Ann. Inst. Henri Poincaré Anal. nonlinear 18, 1 (2001) 97–133
 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved
S0294-1449(00)00056-1/FLA
EXISTENCE OF SOLUTIONS FOR COMPRESSIBLE
FLUID MODELS OF KORTEWEG TYPE
Raphaël DANCHIN a , Benoît DESJARDINS b
a Laboratoire d’Analyse Numérique, Université Paris – VI, 4, place Jussieu,
75252 Paris cedex 05, France
b D.M.I., E.N.S., 45, rue d’Ulm, 75005 Paris, France
Received 30 March 2000
A BSTRACT. – The purpose of this work is to prove existence and uniqueness results of suitably
smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn
and J. Serrin (1985), which can be used as a phase transition model.
We first study the well-posedness of the model in spaces with critical regularity indices with
respect to the scaling of the associated equations. In a functional setting as close as possible to
the physical energy spaces, we prove global existence of solutions close to a stable equilibrium,
and local in time existence for solutions when the pressure law may present spinodal regions.
Uniqueness is also obtained.
Assuming a lower and upper control of the density, we also show the existence of weak
solutions in dimension 2 near equilibrium. Finally, referring to the work of Z. Xin (1998) in
the non-capillary case, we describe some blow-up properties of smooth solutions with finite total
mass.  2001 Éditions scientifiques et médicales Elsevier SAS
R ÉSUMÉ. – On s’intéresse ici à des résultats d’existence et d’unicité de solutions pour un
modèle de fluides compressibles isothermes avec capillarité. Ce modèle de transition de phase a
été dérivé par J.E. Dunn et J. Serrin (1985).
Pour commencer, on montre que le problème de Cauchy est bien posé dans des espaces
à régularité critique pour le scaling des équations. Pour des données initiales proches d’un
état d’équilibre stable, on obtient l’existence globale (et l’unicité) de solutions dans un cadre
fonctionnel aussi proche que possible de l’espace d’énergie physique. Pour des lois de pression
plus générales (pouvant être décroissantes), on prouve des résultats locaux en temps.
En supposant que l’on dispose d’un minorant strictement positif et d’une borne supérieure pour
la densité, on obtient l’existence de solutions faibles en dimension 2 pour des données initiales
proches de l’équilibre. Enfin, en adaptant un travail de Z. Xin pour les fluides sans capillarité, on
établit l’explosion de solutions régulières à masse totale finie.  2001 Éditions scientifiques et
médicales Elsevier SAS
E-mail addresses: [email protected] (R. Danchin),
[email protected] (B. Desjardins).
98 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
1. Introduction
Let us consider a fluid of density ρ > 0, velocity field u ∈ Rd (d > 2), entropy
density s, energy density e, and temperature θ = (∂e/∂s)ρ . We are interested in the
following model of compressible capillary fluid, which can be derived from a Cahn–
Hilliard like free energy (see the pioneering work by J.E. Dunn and J. Serrin in [11], and
also [1,6,12])
∂t ρ + div(ρu) = 0,
(1)
∂t (ρu) + div(ρu ⊗ u) = div(S + K),
(2)
u2
u2
∂t ρ e +
+ div ρu e +
= div(α∇θ) + div (S + K) · u ,
2
2
where the viscous stress tensor S and the Korteweg stress tensor K read as
Si,j = λ divu − P (ρ, e) δi,j + 2µD(u)i,j ,
(3)
(4)
κ
1ρ 2 − |∇ρ|2 δi,j − κ∂i ρ∂j ρ,
(5)
2
D(u)i,j = (∂i uj + ∂j ui )/2 being the strain tensor, and (λ, µ) the constant viscosity
coefficients of the fluid. We require that λ and µ satisfy µ > 0 and λ + 2µ > 0, which in
particular covers the case when λ and µ satisfy Stokes’ law dλ + 2µ = 0. The thermal
conduction coefficient α is a given non negative function of the temperature θ and the
surface tension coefficient κ > 0 is assumed to be constant. In view of the first principle
of thermodynamics, the entropy density s solves
Ki,j =
∂t (ρs) + div(ρsu) =
1
div(α∇θ) + K : D(u) + 2µD(u) : D(u) + λ|div u|2 .
