic equations:


 b(v)t = div ap (Dv) on Q = (0, T ) × Ω,
(1)
v=0
on Σ = (0, T ) × ∂Ω,


0
b(v)(0, ·) = u
on Ω,
where 1 < p < ∞ and div ap (Dv) = div (|Dv|p−2Dv) is the N -dimensional p -Laplacian,
N ≥ 1 , i.e.,
ap : ξ = (ξ1 , . . . , ξN ) ∈ (Rd )N 7→ |ξ|p−2ξ =
P
p/2−1
j 2
|ξ
|
(ξ1 , . . . , ξN ) ∈ (Rd )N .
i,j i
We assume that

N
N

is continuous, monotone with b(0) = 0, i.e.,
 b:R →R
there exists a convex differentiable function Φ : RN → R


with Φ(0) = 0 such that b = ∇Φ,
(2)
(3)
and
u0 ∈ L1 (Ω)N
with Ψ(u0) ∈ L1 (Ω),
where Ψ is the Legendre transform of Φ given by
Z 1
N
Ψ : z ∈ R 7→ sup
(z − b(sσ)) · σ ds.
σ∈RN
†
0
The results of this chapter are being prepared upon publication [BAGuW]
(4)
126
Variational Approach for a Finite Volume Method
Systems of elliptic-parabolic equations of type (1) arise as a model of flow of (several) fluids
through porous media (cf. e.g. [Bear, DiDT94]). They have already been studied extensively
in the literature in the last decade from a theoretical point of view (cf. e.g. [ALpr, AL83,
Kac90, DiDT94, BW96, Ot96, Bou97, BW99, CaW99, BAB]). Existence of weak solutions of
general systems of elliptic-parabolic equations has been proved in [ALpr, AL83], using Galerkin
approximations and time-discretization. Similar results have been obtained later by other
authors using different methods (e.g., using a semigroup approach as in [BW96, Bou97] in the
case N = 1 ).
In particular, it is known that in the case of the system (1), for any u0 satisfying (4),
there exists a weak solution of (1), where the weak solution is defined as follows.
Definition 1 (Weak solution) A function v ∈ E = Lp (0, T ; W01,p(Ω))N is a weak solution
′
′
of the problem (1), if b(v) ∈ L∞ (0, T ; L1 (Ω))N , the function b(v)t ∈ E ′ = Lp (0, T ; W −1,p (Ω))N
(where p′ denotes the conjugate exponent of p ) satisfies
ZZ
< b(v)t , φ >E ′ ,E +
ap (Dv) · Dφ = 0
(5)
Q
for all φ ∈ E , where < ·, · >E ′ ,E denotes the duality pairing between E ′ and E , and
ZZ
Z
− < b(v)t , ξ >E ′,E =
b(v)ξt + u0 ξ(0)
(6)
Q
Ω
for all ξ ∈ E with ξt ∈ L∞ (Q)N and ξ(T ) = 0 .
Moreover, if v is a weak solution of (1), then, by the “chain rule” lemma of [AL83],
B(v) ∈ L∞ (0, T ; L1 (Ω))N , where
Z 1
N
B : z ∈ R 7→ b(z) · z − Φ(z) ≡
(b(z) − b(sz)) · zds ≡ Ψ(b(z)) ∈ R.
(7)
0
From the results of [Ot96, CaW99] it also follows that, in the scalar case N = 1 , there is
uniqueness of a weak solution of (1). To our knowledge, the question of uniqueness is open
in the case N ≥ 2 .
The variational approach of [AL83] for elliptic-parabolic problems has been revisited in
Chapter 2.I. Beyond an extension of different existence results, in Chapter 2.I a concise variant
of the techniques of [AL83], applied to prove continuity of weak solutions of general ellipticparabolic problems with respect to the data and coefficients, has been presented. In this
chapter we are interested in proving, with the same techniques, the convergence of approximations by finite volumes numerical schemes for the model nonlinear elliptic-parabolic problem
(1).
Finite volumes methods are well suited for numerical simulation of processes where extensive quantities are conserved, and it is a very popular method among engineers in hydrology
where systems of equations of this type arise. Therefore justification of convergence of this
2.II.0.
Introduction
127
numerical approximation process is of particular interest. In [EGH98] the finite volume methods has been studied and convergence of this approximation procedure has been proved for
problem (1) in the particular case p = 2 , N = 1 . The same method has also been studied
for this equation (i.e. p = 2, N = 1 ) in the presence of an additional convection term (cf.
[EGH99]), and for a nonlinear diffusion problem in [EGHNS98].
In this chapter we deal with convergence for time implicit finite volume approximations of
(1). Let us emphasize that our main object is not only to prove the convergence of the finite
volume method for (1), but to develop the variational approach for this proof. The main idea of
this adaptation is to rewrite the discrete finite volume scheme under an equivalent continuous
form and to apply known stability techniques for the continuous equation (cf. [AL83, BAB])
in order to get convergence of the discrete approximation scheme.
In Section 1, we introduce the finite volume scheme for (1). We specify a class of admissible
partitions T h of Ω and (T h , k h ) of Q (cf. Definitions 2 and 3) and define the approximate
discrete problems (P h ) ( h > 0 being the discretization parameter). To this end we have to
introduce a finite volume approximate D h of the gradient operator D . A class of admissible
discrete gradient approximations is defined (see (14),(16) and Definition 4 below) and an
example of an admissible gradient approximation is given (cf. Remark 2).
In Section 1 we also state the main result of the chapter: for any admissible family of
grids {T h , k h } and gradient approximations D h , there exists a sequence h → 0 such that
solutions vh of the discrete problems (P h ) (cf. (24)-(26)) converge weakly in L1 (Q)N to
a weak solution of (1).
In Section 2, existence of solutions v h of the discrete equations is established. Note that,
at least for special choices of the gradient approximation, it is also possible to prove uniqueness
of a discrete solution. We derive a discrete analog of Lp (0, T ; W01,p(Ω))N a priori estimate
for v h and obtain a bound of B(v h ) in L∞ (0, T ; L1(Ω))N .
Section 3 is devoted to the problem of rewriting the discrete scheme under the form of an
equivalent equation
ũht = div ap (Dv̂ h )
(8)
in D ′(Q) for appropriate approximations ũh ∈ L1 (Q)N of b(v h ) and Dv̂ ∈ Lp (Q)N of
Dhvh .
In Section 4 we give the proof of the main theorem. The proof is essentially based on
three arguments: the a priori estimates obtained by using the chain rule argument of [AL83], a
lemma of Kruzhkov (cf. [K69a]) to get strong compactness in the parabolic part, and a MintyBrowder argument (cf. e.g. [JLL]) to get convergence in the elliptic part of the continuous
form of the approximate equations.
128
Variational Approach for a Finite Volume Method
Sections 5 and 6 are devoted to the proof of several auxiliary results and technical details
used in the proof of the main theorem. In particular, in Section 5, in addition to the continuous approximation Dv̂ h of the discrete gradients D h vh defined in Section 3, we construct
auxiliary continuous approximations v h of the discrete solutions v h themselves. In this step,
it is convenient to impose the assumption (10) of proportionality of the mesh. Moreover, this
assumption is essentially used in the proof of the discrete Poincaré inequality (cf. Lemma 9 in
the Appendix) when p > 2 .
Section 7 is devoted to the proof of consistency of the finite volume approximation of
the elliptic operator in (1) in the sense of Definition 5 (cf. Section 7). Here we develop an
additional series of arguments. The restriction (iv) of Definition 4 (cf. Section 1) is used at
this stage.
In the Appendix we have collected several auxiliary results form the theory of Sobolev
spaces and their discrete analogues.
Throughout the chapter we keep the notations introduced in this section. Moreover, as an
abuse of notation, in the following sections, the spaces Lp (Q)N , Lp (Ω)N etc. are written
as Lp (Q) , Lp (Ω) etc.
1
The numerical scheme
In order to construct approximate solutions to Problem (1), we use in this section the implicit
discretization in time and a finite volume scheme in space.
First we introduce a notion of admissible mesh of Ω (see also [EyGaHe] and [EGH99]).
Definition 2 (Admissible mesh) Let Ω be an open bounded polygonal subset of Rd . An
admissible finite volume mesh T of Ω is given by : a family of open polygonal convex
subsets with positive measure of Ω called ”control volumes” (for the sake of simplicity, we
shall denote by T the family of control volumes); a family E of subsets of Ω contained in
hyperplans of Rd , with positive (d−1) -measure (these are the edges of the control volumes);
a family of points of Ω , where these families satisfy the following properties:
(i) The closure of the union of all the control volumes is Ω ;
(ii) For any (K, L) ∈ (T )2 with K 6= L , either the length of K ∩ L is 0 or K ∩ L = σ
for some σ ∈ E . Then we will denote σ = K|L .
(iii) For any K ∈ T , there exists a subset EK of E such that ∂K = K \ K = ∪σ∈EK σ .
Furthermore, E = ∪K∈T EK and we will denote by N (K) the set of boundary volumes
of K that is N (K) = {L ∈ T , K|L ∈ EK } .
(iv) The family of points (xK )K∈T is such that xK ∈ K (for all K ∈ T ) and, if
σ = K|L , it is assumed that the straight line (xK , xL ) is orthogonal to σ .
2.II.1.
Numerical scheme
129
In the sequel we will use the following notation. The size of the mesh (or space step)
is defined by: size(T ) := maxK∈T δ(K) , where δ(K) denotes the diameter of the control
volume K . For any K ∈ T and σ ∈ E , m(K) is the d -dimensional Lebesgue measure
of K and m(σ) the (d − 1) -dimensional Lebesgue measure of σ . The set of adjacent
couples (K, L) is denoted by Υ . The set of interior (resp. boundary) edges is denoted by
Eint (resp. Eext ), that is Eint = {σ ∈ E; σ 6⊂ ∂Ω} (resp. Eext = {σ ∈ E; σ ⊂ ∂Ω} ).
The set of external control volumes is denoted by Text . For all K ∈ T , L ∈ N (K) and
σ ∈ EK , we denote by xσ the orthogonal projection of xK on σ (thanks to assumption
(iv), this orthogonal projection is the same from xK or xL if σ = K|L ), by dK,L (resp.
dK,σ ) the Euclidean distance between xK and xL (resp. xσ ) and by νK (resp. νK,L
and νK,σ ) the outside unit normal to K (resp. with respect to L and with respect to σ ).
Remark that νK |σ = νK,σ for all σ ∈ EK and that thanks to assumption (iv), νK,L = νK,σ
if σ = K|L . Finally for all K ∈ T and σ ∈ EK we denote by S(K, σ) (resp. S(σ) ) the
”half-diamond” (resp. ”diamond”) associated to K and σ (resp. to σ ) that is the smallest
convex set that contains σ and xK (resp. S(σ) = S(K, σ) ∪ S(L, σ) ).
Further, let T be an admissible mesh in the sense of Definition 2. Together with k ∈
(0, T ) , it generates the space-time grid {((n − 1)k, nk) × K}K∈T ,n=1,...,[T /k]+1 , which we
denote by (T , k) . We define the grid step h by h := max{k, size(T )} . We denote by
QnK the space-time volumes, i.e., QnK = ((n − 1)k, nk) × K for all n = 1, . . . , [T /k] + 1
and all K ∈ T . The lateral boundary of QnK is denoted by ΣnK . In addition, we introduce
n
the notation ν = 1/2 minK,σ∈N (K) dK,σ and denote by CK
the cartesian product of some
√
fixed d -dimensional cube inside K , with edge 2ν/ d and center xK , and the interval
((n − 1)k, nk) . Furthermore, we will use the notation Υκ (QnK ) for the union of all spacetime volumes of (T , k) that are separated from QnK by at most κ ∈ N (space or time)
interfaces.
Definition 3 (Admissible families of meches and grids) A family of meshes {T } is admissible if each mesh is admissible and the following assumptions hold with some fixed M, ζ :
(v) there exists M such that
M ≥ max card(EK );
K∈T
(9)
(vi) there exists ζ > 0 such that
ζ size(T ) ≤ min dK,σ .
K∈T ,
σ∈EK
(10)
A family of space-time grids (T h , k h ) , parametrized with h ∈ (0, 1) , is admissible if the
family {T h } of space meshes is admissible and max{k h , size(T h )} ≤ h . In the sequel, we
write k for k h and (T h , k) for (T h , k h ) .
130
Variational Approach for a Finite Volume Method
Remark 1 While Definition 2 and assumption (9) impose standard assumptions on the grid
(cf. [EyGaHe]), the hypothesis (10) is a strong proportionality condition. In fact, for p ≤ 2
it is only a technical assumption which permits to prove Lemma 5 (cf. Section 5). Lemma 5
is a pure qualitative result, in the sense that no estimation on the functions v h constructed
in this lemma is used in the sequel, but only the fact of their existence. Elsewhere, we can
replace (10) by the usual (cf. [EyGaHe]) assumption
ζ ≤ min
K∈T h ,
σ∈EK
dK,σ
,
δ(K)
(11)
which should be the only condition taken into account from the numerical point of view.
Nevertheless, in the case p > 2 the restriction (10) is essentially used in the proof of the
discret Poincaré inequality (cf. the Appendix).
When using a finite volume method to approximate problem (1), we consider an approximate solution (if it exists) which is piecewise constant. Then we need to construct an
n
approximation of the gradient. Let us consider a set of values (vK
)K,n ⊂ RN (for the sake of
simplicity, we will make an abuse of notation by omitting K ∈ T h , n = 1, . . . , [T /k] + 1 )
and the dicrete solution defined by
n
v h |QnK = vK
.
(12)
We construct the approximation of the gradient in the following way. The normal component
of the ”gradient” of the discrete solution vh is approximated by the operator G⊥h defined by