θ
(6)
As a reasonable starting point of our analysis, we consider the scaled Van der Waals
equation of state
8
3ρ
,
(7)
−
3−ρ
θ
where a is a positive constant, and the critical density ρc and temperature θc are equal
to 1. Depending on the fixed temperature θ , the pressure is a nondecreasing function of
the density ρ or may present decreasing regions (spinodal regions) for some values of
ρ, which are thermodynamically unstable. The above equation of state (7) ensures the
presence of two basic states, a “liquid” one, and a “gaseous” one. Let us as in [20] put
emphasis on the existence of steady solutions connecting a gas phase to a liquid phase
through a smoothly varying density profile. When initial conditions involve densities in
the unstable (spinodal) region, the two phases are expected to spontaneously separate.
For details on the derivation of the above Korteweg like model, we refer to [1,11,12,16].
In what follows, we do not consider thermal fluctuations so that the pressure p is a
function of ρ only. The corresponding isothermal model which was also considered in
[13,20] then reads as
P (ρ) = aρθ
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
99
∂t ρ + div(ρu) = 0,
(8)
∂t (ρu) + div(ρu ⊗ u) − µ1u − (λ + µ)∇div u + ∇P (ρ) = div K + ρf,
(9)
where f is an exterior forcing term, supplemented with initial conditions
ρ|t =0 = ρ0 > 0 and
ρu|t =0 = m0 .
(10)
In a bounded domain Ω, we would have to precise the boundary conditions, namely
homogeneous Dirichlet conditions for the velocity: u|∂Ω = 0 and Neumann conditions
for the density: ∂n ρ|∂Ω = 0. In order to simplify the presentation, we will focus on the
whole space case Rd (d > 2) and study the well-posedness of (8) (9) for an initial density
close enough to an equilibrium density ρ̄ > 0, or at least bounded away from vacuum,
which is a major difficulty in most of compressible fluid models.
Before getting into the heart of mathematical results, we first derive the physical
energy bounds of the above system in the case f ≡ 0 to simplify the presentation. Let
ρ̄ > 0 be a constant reference density, and π defined by
π(s) = s
Zs
ρ̄
!
P (z)
P (ρ̄)
dz −
,
2
z
ρ̄
(11)
so that P (s) = sπ 0 (s) − π(s), π 0 (ρ̄) = 0, and
∂t π(ρ) + div uπ(ρ) + P (ρ)div u = 0 in D 0 (0, T ) × Rd .
(12)
Notice that π is convex as far as P is non decreasing (since P 0 (s) = sπ 00 (s)), which is
the case for γ -type pressure laws. Multiplying the equation of momentum conservation
by u and integrating by parts over Rd , we obtain the following energy estimate
Z Rd
κ
1 2
ρu + π(ρ) − π(ρ̄) + |∇ρ|2 (t) dx
2
2
+
Zt
Z
ds
0
6
µ D(u) : D(u) + (λ + µ)|div u|2 dx
Rd
Z Rd
|m0 |2
κ
+ (π(ρ0) − π(ρ̄)) + |∇ρ0 |2 dx.
2ρ0
2
(13)
Indeed, in order to compute formally the contribution to energy of the capillary tensor
K, we observe that
div K = κ ρ∇1ρ.
(14)
In view of the above expression, this model can be understood as a diffuse interface
model, in which surface tension takes place between level sets of the continuously
varying density. As a matter of fact, the right hand side (14) can be rewritten up to a
gradient term as the product between ∇ρ and 1ρ, which, roughly speaking, respectively
100 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
represent the normal direction and the curvature of the level sets of the density. As
observed for instance in [1], formal analyses show that the sharp interface limit leads
to the classical two-fluid problem. We obtain indeed
−
Z
u · div K dx =
Rd
Z
κ div(ρu) 1ρ dx = κ
Rd
Z
Rd
d
∂t ∇ρ · ∇ρ dx = κ
dt
Z
Rd
|∇ρ|2
dx.
2
It follows that assuming that the total energy is finite
E0 =
Z R2
κ
1
ρ0 u20 + π(ρ0) − π(ρ̄) + |∇ρ0 |2 < +∞,
2
2
(15)
we have the a priori bounds
π(ρ) − π(ρ̄)
and
∇ρ ∈ L∞ 0, ∞; L2 Rd
d
ρ|u|2 ∈ L∞ (0, ∞; L1 (Rd )),
and
∇u ∈ L2 (0, ∞) × Rd
(16)
d 2
.
(17)
Let us emphasize at this point that the above a priori bounds do not provide any L∞
control on the density from below or from above. Indeed, even in dimension d = 2,
H 1 (Rd ) functions are not necessarily locally bounded. Thus, vacuum patches are likely
to form in the fluid in spite of the presence of capillary forces, which are expected to
smooth out the density.
2. Mathematical results
We wish to prove existence and uniqueness results of solutions to (8) (9) in functional
spaces very close to energy spaces. In the case κ = 0 and p(ρ) = aρ γ , with a > 0 and
γ > 1, P.-L. Lions proved in [17,18] the global existence of weak solutions “à la Leray”
(ρ, u) to (8) (9) for γ > 3d/(d + 2) and initial data (ρ0 , m0 ) such that
π(ρ0) − π(ρ̄)
and
|m0 |2
∈ L1 (Rd ),
ρ0
(18)
where we agree that m0 = 0 on {x ∈ Rd /ρ0 (x) = 0}. More precisely, he obtains the
existence of global weak solutions (ρ, u) to (8)–(10) such that
• ρ − ρ̄ ∈ L∞ (0, ∞; L2γ (Rd )) (where L2p (Rd ) spaces are Orlicz spaces defined in
[18]),
• u ∈ L2 (0, ∞; Ḣ 1 (Rd ))d (Ḣ s being defined in Section 3),
with in addition
p
• ρ ∈ C([0, ∞); Lloc (Rd )) if 1 6 p < γ ,
2γ /(γ +1)
• ρ|u|2 ∈ L∞ (0, ∞; L1 (Rd )), ρu ∈ C([0, ∞); Lloc
(Rd )-weak),
q
d
• ρ ∈ Lloc ([0, ∞) × R ) for q = γ − 1 + 2γ /d.
Moreover, the energy inequality (13) holds for almost every t > 0.
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
101
Notice that the main difficulty for proving Lions’ theorem consists in strong
p
compactness properties of the density ρ in Lloc spaces required to pass to the limit in the
pressure term p(ρ) = aρ γ . In the capillary case κ > 0, more a priori bounds are available
for the density, which belongs to L∞ (0, ∞; Ḣ 1 (Rd )). Hence, one can easily pass to the
limit in the pressure term. However, in the remaining quadratic terms involving gradients
of the density ∇ρ ⊗ ∇ρ (see (5)), we have been unable to pass to the limit.
Let us mention now that the existence of strong solutions is known since the works
by H. Hattori and D. Li [13,14]. Notice that high order regularity in Sobolev spaces H s
is required, namely the initial data (ρ0 , m0 ) are assumed to belong to H s × H s−1 with
s > d/2 + 4. Moreover, they considered convex pressure profiles, which cannot cover
the case of Van der Waals’ equation of state.
Here we want to investigate the well-posedness of the problem in critical spaces, that
is, in spaces which are invariant by the scaling of Korteweg’s system. Recall that such an
approach is now classical for incompressible Navier–Stokes equations (see, for example,
[7] and the references therein) and yields local well-posedness (or global well-posedness
for small data) in spaces with minimal regularity.
Let us explain precisely the scaling of Korteweg’s system. We can easily verify that,
if (ρ, u) solves (8) (9), so does (ρλ , uλ ), where
ρλ (t, x) = ρ(λ2 t, λx)
and
uλ (t, x) = λu(λ2 t, λx),
provided the pressure law P has been changed into λ2 P .
D EFINITION 1. – We will say that a functional space is critical with respect to the
scaling of the equation if the associated norm is invariant under the transformation
(ρ, u) 7→ (ρλ , uλ ) (up to a constant independent of λ).
This suggests us to choose initial data (ρ0 , u0 ) in spaces whose norm is invariant by
(ρ0 , u0 ) 7→ (ρ0 (λ·), λu0 (λ·)).