h
n
m

 G⊥ : (vK )K,n 7→ (g⊥,σ )σ,m ,
(13)
vm − vm m

K ∈ [0, +∞)
 g⊥,σ = L
for
σ
=
K|L,
dK,L while the whole of the ”gradient” of the discrete solution vh on interfaces of the control
volumes is approximated by an operator G h defined by

n
 G h : (vK
)K,n 7→ (gσm )σ,m ,
(14)
 g m ∈ (Rd )N for all σ ∈ E, m = 1, . . . , [T /k] + 1.
σ
Then we extend G h to the whole of Q by the lifting operator Lh defined by

 Lh : (gσm )σ,m 7→ Lh ((gσm )σ,m ) ∈ Lp (Q),
m
 Lh ((g m) ) |
σ σ,m ((m−1)k,mk)×S(σ) = gσ
for all σ ∈ E, m = 1, . . . , [T /k] + 1.
(15)
We define the discrete gradient operator D h by
n
n
D h : (vK
)K,n 7→ (Lh ◦ G h )((vK
)K,n ) ∈ Lp (Q).
(16)
2.II.1.
Numerical scheme
131
It is convenient to extend the gradient approximation procedure to functions in E . Let
introduce the averaging operator Mh defined on E = Lp (0, T ; W 1,p(Ω))N by
ZZ
1
h
h
n
n
M : η 7→ M η = (ηK )K,n , ηK = n
η;
(17)
|CK | CKn
Then by an abuse of notation, we also write D h for the operator
D h : η ∈ E 7→ (Lh ◦ G h ◦ Mh )(η) ∈ Lp (Q),
(18)
h
and D⊥
for the operator
h
D⊥
: η ∈ E 7→ (Lh ◦ G⊥h ◦ Mh )(η) ∈ Lp (Q).
(19)
n
Taking in (17) averages over cylinders CK
is convenient for expressing the consistency of the
gradient approximation for affine in x functions (cf. (iv) in the definition below). The crucial
property is the symmetry with respect to the axis x = xK .
Definition 4 (Admissible family of discrete gradient operators) For a given family of
meshes {T h } , a family of corresponding discrete gradient operators {D h } is admissible, if
D h and the corresponding operators G h have the following properties:
(i) For each h , G h is linear, i.e.,
n
n
n
n
G h ((vK
+ wK
)K,n ) = G h ((vK
)K,n ) + G h ((wK
)K,n )
(20)
(ii) For each h , G h is consistent with G⊥h , i.e.,
m
m
m
gK|L
νK,L = sign(vLm − vK
) g⊥,K|L
for all (K|L) ∈ Υ, m = 1, . . . , [T /k] + 1.
(21)
(iii) The family D h is uniformly local, i.e., there exists κ ∈ N independent of h such that
n
for all K ∈ T h , all n = 1, . . . , [T /k] + 1 and all set of values (vK
)K,n of RN , there
exists a constant C which only depends on p , d and M , ζ in (R2 ) , (R3 ) such
that
h n
D ((v )K,n ) p n ≤ C D h ((v n )K,n ) p
.
K
⊥
K
L (Q )
L (Υκ (Qn ))
K
K
(22)
(iv) For each h , G h is consistent with affine functions. More exactly, assume that, for
K ∈ T h and n ∈ N given, there exists a constant c ∈ (Rd )N and a function w ∈ E
RR
such that Dw ≡ c on Υκ (QnK ) and that vLl = |C1n | C n w whenever QlL ⊂ Υκ (QnK ) .
K
K
Then
gσn = c for all σ ∈ EK .
(23)
132
Variational Approach for a Finite Volume Method
Below we provide an example of gradient approximation that complies with the properties
(i)-(iv) above. For simplicity, we restrict our attention to the 2D case.
Remark 2 (Example of admissible discrete gradients in 2D) Let {T h } be a family of
admissible meshes of Ω in the sense of Definition 2. For T ∈ {T h } , for each σ0 ∈ E , let
σ1 , σ2 , σ3 and σ4 be the four adjacent edges (see figure (1)). Let x1 (resp. x2 ) be
the intersection point of σ0 with σ1 and σ2 (resp. σ3 and σ4 ). Then we construct the
approximate gradient on σ0 in the following way.
n
n
We take the standard values g⊥,σ
, g⊥,σ
of the normal components of the discrete
0
1
gradient on σ0 and σ1 , respectively. The edges are not colinear, so that there exists a
unique vector that has these values as projections on the normal directions to σ0 and σ1 ,
respectively. We draw this vector in x1 and denote it by gσn0 1 . In the same way, we can
reconstruct gσn0 2 (resp. gσn0 3 , gσn0 4 ) in x1 (resp. x2 ) from the normal components of the
gradients on σ0 and σ2 (resp. σ3 , σ4 ).
σ4
x2
σ3
gσn0 3
n
g⊥,σ
0
σ0
gσn0 4
gσn0
σ2
x1
gσn0 2
gσn0 1
σ1
n
g⊥,σ
1
Figure 1: Reconstruction of gradient on σ0
Then the approximate gradient on σ0 (or in S(σ0 ) , according to (16)) is given by
gσn0 =
1 n
(g + gσn0 2 + gσn0 3 + gσn0 4 ).
4 σ0 1
It is easy to show that, if the family {T h } is admissible in the sense of Definition 3, this
approximation of the gradient is admissible in the sense of Definition 4.
2.II.2.
Existence of a discrete solution and a priori estimates
133
We are now in order to write the scheme. The equation for the scheme is given by
n−1
n
X
b(vK
) − b(vK
)
n
m(K)
=
m(K|L) ap (gK|L
) νK,L
k
L∈N (K)
for all K ∈ T , n ∈ N. (24)
The initial condition is given by some values
0
u0K = b(vK
)
for all K ∈ T ,
(25)
and the homogeneous Dirichlet boundary condition is taken into account in the following way:
n
vK
=0
for all K ∈ Text , n ∈ N.
(26)
The discrete Problem (P h ) corresponding to a grid (T h , k) , where T h is an admissible
mesh of Ω in the sense of Definition 2, is given by the discrete equation (24), the initial
condition (25) and the boundary condition (26). In Section 2 below, we prove that there
exists a solution v h to the discrete Problem (P h ) .
The values (u0K )K will be chosen in order to comply, at the limit h → 0 , with the initial
condition in (1) and the restriction (4) on it, i.e.,
(
u0,h → u0
in L1 (Ω)N as h → 0,
(27)
Ψ(u0,h ) → Ψ(u0 ) in L1 (Ω)
as h → 0,
where u0,h |K = u0K . Our main result is the following theorem.
Theorem 1 (Convergence) Let (T hm , k hm )m∈N be a sequence of admissible grids in the
sense of Definition 3 such that hm = max k hm , size(T hm ) < 1 , hm → 0 as m → +∞ .
Assume that (3),(4), and the analogue of (27) hold. For each m ∈ N , let (vhm )m∈N
be a discrete solution to the problem (P hm )m∈N , where (D hm )m∈N is an admissible family
of discrete gradient operators in the sense of Definition 4. Then there exists a subsequence
(hmk )k∈N , hmk → 0 as k → ∞ , such that vhmk ⇀ v in L1 (Q) as hmk → 0 , where
v ∈ E = Lp (0, T ; W01,p(Ω))N is a weak solution of the problem (1) in the sense of Definition 1.
In the sequel, we will omit subscripts in sequences (hm ) , (hmk ) .
2
Existence of a discrete solution and a priori estimates
We will repeatedly use the following remark.
Remark 3 (Discrete integration by parts) Let T be an admissible mesh of Ω in the
sense of Definition 2. Let (vK )K∈T ⊂ RN and (FK,L )(K,L)∈Υ ⊂ RN be real values such
that vK = 0 for all K ∈ Text and FK,L = −FL,K for all (K, L) ∈ Υ . Then
X
X
X
vK
FK,L =
(vK − vL ) FK,L .
(28)
K∈T
L∈N (K)
(K,L)∈Υ
134
Variational Approach for a Finite Volume Method
Now we can state the result for existence of a discrete solution.
Theorem 2 (Existence) Let (T , k) be a space-time grid, where T is an admissible mesh
of Ω in the sense of Definition 2. Let D h be a discrete gradient operator having the properties
(i)-(iii) of Definition 4. Assume that (3) holds; then there exists a solution vh to the discrete
problem (P h ) .
Remark 4 (Uniqueness) While uniqueness of a waek solution of the problem (1) itself for
N ≥ 2 seem to be an open problem, for special choices of the gradient approximation we can
also prove the uniqueness of a solution to the discrete problem (P h ) .
n−1
Proof of Theorem 2: Fix n ∈ {1, . . . , [T /k] + 1} . Assume that the values (vK
)K∈T
N cardT
n
are already found. We denote by V the vector of (R )
whose entries (vK )K∈T satisfy
the condition (26). Let us consider the operator S that associates to a “vector” V the
“vector” given by (24), i.e.,