A natural candidate is the homogeneous Sobolev space Ḣ d/2 × (Ḣ d/2−1)d , but since
d/2
Ḣ
is not included in L∞ , we cannot expect to get L∞ control on the density when
ρ0 ∈ Ḣ d/2 . This is the reason why, instead of the classical homogeneous Sobolev spaces
Ḣ s (Rd ), we will consider homogeneous Besov spaces with the same derivative index
s
B s = B2,1
(Rd ) (for the corresponding definitions, we refer to Section 3). One of the nice
property of B s spaces for critical exponents s is that B d/2 is an algebra embedded in L∞ .
This allows to control the density from below and from above, without requiring more
regularity on derivatives of ρ.
Since a global in time approach does not seem to be accessible for general data, we
will mainly consider the global well-posedness problem for initial data close enough to
stable equilibria (Section 4). More precisely, we will state the following theorem:
T HEOREM 1. – Let ρ̄ > 0 be such that P 0 (ρ̄) > 0. Suppose that the initial density
fluctuation ρ0 − ρ̄ belongs to B d/2 ∩ B d/2−1 , that the initial velocity u0 is in (B d/2−1)d
and that the exterior forcing term f is in L1 (R+ ; B d/2−1)d . Then there exists a constant
η > 0 depending only on κ, µ, λ, ρ, P 0 (ρ̄) and d, such that, if
kρ0 − ρ̄kB d/2−1 ∩B d/2 + ku0 kB d/2−1 + kfkL1 (B d/2−1 ) 6 η,
102 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
then (8)–(10) has a unique global solution (ρ, u) such that the density fluctuation
(ρ − ρ̄) ∈ C(R+ ; B d/2−1 ∩ B d/2 ) ∩ L1 (R+ ; B d/2+1 ∩ B d/2+2 ) and the velocity u ∈
C(R+ ; B d/2−1)d ∩ L1 (R+ ; B d/2+1)d .
In Section 5, we get a local in time existence result for initial densities bounded away
from zero, which does not require any stability assumption on the pressure law, and thus
applies to Van der Waals’ law. The precise statement reads as follows:
T HEOREM 2. – Suppose that the forcing term f belongs to L1loc (R+ ;
B
) , that the initial velocity u0 belongs to (B d/2−1 )d , and that the initial density ρ0 satisfies (ρ0 − ρ̄) ∈ B d/2 and ρ0 > c for a positive constant c. Then there
exists T > 0 such that (8)–(10) has a unique solution (ρ, u) satisfying (ρ − ρ̄) ∈
C([0, T ]; B d/2 ) ∩ L1 ([0, T ]; B d/2+2 ) and u ∈ C([0, T ]; B d/2−1 )d ∩ L1 ([0, T ]; B d/2+1 )d .
d/2−1 d
In Section 6, we show that the problem is still locally well-posed in more general
s
scaling invariant Besov spaces of type Bp,1
which are not related to energy spaces
d/p
d/p−1
(namely ρ − ρ̄ is assumed to be in Bp,1 and u0 to be in (Bp,1 )d ). No stability
assumption on the pressure is required, but we have to suppose that the density is close to
a constant (see Theorem 5). Let us observe that working with p > d allows to consider
s
initial velocities in Bp,1
spaces with negative exponents s, which is in particular relevant
for oscillating initial data.
Finally, we will investigate blow-up properties of smooth solutions without smallness
assumptions on the data, like in the work of Z. Xin [22], and study sufficient conditions
for the existence of weak solutions close to equilibria in dimension d = 2.
Notation. In all the paper, C will stand for a “harmless” constant, and we will
sometimes use the notation A . B equivalently to A 6 CB.
3. Littlewood–Paley theory and Besov spaces
3.1. Littlewood–Paley decomposition
The homogeneous Littlewood–Paley decomposition relies upon a dyadic partition of
def
unity. We can use for instance any ϕ ∈ C ∞ (Rd ), supported in C ={ξ ∈ Rd , 3/4 6 |ξ | 6
8/3} such that
X
ϕ 2−` ξ = 1
if ξ 6= 0.
`∈Z
Denoting h = F −1 ϕ, we then define the dyadic blocks by
def
−`
1` u = ϕ 2 D u = 2
Z
`d
h 2` y u(x − y) dy and
S` u =
X
1k u.
k6`−1
Rd
The formal decomposition
u=
X
`∈Z
1` u
(19)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
103
is called homogeneous Littlewood–Paley decomposition. Let us observe that the above
formal equality does not hold in S 0 (Rd ) for two reasons:
(i) The right-hand side does not necessarily converge in S 0 (Rd ).
(ii) Even if it does, the equality is not always true in S 0 (Rd ) (consider the case u = 1).
Nevertheless, (19) holds true modulo polynomials (see [21]).
Furthermore, the above dyadic decomposition has nice properties of quasi-orthogonality: with our choice of ϕ, we have
1k 1` u ≡ 0 if |k − `| > 2,
and
1k (S`−1 u1` u) ≡ 0 if |k − `| > 5.
(20)
3.2. Homogeneous Besov spaces
D EFINITION 2. – For s ∈ R, p ∈ [1, +∞], q ∈ [1, +∞] and u ∈ S 0 (Rd ), we set
def
s
kukBp,q
=
X
2s` k1` ukLp
q
1/q
.
`∈Z
A difficulty due to the choice of homogeneous spaces arises at this point. Indeed,
s
s
s
k·kBp,q
cannot be a norm on {u ∈ S 0 (Rd ), kukBp,q
< +∞} because kukBp,q
=0
means that u is a polynomial. This enforces us to adopt the following definition for
homogeneous Besov spaces (see [5] for more details):
D EFINITION 3. – Let s ∈ R, p ∈ [1, +∞] and q ∈ [1, +∞]. Denote m = [s − d/p] if
s
s − d/p ∈
/ Z or q > 1 and m = s − d/p − 1 otherwise. If m < 0, then we define Bp,q
as
s
Bp,q
0
= u∈S R
d
s
| kukBp,q
< ∞ and u =
X
0
1` u in S R
d
.
`∈Z
If m > 0, we denote by Pm [Rd ] the set of polynomials of degree less than or equal to m
and we set
s
Bp,q
0
d
= u ∈ S R /Pm R
d
s
| kukBp,q
< ∞ and u =
X
0
d
1` u in S R Pm R
d
.
`∈Z
Remark 1. – The above definition is a natural generalization of the homogeneous
s
Sobolev or Hölder spaces: one can show that B∞,∞
is the homogeneous Hölder space
s
s
Ċ and that B2,2 is the homogeneous Sobolev space Ḣ s .
s
In the sequel, we will use only Besov spaces Bp,q
with q = 1 and we will denote them
s
s
by Bp or even by B if there is no ambiguity on the index p.
3.3. Basic properties of Besov spaces
P ROPOSITION 1. – The following properties hold:
(i) Density: if p < +∞ and |s| 6 d/p, then C0∞ is dense in Bps .
(ii) Derivation: there exists a universal constant C such that
C −1 kukBps 6 k∇ukBps−1 6 CkukBps .
104 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
def √
(ii0 ) Fractional derivation: let Λ = −1 and σ ∈ R. Then the operator Λσ is an
isomorphism from Bps to Bps−σ .
1 −1/p2 ) (where ,→ means
(iii) Sobolev embeddings: if p1 < p2 then Bps 1 ,→ Bps−d(1/p
2
continuous embedding).
(iv) Algebraic properties: for s > 0, Bps ∩ L∞ is an algebra.
(v) Interpolation: (Bps1 , Bps2 )θ,1 = Bpθs1 +(1−θ)s2 .
In Section 6, we will make extensive use of the space Bpd/p . Note that, if p < +∞,
then Bpd/p is an algebra included in the space C0 of continuous functions which tend to
0 at infinity. Note also that Bpd/p × (Bpd/p−1 )d is invariant by the scaling of Korteweg’s
system.
In Sections 4 and 5, we will focus on the case p = 2. Note that the following inclusion
chain
d/2
d/2
d/2
B2,1 ,→ Ḣ d/2 = B2,2 ,→ B2,∞
d/2
d/2
shows us that Ḣ d/2 is very close to B2,1 . But B2,1 has two additional nice properties: to
be an algebra and to be a subset of C0 .