n−1
n
X
b(v ) − b(vK )
n
S(V) = m(K) K
−
m(K|L) ap (gK|L
) · νK,L 
.
k
L∈N (K)
K∈T
We are looking for a solution to the equation S(V) = 0 . Consider the scalar product
(S(V), V) in (RN )cardT . We have
X
X
n−1
1
n
n
n
m(K) b(vK
) · vK
−1
m(K) b(vK
) · vK
−
k
k
K∈T
K∈T
X X
(29)
n
n
−
m(K|L) vK
ap (gK|L
)νK,L = 0.
K∈T L∈N (K)
In view of hypothesis (3), we have that the first term on the left-hand side of (29) is nonnegative. Since all the norms are equivalent on (RN )cardT , for the second term on the
left-hand side of (29) we have
1X
1X
n−1
n
n−1
m(K) b(vK
) · vK
≤ |V|
m(K) b(vK
) = C |V| ;
k K∈T
k K∈T
here and in the sequel of the proof, C denotes a positive constant independent of V , and
|V| denotes the euclidean norm of V . We then handle the last term on the left-hand side of
(29), which we denote by AV . Using the discrete integration by parts (Remark 3), we obtain
that
X
n
n
−AV =
m(K|L) (vLn − vK
) ap (gK|L
)νK,L .
(K,L)∈Υ
In view of the definition (2) of ap and thanks to hypothesis (21), we obtain
−AV =
X
(K,L)∈Υ
n 2
n p−2 (vLn − vK
)
m(K|L) gK|L
.
dK,L
(30)
2.II.2.
Existence of a discrete solution and a priori estimates
135
For p ≥ 2 , we have in view of (21) that
n p−2
n p−2 n
p−2 vLn − vK
g ,
≥ g⊥,K|L
= K|L
dK,L which together with (30) yields
−AV ≥
X
m(K|L) dK,L
(K,L)∈Υ
n
n p
vL − vK
dK,L .
Thanks to the discrete Poincaré inequality (cf. Lemma 9 in the Appendix), we finally obtain
−AV ≥ α
X
K∈T
n p
m(K) |vK
| ≥ C |V|p ,
thanks to the equivalence of the norms on (RN )cardT .
For 1 0 such that, for all
(K, L) ∈ Υ ,
n ≤ |V| .
C gK|L
Together with (30), this yields
−AV ≥ C |V|
p−2
X
m(K|L) dK,L
(K,L)∈Υ
n
n 2
vL − vK
dK,L .
Thanks to the discrete Poincaré inequality (cf. Lemma 9 in the Appendix) for p = 2 , we
finally obtain
−AV ≥ C |V|p−2
X
K∈T
n 2
m(K) |vK
| ≥ c |V|p
with some constant c > 0 independent of V , thanks to the equivalence of the norms on
(RN )cardT .
Returning to (29), we obtain for all 1 0 such
that
(S(V), V) ≥ c |V|p − C |V| ≥ 0,
for |V| large enough. Therefore in view of the Brouwer fixed point theorem (e.g., cf. [JLL,
Lemme 4.3]), there exists a solution to S(V) = 0 , i.e., there exists a solution to (24) with
n−1
condition (26) for (vK
)K given.
⋄
136
Variational Approach for a Finite Volume Method
Proposition 1 Let (T h , k) be an admissible family of grids in the sense of Definition 3.
Let {D h } be a family of discrete gradient operators satisfying (ii) and (iii) of Definition 4.
n
Assume that (3),(4), and (27) hold. Then for all family (vK
)K,n of solutions of the discrete
h
problem (P ) there exists a constant C which only depends on p , Ω , T and kΨ(u0 )kL1 (Ω)
such that
ZZ
h h p
D v ≤ C,
(31)
Q
and
X
K∈T
h
n
m(K) B(vK
) ≤ C,
for all n = 1, . . . , [T /k] + 1.
(32)
i
Proof : Take i ∈ {1, . . . , [T /k] + 1} and multiply each term in (24) by vK
. By (26),
using the discrete integration by parts (Remark 3) and (21), one gets
X
i
i−1
i
m(K)(b(vK
) − b(vK
)) · vK
+
X
+k
(K,L)∈Υ
K∈T h
i p−2 i
i
m(K|L) gK|L
− vLi | = 0.
g⊥,K|L |vK
i−1
i−1
i
i
i
By the convexity of Φ , one has (b(vK
) − b(vK
)) · vK
≥ B(vK
) − B(vK
) . Summation over
i from 1 to n ∈ {1, . . . , [T /k] + 1} , and recalling the definition of the discrete gradient and
its normal component, we infer
Z nkZ X
h h p−2 h h 2 X
n
m(K)Ψ(u0K ).
(33)
m(K)B(vK ) + d
D v D⊥ v ≤
0
K∈T h
Ω
K∈T h
p−2 h h 2
Note that, by (21), for 1 < p < 2 the value D h vh D⊥
v can be set zero whenever
h h
D⊥
v is zero. Therefore the integral in (33) always makes sense.
Now (32) follows directly from (27) and (33). As to (31), there are two cases. For p ≥ 2 ,
one has by (21) and (33)
ZZ ZZ h h p
h h p−2 h h 2
D⊥ v ≤
D v D⊥ v ≤ const.
Q
Q
For 1 < p < 2 , using the “inverse” Hölder inequality with the exponents p/2 < 1 and
p/(p − 2) < 0 , one gets
Z Z 2/p Z Z (2−p)/p Z Z h h p
h h p
h h p−2 h h 2
≤
D v D v D⊥ v .
D⊥ v Q
Q
In the two cases, it follows by (22) that
(22) again, one obtains (31).
Q
RR h h p
D v ≤ const independently of h . Using
Q ⊥ ⋄
2.II.3.
3
Rewriting discrete equations under continuous form
137
Rewriting discrete equations under continuous form
Let vh be the discrete function (12) produced by the finite volume scheme (24)-(26) on a grid
T h , k of size h . We will replace the discrete gradient D h v h by a function Dv̂ h ∈ Lp (Q)
eh ∈ L1 (Q) so that (24) is equivalent to (8), i.e.,
and the function b(v h ) by a function u
ũht = div ap (Dv̂ h )
in D ′(Q) . This representation plays the key role in proving the convergence result of Theorem 1.
Define u
eh as the piecewise affine in t approximation of b(v h ) :
t − kn
n
n−1
(b(vK
) − b(vK
))
(34)
k
Besides, for given K, n and a set Anσ
⊂ (Rd )N , let AnK be defined a.e. on ∂K by
n
u
eh (t, x)|QnK = b(vK
)+
σ∈EK
AnK |σ = Anσ for all σ ∈ EK . Let νK be the exterior unit normal vector to ∂K . Consider
the following Neumann problem in the factor space W = W 1,p (K)/R :
Z

 div a (Dw) = 1
An νK on K
p
m(K) ∂K K
(35)