3.4. Besov–Chemin–Lerner spaces
def
The study of non stationary PDE’s usually requires spaces of type LrT (X) = Lr (0, T ;
X) for appropriate Banach spaces X. In our case, we expect X to be a Besov space, so
that it is natural to localize the equations through Littlewood–Paley decomposition. We
then get estimates for each dyadic block and perform integration in time. But, in doing
so, we obtain bounds in spaces which are not of type Lr (0, T ; Bps ). This approach was
initiated in [9] and naturally leads to the following definitions:
D EFINITION 4. – Let (ρ, p) ∈ [1, +∞]2 , T ∈ ]0, +∞] and s ∈ R. We set
def
kukLeρ (B s ) =
T
X
p
`∈Z
ZT
`s
2
1/ρ
k1` u(t)kρLp
dt
.
0
Noticing that Minkowski’s inequality yields kukLρT (Bps ) 6 kukLeρ (B s ) , we define
p
T
spaces as follows
e ρ (B s )
L
T
p
def e ρ B s = u ∈ Lρ B s | kuk ρ s < +∞ .
L
T
T
p
p
e (B )
L
T
p
e 1 (B s ) but that the embedding L
e ρ (B s ) ⊂ Lρ (B s ) is
Let us observe that L1T (Bps ) = L
T
T
T
p
p
p
strict if ρ > 1.
eT (B s ) the subset of functions of L
e ∞ (B s ) which are continuous
We will denote by C
p
T
p
on [0, T ] with values in Bps .
e ρ (B s ∩ B s 00 )
Throughout
the
paper,
the
notation
L
(respectively
T
p
p
e ρ (B s ×B s 00 )) will stand for L
e ρ (B s )∩ L
e ρ (B s 00 ) (respectively L
e ρ (B s )× L
e ρ (B s 00 )). MoreL
T
T
T
T
T
p
p
p
p
p
p
e ρ (B s ) means L
e ρ (B s ).
over, in the case T = +∞, the T will be omitted. For example, L
+∞
p
p
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
105
We will often use the following interpolation property
1−θ
kuke
6 kukθe
ρ
ρ
s kuk ρ2
L (B s )
e
L 1 (B 1 )
T
p
s
LT (Bp2 )
p
T
with
1
1−θ
θ
+
and s = θs1 + (1 − θ)s2 ,
=
ρ ρ1
ρ2
and the following embeddings
e ρ B d/p ,→ Lρ (C0 )
L
T
T
p
and
eT B d/p ,→ C [0, T ] × Rd .
C
p
e (B s ) spaces suit particularly well to the study of smoothing properties of the heat
The L
T
p
equation. In [7], J.-Y. Chemin proved the following proposition
ρ
P ROPOSITION 2. – Let p ∈ [1, +∞] and 1 6 ρ2 6 ρ1 6 +∞. Let u solve
∂t u − ν1u = f,
u|t =0 = u0 .
Then there exists C > 0 depending only on d, ν, ρ1 and ρ2 such that
kukLeρ1 (B s+2/ρ1 ) 6 Cku0 kBps + Ckf ke
ρ
s−2+2/ρ2 .
L 2 (B
)
p
T
p
T
In Sections 4, 5 and 6, we will point out similar smoothing properties for the linearized
Korteweg system.
e ρ (B s ) spaces with respect to the product.
Let us now state properties of L
T
p
P ROPOSITION 3. – If s > 0, 1/ρ2 + 1/ρ3 = 1/ρ1 + 1/ρ4 = 1/ρ 6 1, u ∈ LρT1 (L∞ ) ∩
e ρ4 (B s ), then uv ∈ L
e ρ (B s ) and
and v ∈ LρT2 (L∞ ) ∩ L
T
T
p
p
e ρ3 (B s )
L
T
p
ρ
kuvke
ρ
. kukLρ1 (L∞ ) kvke
+ kvkLρ2 (L∞ ) kukLeρ3 (Bps ) .
L (Bps )
L 4 (Bps )
T
T
T
T
T
e 1 (B s1 ) and v ∈ L
e 2 (B s2 ),
If s1 , s2 6 d/p, s1 + s2 > 0, 1/ρ1 + 1/ρ2 = 1/ρ 6 1, u ∈ L
T
T
p
p
e ρ (B s1 +s2 −d/p ) and
then uv ∈ L
T
p
ρ
ρ
s
ρ
s
kuvke
s +s −d/p . kuk ρ1
ρ
L (Bp1 ) kvkL 2 (Bp2 ) .
)
L (B 1 2
T
p
T
T
This proposition is a straightforward adaptation of the corresponding results for usual
homogeneous Besov spaces (see [8]).
e ρ (B s ) spaces.
We finally need a composition lemma in L
T
p
e (B s ) ∩ L∞ (L∞ ).
L EMMA 1. – Let s > 0, p ∈ [1, +∞] and u ∈ L
T
p
T
[s]+2,∞
e ρ (B s ). More precisely,
(i) Let F ∈ Wloc
(Rd ) such that F (0) = 0. Then F (u) ∈ L
T
p
there exists a function C depending only on s, p, d and F such that
ρ
kF (u)ke
ρ
6 C kukL∞
.
∞ kuk eρ
LT (Bps )
LT (Bps )
T (L )
e (B s ) ∩ L∞ (L∞ ) and G ∈ W [s]+3,∞ (Rd ), then G(v) − G(u)
(ii) If v also belongs to L
loc
T
p
T
e ρ (B s ) and there exists a function C depending only on s, p, d and G, and
belongs to L
T
p
such that
ρ
106 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
G(v) − G(u)
eT (Bps )
L
ρ
6 C kukL∞
kv − ukLeρ (B s ) (1 + kukL∞
∞ , kvkL∞ (L∞ )
∞ + kvkL∞ (L∞ ) )
T (L )
T
T (L )
T
T
p
+ kv − ukL∞
∞ (kuk ρ
e (B s ) + kvkL
eρ (B s ) ) .
L
T (L )
p
T
T
p
Proof. – For (i), one just has to use the proof of [2] and replace L2 norms with Lp
norms. For (ii), we use the following identity
G(v) − G(u) = (v − u)
Z1
H u + τ (v − u) dτ + G0 (0)(v − u),
0
where H (w) = G0 (w) − G0 (0), and we conclude by using (i) and Proposition 3.
2
4. Global solutions near equilibrium
In this section, we want to prove global existence and uniqueness of suitably smooth
e ρ (B s ) which are very
solutions to the Korteweg system (8) (9) in the functional spaces L
T
2
close to the physical energy spaces. Given a reference density ρ̄ such that the stability
condition P 0 (ρ̄) > 0 is satisfied, we introduce the density fluctuation q = (ρ − ρ̄)/ρ̄
and the scaled momentum m = ρu/ρ̄. We also define the scaled viscosity coefficients
µ̄ = µ/ρ̄ and λ̄ = λ/ρ̄ and the scaled surface tension coefficient κ̄ = ρ̄κ. Assuming that
the density ρ is bounded away from zero, we rewrite the Korteweg system (8) (9) as
follows
∂t q + div m = 0,
(21)
0
∂t m − µ̄1m − (λ̄ + µ̄)∇div m − κ̄∇1q + P (ρ̄)∇q = G(q, m) + f,
(22)
(q, m)|t =0 = (q0 , m0 ),
(23)
where we define G = G1 + G2 + G3 + G4 + G5 by
m⊗m
,
G2 (q, m) = −∇H (q),
1+q
qm
qm
G3 (q, m) = −µ̄1
− (λ̄ + µ̄)∇div
,
1+q
1+q
κ̄
G4 (q, m) = ∇ 1q 2 − |∇q|2 − κ̄ div(∇q ⊗ ∇q) = κ̄q∇1q,
2
G1 (q, m) = −div
and
G5 (q, m) = fq,
H being defined by H (q) = (P (ρ̄(1 + q)) − P (ρ̄) − P 0 (ρ̄)q ρ̄)/ρ̄.