n
ap (Dw)νK |∂K = AK νK .
′
n
Lemma 1 Let AnK ∈ Lp (∂K) . Then there exists a unique distribution solution ŵK
to (35)
in W . This solution gives the global minimum to the functional
Z
Z
Z
1
1
p
n
L : w ∈ W 7→ Lw = kDwkLp (K) −
wAK νK +
w
AnK νK .
(36)
p
m(K)
∂K
K
∂K
R
1/p
Proof : Supply W with the norm kwkW = K |Dw|p
(we will ambiguously denote
by the same symbol an element of W 1,p (K) and the corresponding equivalence class as
element of W ). Consider the operator A : w ∈ W 7→ Aw ∈ W ′ defined by
Z
′
< Aw, ϕ >W ,W =
ap (Dw) : Dϕ
(37)
K
for all ϕ ∈ W , and the functional f ∈ W ′ defined by
Z
Z
Z
1
n
< f, ϕ >W ′ ,W = −
ϕ
AK νK +
ϕAnK νK
m(K) K
∂K
∂K
(38)
for all ϕ ∈ W . Note that f is well defined, since the right-hand side of (38) is invariant
under translation by a constant in W 1,p (K) .
The operator A in (37) is bounded, hemicontinuous, strictly monotone on W , and
R
|Dw|p
< Aw, w >W ′,W
K
=
= kwkp−1
W → ∞
kwkW
kwkW
138
Variational Approach for a Finite Volume Method
as kwkW → ∞ . Thus A is bijective (cf. e.g. [JLL, Chapitre 2,Théorème 2.1]).
Besides, the functional L is well defined, convex on W , and
Z
1
1
p
n
Lw ≥ kDwkLp (K) − kAK kLp′ (∂K) w −
w
≥
p
m(K) K Lp (∂K)
1
≥ kDwkpLp (K) − constkDwkLp (K)
p
by Lemmae 12,11 (cf. the Appendix). Hence Lw → +∞ as kwkW → ∞ , so that L
attains its global minimum on W (cf. e.g. [Br, Corollaire III.20]). Using the relation of the
p-laplacian with the Lp norm, by the standard variational argument we find out that this
n
minimum is attained at the unique solution ŵK
of the equation Aw = f in W ′ .
⋄
n
Let v̂K
be the solution of (35) with
AnK = ap (gσn ) for σ ∈ EK ,
(39)
n
where gσn are taken from (14), and v̂K
is normalized by assigning
1
m(K)
Z
K
n
n
v̂K
= vK
.
(40)
We introduce the discrete-continuous approximation of D h vh by setting
n
Dv̂ h (t, x)|QnK = Dv̂K
(t, x).
(41)
Let us also define v̂ h by
n
v̂ h (t, x)|QnK = v̂K
(t, x).
(42)
Clearly, we have u
eht = div ap (Dv̂ h ) pointwise on QnK . Note that Dv̂ h is only the pointwise
gradient of v̂ h , while the gradient of v̂ h in the sense of distributions contains Dirac masses
concentrated on grid edges. Nevertheless, one has the following result.
n
Proposition 2 (The continuous form of (24)) Assume that (vK
)K,n verifies (24). Let
h
h
h
u
e and Dv̂ be defined by (34) and (41), respectively. Then ũt = div ap (Dv̂ h ) holds in
D ′(Q) .
Proof : By the local conservativity of the scheme, the fluxes are continuous on the
space interfaces: AnK|L (x) = AnL|K (x) for all x ∈ K|L , for all (K, L) ∈ Υ , all n ∈
2.II.4.
Proof of Theorem 1
139
{1, . . . , [T /k] + 1} . Moreover, u
eh is continuous on the time interfaces. Therefore, for all
ph ∈ D(Q) ,
ZZ
XZ Z
h
h
(e
u · ϕt − ap (Dv̂ ) : Dϕ) =
(e
uh · ϕt − ap (Dv̂ h ) : Dϕ) =
Q
Qn
K
K,n
Xn Z Z
X Z kn Z
h
h
=
−
(e
ut − div ap (Dv̂ )) · ϕ +
ϕAnK|L νK,L +
n
QK
K,n Z
L∈N (K) k(n−1) K|L
Z
o
h
h
+
ϕ(kn, ·) · u
e (kn, ·) −
ϕ(k(n − 1), ·) · u
e (k(n − 1), ·) =
K
K
Z
Z
kn
X X
=
ϕAnK|L (νK,L + νL,K )+
k(n−1) K|L
n (K,L)∈Υ
Z
Z
h
+
ϕ(T, ·) · u
e (T, ·) −
ϕ(0, ·) · u
eh (0, ·) = 0,
K
K
so that (8) holds.
⋄
In addition to this result, it will be useful to have in hand some “Lp (0, T ; W01,p(Ω)) version” of v h for each h . We will call v h a family of continuous in x approximations
of v h , if
vh ∈ E
with kv h kE ≤ constkD h vh kLp (Q)
(43)
with a constant independent of h ,
kv h − v h kL1 (Q) → 0 as h → 0,
(44)
and
1
km(K)
ZZ
h
v =
Qn
K
n
vK
1
= n
|CK |
ZZ
vh.
(45)
n
CK
The existence of such approximations is proved in Lemma 5 (cf. Section 5) for the case where
(10) holds.
4
The proof of Theorem 1
In Chapter 2.I, in the context of continuous dependence upon the data of weak solutions to
“general” elliptic-parabolic problems, the convergence proof for weak solutions of approximating problems has been reduced to the three essential arguments:
(A) a priori estimates, by the chain rule argument of Alt-Luckhaus (cf. [AL83, Ot96,
CaW99, BAB]);
(B) strong compactness in the parabolic term, by the Kruzhkov lemma (cf. [K69a] and
Chapter 2.I, Lemma 6);
140
Variational Approach for a Finite Volume Method
(C) convergence in the elliptic term, by the Minty-Browder argument (cf. e.g. [Mi62, Mi63,
Bro63, JLL] and Chapter 2.I, Lemma 7).
Here we will take advantage of the “continuous” form (8) of the system (24) and pass to the
limit in (a subsequence of) vh as h → 0 by applying the same arguments to, respectively,
(A) v h , the continuous in x approximations of vh ;
(B) u
eh , the piecewise affine in t approximations of b(v h ) ;
(C) Dv̂ h , the discrete-continuous approximations of D h vh .
Proof of Theorem 1: We will repeatedly refer to results contained in Sections 5-7
below. The proof consists of three steps.
(A) Let vh be a family of solutions to the family of discrete problems (P h ) . By Lemma 5
(cf. Section 5) there exist v h ∈ E satisfying (43)-(45). In particular, by Proposition 1,
kv h kE ≤ const uniformly in h . Hence there exists a subsequence h → 0 and a function
v ∈ E such that v h ⇀ v in E as h → 0 . By (44), one also has v h ⇀ v ∈ L1 (Q) (cf.
also Remark 5).
(B) We claim that the family {e
uh } is relatively compact in L1 (Q) . Indeed, extend u
eh
by zero on (R × Rd ) \ Q . Let us check the following three conditions:
(i)
{e
uh } is bounded in L1 (Q) ;
(ii)
{ap (Dv̂ h )} is bounded in L1 (Q) ;
(iii) for all ∆ > 0 small enough, one has
ZZ
sup
|e
uh (t, x + ∆x) − u
eh (t, x)| dxdt ≤ ωx (∆)
|∆x|≤∆
(46)
Q
uniformly in h , where ωx (∆) → 0 as ∆ → 0 .
In order to prove (i), note that
|b(z)| ≤ δB(z) + sup |b(ζ)|
(47)
|ζ|≤1/δ
holds for all δ > 0 (cf. e.g. [AL83]). Hence we have by (34),(25)
ZZ
ZZ
X
X
h
h
0
ke
u kL1 (Q) ≤ 2
|b(v )| + k
m(K)|uK | ≤
B(v h ) + const + k
m(K)|u0K |,
Q
K∈T h
Q
which is bounded uniformly in h , by Proposition 1 and (27).
As to (ii), one has
h
kap (Dv̂ )kL1 (Q) =
ZZ
Q
1/p
|Dv̂ h |p−1 ≤ kDv̂ h kp−1
≤ const
Lp (Q) |Q|
K∈T h
2.II.4.
Proof of Theorem 1
141
by the a priori estimate on kDv̂ h kLp (Q) , which is proved in Lemma 2 (cf. Section 5).
The estimate (46) of (iii) is proved in Lemma 6, as a consequence of the a priori estimate
(31) (cf. Section 6).
Now we may conclude by Lemma 6 from Chapter 2.I that
ZZ
sup
|e
uh (t + ∆t, x) − u
eh (t, x)| dxdt ≤ ωt (∆)
|∆t|≤∆
(48)
Q
uniformly in h , where ωt (∆) → 0 as ∆ → 0 . Thus there exists a subsequence h → 0 and
a function u ∈ L1 (Q) such that u
eh → u in L1 (Q) and a.e. on Q . Besides, we have to
establish that u = b(v) , where v is the weak limit of v h in E . The proof, which follows
the idea of [BrSt73], is given in Lemma 7 (cf. Section 6).
(C) First note that u
eh → b(v) in L1 (Q) , so that u
eht → b(v)t in D ′ (Q) . Moreover, by
(8) ke
uht kE ′ = kap (Dv̂ h )kLp′ (Q) = kDv̂ h kp−1
Lp (Q) , which is bounded by Lemma 2 (cf. Section 5).
Therefore {e
uht } is weak- ∗ relatively compact in E ′ .
It follows that there exists a subsequence, which we abusively denote by h → 0 , such that
(i)
v h ⇀ v in E ;
(ii)
−e
uht ⇀ −b(v)t in E ′ .
∗
Moreover, for all h one has
(iii) u
eht = Ah v h , where v h is constructed in Lemma 5 and Ah is the operator that maps
′
η ∈ E to −div Ah η ∈ E ′ , with Ah : E 7→ Lp (Q) defined in (51) below.
Indeed, let us define the finite volume approximate Ah · of ap (D·) . For η ∈ E and all
space-time volume QnK , set
n
D η̂ h (t, x)|QnK = D η̂K
(x),
(49)
n
n
h
h
n
where η̂K is the unique solution to the problem (35) with AK |σ = ap ((G ◦ M )η)σ for
n
all σ ∈ EK . It is convenient to normalize η̂K
by assigning
Z
ZZ
1
1
n
η̂ =
η.
(50)
m(K) K K
k m(K) QnK
Here Mh and G h are the averaging operator and the gradient approximation operator,
respectively, defined by (17) and (14), respectively. Assign
Ah η = ap (D η̂ h ).
(51)
n
From (45) we have G h ◦ Mh v h = G h (vK
)K,n . Therefore, Ah v h = ap (Dv̂ h ) with Dv̂ h
defined in (41), so that (8) yields (iii).
142
Variational Approach for a Finite Volume Method
Arguing as in the proof of Proposition 2, one gets
ZZ
ZZ
h
h
< A η, ϕ >E ′ ,E =
ap (D η̂ ) : Dϕ =
Ah η : Dϕ
Q
(52)
Q
for all ϕ ∈ E . In particular, it follows that Ah is monotone for all h .
Now we apply Lemma 7 from Chapter 2.I to Ah and A : E 7→ E ′ defined by Aη =
−div ap (Dη) , or, equivalently, by
ZZ
< Aη, ϕ >E ′ ,E =
ap (Dη) : Dϕ
(53)
Q
for all ϕ ∈ E . Note that A is hemicontinuous. Two more assumptions of Lemma 7
(Chapter 2.I) have to be checked:
(iv) lim inf h→0 < −e
uht , v h >E ′ ,E ≤ < −b(v)t , v >E ′,E ;
(v) for all η ∈ E , Ah η → Aη in E ′ .
In fact the property (v) expresses the consistency in E ′ of the finite volume approximation
of the operator −div ap (D·) on the space E . We prove in Theorem 3 (cf. Section 7) that
(v) actually holds under the assumptions of Theorem 1.
Before proving (iv), let us show that the initial condition (6) holds. From (34) we have
ZZ
ZZ
X
h
h
0
< −e
ut , ζ >E ′ ,E = −
u
et · ζ =
m(K)uK · ζ(0, ·) +
u
eh · ζt
(54)
Q
Q
K∈T h
for all ζ ∈ E with ζt ∈ L∞ (Q) and ζ(T, ·) = 0 . By (27), we can pass to the limit in (54)
and obtain (6). Now we can apply the usual chain rule argument (cf. [AL83, Lemma 1.5])
and deduce that B(v) ∈ L∞ (0, T ; L1(Ω)) and
Z
Z
< −b(v)t , v >E ′ ,E = − B(v)(T ) + Ψ(u0 ).
(55)
Ω
Ω
(without loss of generality, we assume T to be a Lebesgue point of kB(v)kL1 (Ω) (·) ). Besides,
by (8),(34),(45), and the monotonicity of b(·) , one has
ZZ
1X
h h
n
n−1
< −e
ut , v >E ′ ,E = −
(b(vK ) − b(vK )) ·
vh =
n
k K,n
QK
=
X
K∈T h
[T /k]+1
X
n
n−1
n
) − b(vK
)) · vK
≤
m(K)(b(vK
n=1
X
X
[T /k]+1
−
m(K)B(vK
)+
m(K)Ψ(u0K ).
K∈T h
K∈T h
Together with (55),(27) and the Fatou lemma, this yields (iv).
We are now in position to conclude that −b(v)t = Av in E ′ , so that (5) also holds.
Thus v is a solution of (1) in the sense of Definition 1.
⋄
Remark 5 In fact, one could replace in the requirement (44) the space L1 (Q) by the space
Lp (Q) , and show that v h ⇀ v in Lp (Q) .
2.II.5.
5
Two kinds of continuous approximations
143
Two kinds of continuous approximations
In this section we prove two auxiliary results concerning the approximations Dv̂ h and v h of
D h v h and vh , respectively. We also establish a uniform estimate on the space translates of
v h in Lq (Q) , 1 ≤ q ≤ p .
n
Lemma 2 Let (T h , k) be an admissible family of grids, and (vK
)K,n be a solution of
h
h
(P ) . Assume that the family of discrete gradient operators D satisfy (ii),(iii) of Definition 4, and let Dv̂ h be defined by (41). Then kDv̂ h kLp (Q) ≤ const uniformly in h .
n
Proof : For all K, n fixed, the function v̂K
∈ W 1,p (K) satisfies (35) in D ′ (K) . Take
n
v̂K
for the test function; it follows by (39) that
Z
Z
1 Z
X
1
n
n
n
n p
m(σ)
v̂ −
v̂
ap (gσ )νK,σ .
|Dv̂K | =
m(σ) σ K m(K) K K
K
σ∈E
K
Multiplying and dividing each term of the sum in the right-hand side by dK,σ /d , integrating
in t over (k(n − 1), kn) and summing over K, n , we obtain by the Hölder inequality
X X 1
1/p′
h p
n p′
kDv̂ kLp (Q) ≤ d
k m(K)dK,σ |ap (gσ )|
×
d
K,n σ∈EK
1 R n
R n p
(56)
1
X X 1
m(σ) σ v̂K − m(K) K v̂K 1/p
×
k m(K)dK,σ .
d
dK,σ
K,n σ∈EK
According to (18),(15) and (45), the first term in the right-hand side of (56) equals
Z Z
1/p′
h h p−1 p′
h h p−1
(|D v̂ | )
= kD h v̂ h kp−1
Lp (Q) = kD v kLp (Q) ,
Q
which is bounded by Proposition 1. Besides, by Lemma 10 (cf. the Appendix) the second
term in the right-hand side of (56) is estimated by the value
X
1/p
n p
n
const
kDv̂K kLp (Qn )
= constkDv̂K
kLp (Q) .
K
K,n
Hence (56) yields
kDv̂ h kpLp (Q) ≤ constkDv̂ h kLp (Q)
with a constant independent of h , which completes the proof.
⋄
n
Lemma 3 Let (T h , k) be an admissible family of grids, (vK
)K,n be a solution of (P h ) ,
and v h be defined by (12) on Q and extended by zero on (R+ × R) \ Q . For x, ∆x ∈ Rd ,
let Bx,x+∆x be the broken line that joins the centers of successive space mesh volumes crossed
by [x, x + ∆x] ∩ Ω , and lx,x+∆x be the length of Bx,x+∆x ∩ Ω . Then for 1 ≤ q ≤ p ,
ZZ
|v h (t, x + ∆x) − v h (t, x)|q dxdt ≤ ωx(q) (|∆x|)
(57)
Q
(q)
(q)
uniformly in h , with ωx : R+ 7→ R+ such that ωx (∆) → 0 as ∆ → 0 . Moreover, one
(q)
can take ωx (∆) = const ∆ (l(∆))q−1 , where l(∆) = supx∈Rd sup∆x∈Rd ,|∆x|≤∆ lx,x+∆x .
144
Variational Approach for a Finite Volume Method
S
Proof : Take x, ∆x ∈ Rd and let Bx,x+∆x = {(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø} [xK , xL ] , where
xK , xL are the centers of the volumes K, L , respectively. Since
X
lx,x+∆x =
dK,L ,
{(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø}
convexity of the mapping y ∈ Rd 7→ |y|q yields
h
h
q
|v (t, x + ∆x) − v (t, x)| ≤ (lx,x+∆x)
q−1
X
{(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø}
n
vK − vLn q
dK,L dK,L for t ∈ (k(n − 1), kn) . For (K, L) ∈ Υ fixed, denote by ΩKL (∆x) the set of x ∈ Rd
such that [x, x + ∆x] ∩ (K|L) 6= Ø . Clearly, ΩKL (∆x) is a prism of measure less or equal
than |∆x|m(K|L) . Hence
n
ZZ
X
vK − vLn q
h
h
q
q−1
|ΩKL (∆x)| ≤
|v (t, x + ∆x) − v (t, x)| ≤ (l(|∆x|))
kdK,L d
K,L
Q
K,n
ZZ
h h q
|D⊥
≤ d|∆x|(l(|∆x|))q−1
v | ≤ const|∆x|(l(|∆x|))q−1
Q
by Proposition 1 and the Hölder inequality.
⋄
For q > 1 , this result will be used together with the simple lemma below.
Lemma 4 Let {T h } be a family of admissible meshes of Ω ⊂ Rd parametrized by h =
size(T h ) such that (10) holds. Let x, ∆x ∈ Rd and Bx,x+∆x , lx,x+∆x , l(·) be defined as
in Lemma 3. Then l(∆) ≤ const (∆ + 2h) , where const only depends on d and on ζ in
(10).
∆x
Proof : Let Cx,x+∆x be the cylinder in Rd of radius h , with the segment [x−h |∆x|
, x+
∆x
∆x + h |∆x| ] for the axis. If the segment [x, x + ∆x] crosses a control volume K ∈ T h ,
K is contained in Cx,x+∆x . Note that |Cx,x+∆x | = const (|∆x| + 2h) hd−1 . On the other
hand, for all K ∈ T h , |K| ≥ const (ζh)d , and the maximum length of Bx,x+∆x ∩ K is
2h . Therefore l(|∆x|) is estimated by 2h times the number of control volumes contained
hd−1
in Cx,x+∆x , i.e., l(|∆x|) ≤ 2h const (|∆x|+2h)
, which concludes the proof.
⋄
const hd
Remark 6 In fact, we prove in Lemma 3 that for any mesh T admissible in the sense of
Definition 2, for all (wK )K ⊂ RN given, and w defined by w|K = wK , the norm of
w(· + ∆x) − w(·) in Lq (Ω) is estimated by dC |∆x|(l(|∆x|))q−1 , where C is the discrete
W01,q -norm of w . As it is shown in Lemma 4, for a family {T h } this estimate can be
improved to const dC |∆x|(|∆x| + 2h)q−1 , with const independent of h = size(T h ) ,
provided one imposes the restriction of proportionality (10) on {T h } .
Moreover, calculating more carefully |ΩKL (∆x)| (cf. e.g. [EyGaHe, EGH98]), we can
prove this last estimate in case 1 ≤ q ≤ 2 without the restriction (10).
2.II.5.
Two kinds of continuous approximations
145
n
Lemma 5 Let (T h , k) be an admissible family of grids, (vK
)K,n be a solution of (P h ) ,
and v h be defined by (12). Then there exists a family {v h } ⊂ E such that (43)-(45) hold.
Proof : We first convolute vh in x with a special mollifier, and then restore the average
over each mesh volume.
Let ν = 1/2 minK∈T h ,σ∈EK dK,σ . By (10) we have
h/ν ≤ const.
(58)
d
dist (x, S(0, 1)) , where S(0, r) , B(0, r) are the (d − 1) Take ρ : x ∈ Rd 7→ |B(0,1)|
dimensional sphere and the d -dimensional ball of centre 0 and radius r , respectively. Let
Ων = {x ∈ Ω | dist (x, ∂Ω) > ν} and χΩν be the characteristic function of Ων . We set
1 x
v h,ν = d ρ
∗ (v h χΩν )
(59)
ν
ν
and construct v h as
X
αK ϕ K .
v h = v h,ν +
K∈T
(60)
h
K
π x−x
. Here π : Rd 7→ R is a function with the
Here ϕK (t, x) = ϕK (x) = m(K)
νd
ν
R
properties supp π = {x ∈ Rd | 1 ≤ |x| ≤ 2} , π ≥ 0 , π = 1 , π ∈ C ∞ (Rd ) ; xK is the
center of K , and
ZZ
1
n
n
v h,ν .
(61)
αK (t, x)|QnK = αK = vK −
k m(K) QnK
Note that ϕK ∈ E ; by (10) and (58) |ϕK | ≤ const and |DϕK | ≤ const/h with const
RR
1
independent of h . Moreover, k m(K)
ϕK = 1 . In addition, by the choice of ν each
Qn
K
control volume K ∈ T h contains a ball of radius 2ν . Hence, by (59),(60) and the definition
n
of π , v h |CKn ≡ v h,ν |CKn = vK
. Therefore v h verify both equalities in (45).
By Lemma 3, (v h (·+∆x, ·)−v h (·, ·)) vanish in L1 (Q) uniformly in h and ∆x ∈ B(0, ν)
as ν goes to zero. A fortiori, the same holds with the function v h replaced by v h χΩν .
Since ν → 0 as h → 0 , by the usual property of convolution regularisations (cf. e.g. [Br,
Théorème IV.22]), kv h,ν − vh χΩν kL1 (Q) → 0 as h → 0 . Moreover, it follows by the Hölder
inequality, the discrete Poincaré inequality (cf. Lemma 9 in the Appendix), and Proposition 1
that
Z TZ
Z Z
1/p 1/p′
1/p′
h
h p
|v | ≤
|v |
T |Ω \ Ων |
≤ const |Ω \ Ων |
→0
0
Ω\Ων
Q
as h → 0 . Therefore
kv h,ν − vh kL1 (Q) → 0
(62)
146
Variational Approach for a Finite Volume Method
as h → 0 . Finally,
h
kv − v
h,ν
kL1 (Q) =
X
n
|αK
|
ZZ
Qn
K
ϕK =
Z Z
ZZ
ZZ
1
=
vh −
v h,ν ϕK ≤
QnK
n
n
k
m(K)
Q
Q
K
K
K,n
XZ Z
≤ const
|vh − v h,ν | = const kvh − v h,ν kL1 (Q) → 0
X
K,n
Qn
K
K,n
as h → 0 . Therefore (44) holds.
It remains to prove (43). First, let us estimate v h,ν in E . It is convenient to write
Dρ(·) in the spherical coordinates; indeed, if |x| = r and x/|x| = eθ ∈ S(0, 1) , we
have Dρ(x) = const eθ χB(0,1) a.e. on Rd . Let S + (0, r) = {x ∈ S(0, r) | x1 > 0} and
B + (0, r) = {x ∈ B(0, r) | x1 > 0} . Denote v h χΩν by f . Separating the two hemispheres
S + (0, 1) and S(0, 1) \ S + (0, 1) , we find
Z
x − y 1 |Dv (t, x)| = d+1 Dρ
f (t, y) dy =
ν
ν
Z
1 σ
= d+1 Dρ
f (t, x − σ) dσ =
ν
ν
B(0,ν)
Z
Z ν
const d−1
= d+1 eθ
f (t, x − reθ )r
drdθ =
ν
S(0,1)
0
Z
Z ν
const |f (t, x + reθ ) − f (t, x − reθ )|r d−1 drdθ ≤
= d+1 ν
S + (0,1) 0
ZZ
Z ν
const
≤ d+1
|f (t, x + σ) − f (t, x − σ)| dσ.
ν
B + (0,ν) 0
h,ν
Therefore by the Hölder inequality one has
ZZ
const
|Dv | ≤ p(d+1)
ν
Q
Z
−d−p
≤ const ν
h,ν p
ZZ Z
′
Q B + (0,ν)
B + (0,ν)
≤ const ν −p sup
|σ|≤ν
ZZ
ZZ
Q
Q
|f (t, x + σ) − f (t, x − σ)|p (const ν d )p/p dσdxdt ≤
|v h (t, x + σ) − vh (t, x)|p dxdtdσ ≤
|v h (t, x + σ) − v h (t, x)|p dxdt.
Hence by Lemma 3, (58) and Lemma 4 we finally deduce that kDv h,ν kLp (Q) ≤ const uniformly in h .
Now we are able to estimate (v h − v h,ν ) in E . In fact, we have
ZZ
h
Q
|D(v − v
h,ν
p
)| =
X
K,n
n p
|αK
|
ZZ
Qn
K
|DϕK |p ≤
const X
n p
k m(K)|αK
|.
hp K,n
(63)
2.II.6.
Compactness result for the parabolic term
147
Moreover,
Z Z
p
1
1
n p
h,ν
p/p′
h |αK
| =
(v
−
v
)
≤
(k
m(K))
×
(k m(K))p QnK
(k m(K))p
ZZ
ZZ
h,ν
1
p
h p
×
v −v ≤
const δ(K)
|Dv h,ν |p ,
n
n
k
m(K)
QK
QK
(64)
since Dvh ≡ 0 on QnK . The inequality in (64) holds, because, as we have already seen,
n
v h,ν |CKn ≡ vK
≡ vh |CKn , and the Poincaré inequality for (v h,ν − vh ) in K therefore holds
with const independent of h, ν . Substituting (64) into (63) and using (58), we obtain the
desired estimate and prove (43).
⋄
6
The compactness result for the parabolic term
In this section we prove the uniform estimate (46) on space translates of u
eh in L1 (Q) ,
starting from the result of Lemma 3 in Section 5. Then we identify the function u = limh→0 u
eh
in L1 (Q) (for a subsequence) with b(v) , where v = weak − limh→0 v h in E .
n
Lemma 6 Let (T h , k) be an admissible family of grids, (vK
)K,n be a solution of (P h ) ,
and u
eh be defined by (34). Then (46) holds.
Proof : It is a modification of the corresponding proofs in [AL83] and Chapter 2.I.
First note that
ZZ
Z
h
h
|e
u (t, x + ∆x) − u
e (t, x)| dxdt ≤ k |u0,h (x + ∆x) − u0,h (x)| dx+
Q
Ω
ZZ
h
h
+2
|b(v (t, x + ∆x)) − b(v (t, x))|.
(65)
Q
By (27), the set {u0,h } is compact in L1 (Q) , so that
Z
sup
|u0,h (x + ∆x) − u0,h (x)| dx ≤ ω 0 (∆)
|∆x|≤∆
Ω
with ω 0 : R+ 7→ R+ such that ω 0(∆) → 0 as ∆ → 0 , uniformly in h . Moreover, note that
b(v h ) are equiintegrable on Q . Indeed, for all set F ⊂ Q one has by (47) and Proposition 1
ZZ
ZZ
h
|b(v )| ≤ inf δ
B(v h ) + C(δ)|F | ≤ inf const δ + C(δ)|F | = ω 1 (|F |),
F
δ>0
Q
δ>0
and ω 1 (|F |) → 0 as |F | → 0 .
h
Further, for M > 0 let us introduce RM
= {(t, x) ∈ Q||vh (t, x)| ≤ M, |v h (t, x+∆x)| ≤
M} . It follows from Proposition 1 and the discrete Poincaré inequality (Lemma 9 in the
h
Appendix) that kv h kL1 (Q) ≤ const uniformly in h . Hence |Q \ RM
| → 0 as M → +∞
uniformly in h , by the Chebyshev inequality. Let ωb,M (·) be the modulus of continuity of
148
Variational Approach for a Finite Volume Method
h
h
b(·) on [−M, M]N . Integrating separately over Q \ RM
and RM
in the last term in (65),
we get
ZZ
h
|e
uh (t, x + ∆x) − u
eh (t, x)| ≤ ω 0 (|∆x|) + 2ω 1 (|Q \ RM
|)+
Q
ZZ
+
ωb,M (|vh (t, x + ∆x) − v h (t, x)|) dxdt.
Rh
M
It follows by the concavity of ωb,M (·) and Lemma 3 that
ZZ
n
h
sup
|e
uh (t, x + ∆x) − u
eh (t, x)| dxdt ≤ inf
ω 0 (|∆x|) + 2ω 1 (sup |Q \ RM
|)+
M
>0,δ>0
h
|∆x|≤∆
Q
o
1 ZZ
+|Q| ωb,M
= ωx (∆),
|v h (t, x + ∆x) − v h (t, x)| dxdt
|Q| Q
and ωx (∆) → 0 as ∆ → 0 uniformly in h .
⋄
n
Lemma 7 Let (T h , k) be an admissible family of grids and (vK
)K,n be a solution of
h
h
h
1
(P ) . Assume that v ⇀ v in E and u
e → u in L (Q) for a sequence h → 0 , where
u
eh are defined by (34) and v h satisfy (43),(44). Then u = b(v) .
Proof : We claim that v h ⇀ v in L1 (Q) and b(v h ) → u in L1 (Q) , and then apply
the usual monotonicity argument (cf. [BrSt73]).
Since v h ⇀ v also in L1 (Q) , the first claim follows from (44). Further, let us show that
u
eh → u in L1 (Q) implies ke
uh − b(v h )kL1 (Q) → 0 as h → 0 . We have
XZ Z t
−
kn
h
h
n
n
n−1
b(v ) +
dxdt =
ke
u − b(v )kL1 (Q) =
(b(v
)
−
b(v
))
K
K
K
n
k
Q
K
K,n
Z
Z
(66)
X
1X
n
n−1
h
h
k m(K)|b(vK ) − b(vK )| ≤ 2
|e
u (t + k, x) − u
e (t, x)| dxdt.
=
2 K,n
Qn
K
K,n
The last inequality follows form the “geometrical” observation that

Z k

k|β − α| + k|γ − β| ≤ 4
|α + θ(β − α) − β − θ(γ − β)| dθ
0

for all k > 0, α, β, γ ∈ R.
(67)
Indeed, it is easily checked that for L = |β −α|+|γ −β| fixed, the minimum of the right-hand
side of (67) is attained at (β − α) = −(γ − β) and equals 14 kL . Recall (34); it is sufficient
n−1
n+1
n
to apply (67) to α = b(vK
) , β = b(vK
) , γ = b(vK
) in order to obtain (66). Note that,
h
1
by the compactness of {e
u } in L (Q) , (48) holds. Therefore the right-hand side of (66)
vanishes as h → 0 .
Without loss of generality, we can assume that b(v h ) → u a.e. on Q . For each ε > 0 ,
choose a set Rε ⊂ Q , with |Rε | < ε , such that v ∈ L∞ (Q \ Rε ) and b(v h ) → u in
L∞ (Q \ Rε ) . This is always possible, by the Chebyshev inequality and the Egorov Theorem.
2.II.7.
Consistency of the finite volume approximation
149
Consequently, we have b(v + λζ) → b(v) in L∞ (Q \ Rε ) as λ ∈ R tends to zero, for all
ζ ∈ L∞ (Q) . Then for all η ∈ L∞ (Q)
ZZ
ZZ
u · (v − η) = lim
b(v h ) · (vh − η) ≥
h→0
Q\Rε
Q\Rε
ZZ
ZZ
(68)
h
≥ lim
b(η) · (v − η) =
b(η) · (v − η).
h→0
Q\Rε
Q\Rε
The inequality in (68) is due to the monotonicity of b(·) :
RR
Q\Rε
(b(v h ) − b(η)) · (vh − η) ≥ 0 .
Now it is sufficient to take η = v + λζ with λ ↑ 0 and λ ↓ 0 in order to deduce that
RR
± Q\Rε (u − b(v)) · ζ ≥ 0 . Since ζ ∈ L∞ (Q) and ε > 0 are arbitrary, u = b(v) a.e. on
Q.
⋄
7
Consistency of the finite volume approximation
In this section we prove that, for an arbitrary function η ∈ E , the finite volume approximation
Ah η = −div Ah η defined by (52),(49) is a good approximation for Aη = div ap (Dη) in E ′ .
More exactly, we have the following definition.
Definition 5 (Consistent approximations) Let (T h ) be a family of admissible meshes,
parametrized by h → 0 , (T h , k h ) be the corresponding grids with max{size(T h ), k h } ≤
h , and {D h } be a family of discrete gradient operators. Let Ah : E 7→ E ′ be the operator
defined by (52),(49).
We say that Ah is the approximation of the elliptic operator A · = −div ap (D ·) corre
sponding to (T h , k h ) and {D h } . This approximation is consistent if for all η ∈ E one
has Ah η → Aη in E ′ as h → 0 .
7.1 Properties of finite volume approximations
of the elliptic term and the consistency
Theorem 3 (Consistency) Let the family of grids and the family of discrete gradient operators be admissible in the sense of Definitions 3 and 4, respectively. Then the corresponding
approximation of −div ap (D ·) is consistent.
The main ingredient of the proof is the following result. Recall that Υκ (QnK ) denotes
the union of all space-time volumes of (T h , k) that are separated from QnK by at most κ
(space or time) interfaces.
Proposition 3 Let (T h , k) be an admissible family of grids, and {D h } be an admissible
′
family of discrete gradient operators. Then the operators Ah : E 7→ Lp (Q) defined by
(51),(49) have the following properties:
150
Variational Approach for a Finite Volume Method
(i) The operators Ah are uniformly local, i.e., there exists a constant C , independent of
S
ni
h , such that for all η ∈ E , for all set H ⊂ Q such that H = m
i=1 QKi , one has
′
kAh ηkpLp′ (H) = kD η̂ h kpLp (H) ≤ C kDηkpLp (Υκ+1 (H)) ,
where Υκ+1 (H) =
Sm
ni
i=1 Υκ+1 (QKi )
.
(ii) The operators Ah are locally Hölder equi-continuous, i.e., for all R > 0 there exists a
constant C(R) , independent of h , such that
kAh η − Ah µkLp′ (Q) ≤ C(R)kη − µkαE
whenever kηkE ≤ R , kµkE ≤ R . Here α = 1/2 for p ≥ 2 and α = p/(p′)2 for
1<p≤2.
Proof of Theorem 3: We have to prove that kap (Dη) − Ah ηkLp′ (Q) → 0 as h → 0 .
Let us first prove the theorem for the case of η ∈ E that is piecewise constant in t and
piecewise affine in x . Let J ⊂ Q be the set of discontinuities of Dη . Clearly, J is of finite
S
d -dimensional Hausdorff measure Hd (J) . Let us introduce H h = {K,n | Υκ (Qn )∩J6=Ø} QnK .
K
Note that |H h | ≤ (κ + 1)h Hd (J) → 0 as h → 0 ; likewise, |Υκ+1(H h )| → 0 as h → 0 .
Therefore by (2) and Proposition 3(i) we have
ZZ
ZZ
ZZ
h p′
p
|ap (Dη) − ap (D η̂ )| ≤
|Dη| + C
|Dη|p → 0
Hh
Hh
Υκ+1 (H h )
as h → 0 . Besides, for all QnK such that QnK ∩ H h = Ø we have D η̂ h ≡ Dη on QnK .
Indeed, we have Dη ≡ const on Υκ+1 (QnK ) . Therefore D h η|QnK ≡ Dη = const by (23).
Hence Dw = Dη satisfies the boundary condition in (35); the equation is also satisfied, since
RR
R
1
h
div ap (D h η) ≡ 0 on QnK and k m(K)
a
(D
η)ν
=
a
(Dη)
ν =0.
n
p
K
p
Σ
∂K K
K
It follows that
h
kap (Dη) − A ηkLp′ (Q) =
Z Z
as h → 0 , which was our claim.
h
Hh
|ap (Dη) − ap (D η̂ )|
p′
1/p′
→0
Now let us approximate an arbitrary function η in E by functions µl that are piecewise
constant in t and piecewise affine in x . More exactly, there exists a sequence µl in E such
that µl → η in E and a.e. on Q as l → ∞ , and |Dµl |p are dominated by an L1 (Q)
function independent of l . We have
kap (Dη) − Ah ηkLp′ (Q) ≤ kap (Dη) − ap (Dµl )kLp′ (Q) +
+kap (Dµl ) − Ah µl kLp′ (Q) + kAh µl − Ah ηkLp′ (Q) .
(69)
2.II.7.
Consistency of the finite volume approximation
151
As l → 0 , the first term in the right-hand side of (69) converges to zero by the Lebesgue
dominated convergence theorem, independently of h . The second one converges to zero as
h → 0 , for all l fixed. Finally, by Proposition 3(i) and (ii), the third one converges to zero
as l → ∞ , uniformly in h . Hence the left-hand side of (69) can be made as small as desired
for h sufficiently small. This concludes the proof.
⋄
7.2 Proof of Proposition 3
The main ingredient is the following lemma.
Lemma 8 Let (T h , k) be an admissible family of grids, and {D h } be an admissible
S
ni
family of discrete gradient operators. Let η, µ ∈ E and H ⊂ Q such that H = m
i=1 QKi .
Then
(i) for all R > 0 there exists a constant C(R) , independent of h , such that
m
X
i=1
′
kap (D h η) − ap (D h µ)kpLp′ (Σni ) ≤
Ki
C(R)
min{p,p′ }
kDη − DµkLp (Υκ+1 (H))
h
whenever kηkE ≤ R , kµkE ≤ R ; here Υκ+1 (H) =
(ii) in case µ = 0 , one has
m
X
i=1
′
kap (D h η)kpLp′ (Σni ) ≤
Ki
Sm
i=1
Υκ+1 (QnKii ) ;
C
kDηkpLp (Υκ+1 (H))
h
with a constant C independent of kηkE .
Proof of Lemma 8: We consider separately the two cases 1 2 . Note
the following inequalities, valid for all y1 , y2 ∈ (Rd )N (cf. e.g. [DiDT94, Bou97]):

 |ap (y1 ) − ap (y2 )|p′ ≤ const |y1 − y2 |p , 1 < p ≤ 2;
(70)
 |ap (y1 ) − ap (y2 )|p′ ≤ const |y1 − y2 |p′ |y1 |(p−2)p′ + |y2|(p−2)p′ , p ≥ 2.
a) 1 < p ≤ 2 . We first claim that, i = 1, . . . , m , (70) and (20) yield
′
kap (D h η) − ap (D h µ)kpLp′ (Σni ) ≤
Ki
const h
kD (η − µ)kpLp (Qni ) .
Ki
h
Indeed, note that for arbitrary n and K , for all σ ∈ EK one has
Z nk Z
Z nk Z
d
h
p
|D (η − µ)| =
|D h (η − µ)|p
d
K,σ
(n−1)k σ
(n−1)k S(K,σ)
by (18); besides, 1/dK,σ ≤ ζ/h by (10).
(71)
152
Variational Approach for a Finite Volume Method
Combining (71) with (22), Corollary 1 (cf. the Appendix) and (9), we obtain
′
kap (D h η) − ap (D h µ)kpLp′ (Σni ) ≤
Ki
const h
const
p
≤
kD⊥ (η − µ)kLp (Υκ (Qni )) ≤
kDη − DµkpLp (Υ (Qni )) .
κ+1
K
Ki
h
h
i
Summing over i from 1 to m and using (9) once again, we get (i). Note that the constant
is independent of kηkE , kµkE , so that (ii) holds in this case.
b) p > 2 . As in case a), we get from (18) and (10)
m
X
i=1
m
′
kap (D h η) − ap (D h µ)kpLp′ (Σni ) ≤
Ki
′
const X
kap (D h η) − ap (D h µ)kpLp′ (Qni ) .
Ki
h i=1
Further, by (70) and the Hölder inequality with q = p/p′ and q ′ = (p − 1)/(p − 2) we get
m
X
i=1
p′/p
const X h
p′ /p
h
kap (D η) − ap (D
≤
kD η − D µkLp′ (Qni )
×
Ki
h
i
i=1
m
m
X
(p−2)/(p−1)
X
×
kD h ηkpLp′ (Qni ) +
kD h µkpLp′ (Qni )
≤
m
h
h
′
µ)kpLp′ (Σni )
K
Ki
i=1
≤
i=1
const p(p−2)/(p−1)
R
h
Ki
m
X
i=1
p′ /p
kD h η − D h µkLp′ (Qni )
Ki
p′ /p
.
As in case a), (i) follows by (22), Corollary 1 and (9):
m
X
kap (D h η) − ap (D h µ)kpLp′ (Σni ) ≤
m
X
kap (D h η)kpLp′ (Σni ) ≤
i=1
′
Ki
′
const p(p−2)/(p−1)
R
kDη − DµkpLp (Υκ+1 (H)) .
h
Moreover, in case µ = 0 we have directly
i=1
m
′
Ki
const X h p
const
kD ηkLp (Qni ) ≤
kDηkLp (Υκ+1 (H)) ,
Ki
h i=1
h
so that (ii) also holds, which ends the proof.
Proof of Proposition 3:
(i) Take a grid volume QnKii ⊂ H . We have
ni
Ah η|QnKi = ap (D η̂ h )|QnKi = ap (D η̂K
),
i
i
i
ni
where D η̂K
is the solution of the analogue of (35) with AnKii = ap (D h η)|∂Ki .
i
Let η h be the time average of η in each grid volume:
Z
1 nk
h
η (t, ·)|QnK =
η(τ, ·) dτ for all K, n.
k (n−1)k
Note that on each QnK , both η̂ h , η h do not depend on t . It follows from Lemma 1 that
ZZ
ZZ
1
1
h p
h
h
h p
kD η̂ kLp (Qni ) −
η̂ ap (D η)νKi ≤ kDη kLp (Qni ) −
η h ap (D h η)νKi .
n
n
Ki
Ki
p
p
ΣKi
ΣKi
i
i
⋄
2.II.7.
Consistency of the finite volume approximation
153
Summing over i from 1 to m , by the Hölder inequality we obtain
kD η̂ h kpLp (H)
≤
kDη h kpLp (H)
+p
m
X
i=1
h
kη̂ −
η h kpLp (Σni )
K
i
m
1/p X
i=1
kap (D
h
′
η)kpLp′ (Σni )
K
i
1/p′
(72)
in the sense of traces. Applying the Poincaré inequality and the imbedding theorem (cf.
Lemma 11 and Lemma 12 in the Appendix, respectively), we obtain
1
kη̂ h − η h kpLp (Σni ) ≤ const
δ(K)p kD η̂ h − Dη h kpLp (Qni ) +
Ki
Ki
δ(K)
+δ(K)p−1 kD η̂ h − Dη h kpLp (Qni ) ≤ const hp−1kD η̂ h − Dη h kpLp (Qni ) .
Ki
Ki
Note that kDη h kLp (H) ≤ kDηkpLp (H) . Therefore
m
X
i=1
kη̂ h − η h kpLp (Σni )
Ki
1/p
1/p
′
≤ const h1/p kD η̂ h kpLp (H) + kDηkpLp (H)
.
(73)
′
Taking into account Lemma 8(ii) and applying twice the Young inequality ab ≤ δap +C(δ)bp ,
valid for all a, b ∈ R+ , δ > 0 and the corresponding C(δ) , we get from (72),(73)
kD η̂ h kpLp (H) ≤ kDηkpLp (H) +
1/p
′
′
p/p′
+const h1/p kD η̂ h kpLp (H) + kDηkpLp (H)
h−1/p kDηkLp (Υκ+1 (H)) ≤
1
≤ kD η̂ h kpLp (H) + const kDηkpLp (Υκ+1 (H)) ;
2
whence (i) is immediate.
(ii) First note that, by (i), we can assume
kDηkLp (Q) , kD η̂ h kLp (Q) , kDµkLp (Q) , kD µ̂h kLp (Q) ≤ R
(74)
for some R > 0 , uniformly in h .
Fix a grid volume QnK . Taking (η̂ h − µ̂h ) as a test function in (35) written for η̂ h , then in
(35) written for µ̂h , subtracting the two identities and integrating in t over ((n − 1)k, nk) ,
we get
ZZ
ZZ
1
h
h
h
h
(ap (D h η) − ap (D h µ)) :
(ap (D η̂ ) − ap (D µ̂ )) : (D η̂ − D µ̂ ) = −
k m(K) ΣnK
Qn
ZZ
KZZ
(75)
:
(η̂ h − µ̂h ) +
(η̂ h − µ̂h )(ap (D h η) − ap (D h µ))νK .
Qn
K
Σn
K
Consider separately the two cases 1 2 . Note the following inequalities,
valid for all y1 , y2 ∈ (Rd )N (e.g., cf. [DiDT94, Bou97]):

′

|ap (y1 ) − ap (y2 )|p ≤ const (ap (y1 ) − ap (y2 )) : (y1 − y2 ), 1 < p ≤ 2;



′
|ap (y1 ) − ap (y2 )|p ≤
(76)
h
ip′ /2 h
i(2−p′ )/2




≤ const (ap (y1 ) − ap (y2 )) : (y1 − y2 )
|y1 |p + |y2 |p
, p ≥ 2.
154
Variational Approach for a Finite Volume Method
h
a) 1 < p ≤ 2 . This time, let us normalize η̂ , µ̂
h
on
QnK
so that
By (75) and (76), we obtain
ZZ
ZZ
Qn
K
(η̂ h − µ̂h ) = 0 .
′
|ap (D η̂ h ) − ap (D µ̂h )|p ≤
Q
X
1/p′ X
1/p′
p′
h
h
h
h p
≤ const
kap (D η) − ap (D µ)kLp′ (Σn )
kη̂ − µ̂ kLp (Σn )
.
K
K,n
K
K,n
As in the proof of (i), we find out that
X
K,n
′
kη̂ h − µ̂h kpLp (Σn ) ≤ const h1/p kD η̂ h − D µ̂h kpLp (Q) .
(77)
K
On the other hand, Lemma 8(ii) yields
X
K∈T h
′
kap (D η̂ h ) − ap (D µ̂h )kpLp (Σn ) ≤
K
C(R)
kDη − DµkpLp (Q) .
h
(78)
Substituting (77),(78) into (75) and taking into account (74), we deduce that
′
p/p′
′
kap (D η̂ h ) − ap (D µ̂h )kpLp′ (Q) ≤ C(R) h−1/p kDη − DµkLp (Q) ×
′
p/p′
×h1/p kD η̂ h − D µ̂h kLp (Q) ≤ C(R)kDη − DµkLp (Q) .
Thus Ah are locally Hölder equi-continuous with α = p/p′ 2 .
b) p > 2 . Using (76),(75),(74), and applying the Hölder inequality with q = 2/p′ ,
q ′ = 2/(2 − p′ ) , we infer
ZZ
Z Z
p′ /2
|ap (D η̂ ) − ap (D µ̂ )| ≤
(ap (D η̂ h ) − ap (D µ̂h )) : (D η̂ h − D µ̂h )
×
Q
Q
(2−p′ )/2
X
1/p′ ×p′ /2
p′
h p
h p
h
h
× |D η̂ | + |D µ̂ |
≤ C(R)
kap (D η) − ap (D µ)kLp′ (Σn )
×
h
h
p′
X
1/p×p′ /2
×
kη̂ h − µ̂h kpLp′ (Σn )
.
K
K,n
K
K,n
As in case a), we deduce
′
p′ /2
kap (D η̂ h ) − ap (D µ̂h )kpLp′ (Q) ≤ C(R)kDη − DµkLp (Q) .
Thus Ah are locally Hölder equi-continuous with α = 1/2 .
⋄
2.II.8.
Appendix
155
Appendix: auxiliary results
In this appendix, we give some useful auxiliary results. First, we prove a discrete version of
the Poincaré inequality for proportional meshes.
Lemma 9 (Discrete Poincaré inequality) Let T be an admissible mesh of Ω ⊂ Rd in
the sense of Definition 2, δ(Ω) = diam Ω , h = size(T ) , and ζ be defined by (10). Let
(wK )K be a set of values in RN such that wK = 0 for all K ∈ Text . Then there exists
α > 0 depending only on p , d and ζ such that
X
X 1
wL − wK p
1
p
p
.
m(K) |wK | ≤ δ(Ω)
m(K|L) dK,L α
d
d
K,L
h
K∈T
(K,L)∈Υ
Proof : We follow the proof of the Poincaré inequality for W01,p (Ω) . Without loss of
generality, one can assume that Ω ⊂ [0, δ(Ω)]d . As in the proof of Lemma 3, for all x ∈ Ω
consider the intersection of the segment [x, x + ∆x] with the control volumes of T , where
∆x is chosen to be (−δ(Ω), 0, . . . , 0) ∈ Rd . Let Bx,x+∆x be the corresponding broken line,
and lx,x+∆x its length; one has lx,x+∆x ≤ l(δ(Ω)) in the notation of Lemma 3. Define w
by w|K = wK . Since the boundary condition on w is zero, as in Lemma 3 we have
X
wK − wL p
p
p−1
.
dK,L |w(x)| ≤ l(δ(Ω))
dK,L {(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø}
This time, |ΩKL | ≤ δ(Ω) m(K|L) , where ΩKL = {x ∈ Rd | [x, x + ∆x] ∩ (K|L) 6= Ø} .
Hence
Z
X
wK − wL p
p
p−1
|ΩKL | ≤
|w| ≤ l(δ(Ω))
dK,L dK,L Ω
(K,L)∈Υ
X 1
wK − wL p
p−1
.
≤ δ(Ω)l(δ(Ω)) d
m(K|L) dK,L d
dK,L (K,L)∈Υ
Our claim now follows from Lemma 4.
⋄
Remark 7 As in Remark 6, a more careful estimate of |ΩKL | in the proof of Lemma 9 allows
to bypass (10), in case p ≤ 2 . For p = 2 , this yields the result of [YCGH, Remark 7].
Next, we prove another kind of Poincaré inequality, under the form convenient for application in finite volume schemes (cf. [EGGHH00, Lemma 6.1]).
Lemma 10 Let K be a volume of an admissible mesh of Ω ⊂ Rd in the sense of Definition 2,
and QnK = ((n − 1)k, nk) × K be the corresponding element of the space-time grid. Then
there exists a constant C that only depends on p , d , and ζ in (11) such that for all
σ ∈ EK , for all w ∈ Lp ((n − 1)k, nk; W 1,p(K)) one has
w K − wσ p
1
≤ C kDwkp p n ,
k m(σ) dK,σ (79)
L (QK )
d
dK,σ 156
Variational Approach for a Finite Volume Method
RR
R nk R
1
1
where wK = k m(K)
w and wσ = k m(σ)
w , in the sense of traces. Besides,
Qn
(n−1)k
σ
K
RR
1
n
is defined in Section 1.
the same holds with w K replaced by w C = |C n | C n w , where CK
K
K
√
Proof : Set h = δ(K) ; one has ν = ζδ(K)/(2 d) . First, note that
wK − w σ p
1
≤ C0 (p, d, ζ) m(σ) k |wK − wσ |p .
k m(σ) dK,σ d
dK,σ hp−1
Further, it follows by the Jensen inequality that
Z nk
p
k |wK − w σ | ≤
|wK (t) − wσ (t)|p ,
(n−1)k
where wK (t) , wσ (t) are the averages of w(t, ·) over K and σ , respectively, for a.a.
t ∈ ((n − 1)k, nk) . Thus it is sufficient to prove that for all w ∈ W 1,p (K) one has
hp
|wK − wσ | ≤ const
kDwkpLp (K)
m(K)
p
(80)
(we abusively keep the same notation, dropping the dependence of w on t ).
Furthermore, without loss of generality we can assume that σ is parallel to the hyperplane
{x1 = 0} of Rd . Let C = C 1 × C d−1 be a d -dimensional cube with edge 2ν and sides
parallel or perpendicular to σ , contained in K , and such that dist (C, σ) ≥ ν (see fig.2).
R
1
Let m(C) denote the d -dimensional measure of C ; set wC = m(C)
w . It is known that
C
hp
p
|wK − wC | ≤ const m(K) kDwkLp (K) (cf. (84) in Lemma 11 below). Therefore it is sufficient
to prove (80) with wK replaced by wC ; this will also prove the last statement of the lemma.
We denote a point of σ by s and a point of C by x = (ρ, l) , ρ ∈ C 1 , l ∈ C d−1 . By
the standard density argument, we can assume that w ∈ C 1 (K) . By the Newton-Leibnitz
formula and the Hölder inequality one has
Z Z Z |x−s| p
1
x − s hp−1
p
|wC − wσ | ≤
Dw(s
+
r
)
drdsdx
≤
×
p
(m(C)m(σ))
|x
−
s|
m(C)m(σ)
C
σ
0
p
Z Z Z |x−s| x − s hp−1 const
×
Dw(s
+
r
)
drdsdx
≤
×
|x − s| m(σ) hd
C σ 0 Z Z Z
p
(81)
x
−
s
n
o
×
Dw(s + r |x − s| ) drdsdldρ ≤
(ρ,l,s,r)∈C 1 ×C d−1 ×σ×[0,h]
hp−1 const ≤
I1 + I2 ,
m(σ) hd
x−s p
where I1 , I2 are the integrals of |Dw(s + r |x−s|
)| over M ∩ {r ∈ [0, ν/2]} and M ∩ {r ∈
[ν/2, h]} , respectively, and M = C 1 × C d−1 × σ × [0, h] . In (81) we have extended Dw by
zero outside K , so that all integrals make sense.
x−s
Introduce the change of variables (ρ, l, s, r) ↔ (ρ, s, y) , where y = s + r |x−s|
(cf.
fig.2). Clearly, for all (ρ, s) ∈ C 1 × σ there is a one-to-one correspondence between (l, r) ∈
C d−1 × [0, h] and y in some subset of K . Moreover, the Jacobian det k D(l,r)
k of the
Dy
2.II.8.
Appendix
157
(x2 , . . . , xd ) ∈ Rd−1
σ ν/2
ρ ∈ C1
σ
x∈C
l ∈ C d−1
y∈K
s∈σ
r = |y − s|
ν/2
≥
x1 ∈ R
2ν
σy
y
C
C
K
K
Figure 2: The cube C and the change of variables
Figure 3: The set σy
d−1
transformation (l, r) ↔ y is estimated by const h/|y1|
, where const is independent
of (ρ, l, s, r) ∈ M . Thus we have
Z
ZZ
p
I2 ≤
|Dw(y)|
dsdρ const dy ≤ h m(σ)kDwkpLp (K) .
(82)
{y∈K}
On the other hand, one has
Z
Z
p
I1 ≤
|Dw(y)|
{y∈K}
{s∈σ,ρ∈C 1 }
h d−1
dy
ds
dρ const
→k−
→
|y1 |
{s∈σ | ∃x∈C, −
x−s
y−s}
{ρ∈C 1 }
Z
Note that, when |y1 | < ν/2 , the (d−1) -dimensional measure of the set σy = {s ∈ σ | ∃x ∈
−−−→ −−−→
C, x − s k y − s} is estimated by const |y1 |d−1 , where the constant is absolute (cf. fig.3).
Hence
Z
1
d−1
I2 ≤ const h h
|Dw(y)|p
|y1|d−1 dy ≤ const hd kDwkpLp (K) .
(83)
d−1
|y1|
{y∈K}
Substituting (82),(83) in (81), and taking into account that m(σ) ≤ hd−1 , we finally deduce
(80). This ends the proof.
⋄
Corollary 1 Let (T h , k) be a family of admissible space-time grids in the sense of Definih
tion 3, and let D⊥
be the corresponding operator defined by (19). Let η ∈ E . Then there
exists a constant C which only depends on p , d , and M , ζ such that
h p
D⊥ η p n ≤ C kDηkp p
L (Υ1 (Qn )) .
L (Q )
K
K
158
Variational Approach for a Finite Volume Method
Proof : Let η nσ =
h p
D η p
⊥
L (Qn
K)
1
km(σ)
=
R nk
R
(n−1)k σ
η . By definition, we have
n
n p
X 1
ηL − ηK
k m(σ) dK,σ d
dK,L σ∈E
K
σ=K|L
n
n p
X 1
|η σ − ηK
|
|η nσ − ηLn |p
≤ C
k m(σ) dK,σ
+
d
(dK,σ )p
dK,σ (dL,σ )p−1
σ∈E
K
σ=K|L
n
n p
η σ − ηK
dK,σ K
n
X 1
η σ − ηLn p
+ C
k m(σ) dL,σ d
d
L,σ
σ∈E
X1
≤ C
k m(σ) dK,σ
d
σ∈E
K
σ=K|L
≤ C kDηkpLp (Υ1 (Qn )) ,
K
⋄
by Lemma 10.
Finally, we need the following versions of the Poincaré inequality the trace imbedding
theorem for the spaces W 1,p .
Lemma 11 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a cube C of
R
1
w . Then there exists
edge ζδ(K) > 0 ; let 1 ≤ p < ∞ . Let w ∈ W 1,p (K) , wK = |K|
K
a constant C = C(p, d, ζ) , independent of w , such that
kw − wK kLp (K) ≤ C δ(K) kDwkLp (K) .
In addition, one has
|wK − wC |p ≤ C
where wC =
1
|C|
R
C
δ(K)p
kDwkLp (K) ,
|K|
(84)
w , with a constant C = C(p, d, ζ) , independent of w .
Proof : The proof follows the lines of the proofs of [EgKo, Theorems 59,60], with p = 2
replaced by p ∈ (1, ∞) and the Cauchy inequality replaced by general Hölder inequality. ⋄
Lemma 12 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a ball of radius
ζδ(K) > 0 ; let 1 ≤ p < ∞ . Then there exists a constant C which only depends on p and
ζ such that
1
p
p
p
p−1
kwkLp (∂K) ≤ C
kwkLp (K) + δ(K)
kDwkLp (K) ,
δ(K)
for all w ∈ W 1,p (K) .
2.II.8.
Appendix
159
Proof : We cannot refer to [EgKo, Theorem 76], where the boundary of K is supposed
to be Lipschitz. Nevertheless, it is sufficient to introduce coordinates of the spherical type in
a neighbourhood of ∂K of thickness of order δ(K) . More exactly, let xK be the center
of the ball of radius ζδ(K) inside K . Introduce the family of (d−1) -dimensional surfaces
Sτ ⊂ K homothetic to S0 = ∂K with respect to xK . Parametrize it by τ the maximal
distance between the points of Sτ and S0 lying on the ray emanating from xK that have
the longest intersection with K . Note that the surfaces Sτ are well defined at least for
τ ∈ [0, ζδ(K)] . Note also that the distance between the points of Sτ1 and Sτ2 lying on a
same ray emanating from xK does not exceed |τ1 − τ2 | . The proof of the lemma goes on as
in [EgKo, Theorem 76], upon replacing the original family Sτ by the one constructed above.
⋄
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