In Section 4.1, we study the linearized system around (q, m) = (0, 0), which turns
out to have the same smoothing properties as the heat equation. Finally, we prove in
Section 4.2 our main global theorem, estimating the right-hand side G of (22) in terms
s
of suitable norms of (q, m). Notice that in Sections 4 and 5, B s will stand for B2,1
.
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
107
4.1. Estimates for the linearized system
This section is devoted to the linearized isothermal system of Korteweg type around
(q, m) = (0, 0). This system reads
∂t q + div m = F,
∂t m − µ̄1m − (λ̄ + µ̄)∇div m − κ̄∇1q + β∇q = G.
(LNSK1)
The term β∇q corresponds to the linearized pressure (that is β = P 0 (ρ̄) > 0). Our
purpose is to prove estimates for (LNSK1) in Besov spaces closely related to energy
spaces. We get:
P ROPOSITION 4. – Let s ∈ R, 1 6 r1 6 r 6 +∞ and T ∈ ]0, +∞]. If (q0 , m0 ) ∈
e r1 ((B s−2+2/r1 ∩ B s−3+2/r1 ) × (B s−3+2/r1 )d ) then
(B s ∩ B s−1 ) × (B s−1 )d and (F, G) ∈ L
T
eT ((B s ∩ B s−1 ) × (B s−1 )d ) ∩
the linear system (LNSK1) has a unique solution (q, m) ∈ C
r
s+2/r
s−1+2/r
s−1+2/r d
e
LT ((B
∩B
) × (B
) ). Moreover, there exists a constant C depending
only on r, r1 , µ̄, λ̄, κ̄ and β such that the following inequality holds:
kqke
+ kmke
Lr (B s+2/r ∩B s−1+2/r )
Lr (B s−1+2/r )
T
T
r
6 C kq0 kB s ∩B s−1 + km0 kB s−1 + kFke
+ kGkLer1 (B s−3+2/r1 ) .
L 1 (B s−2+2/r1 ∩B s−3+2/r1 )
T
T
Proof. – Denote by W (t) the semi-group associated to (LNSK1). According to
Duhamel’s formula,
q(t)
m(t)
q0
= W (t)
m0
+
Zt
F (s)
W (t − s)
G(s)
ds.
(24)
0
Let us first consider the case F ≡ 0 and G ≡ 0 and denote (q` (t), m` (t))t =
W (t)(1` q0 , 1` m0 )t . Then, we have the following lemma
L EMMA 2. – There exist two positive constants c and C depending only on λ̄, µ̄, κ̄
and β such that for all ` ∈ Z,
km` (t)kL2 + k∇q` (t)kL2 + kq` (t)kL2
6 Ce−c2
2` t
k1` m0 kL2 + k∇1` q0 kL2 + k1` q0 kL2 .
Proof. – We apply the operator 1` to (LNSK1) in the case F ≡ G ≡ 0 and get
∂t q` + div m` = 0,
(25)
∂t m` − µ̄1m` − (λ̄ + µ̄)∇div m` − κ̄∇1q` − β∇q` = 0.
(26)
In view of Eq. (25), integrations by parts yield
Z
1q` div m` dx =
1 d
k∇q` k2L2
2 dt
q` div m` dx = −
1 d
kq` k2L2 .
2 dt
Rd
Z
Rd
and
108 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
Thus, taking scalar product of (26) with m` , we get
1 d
km` k2L2 + βkq` k2L2 + κ̄k∇q` k2L2 + µ̄k∇m` k2L2 + (λ̄ + µ̄)kdiv m` k2L2 = 0. (27)
2 dt
In order to obtain a second energy estimate, we take the scalar product of m` with the
gradient of (25), which yields
Z
m` · ∂t ∇q` dx − kdiv m` k2L2 = 0.
(28)
Rd
Taking the scalar product of (26) with ∇q` , we obtain
Z
∇q` · ∂t m` dx + κ̄k1q` k2L2 + βk∇q` k2L2 6 Ck∇m` kL2 k∇ 2 q` kL2 .
(29)
Rd
Summing (28) and (29), we deduce
d
dt
Z
Rd
κ̄
m` · ∇q` dx + k∇ 2 q` k2L2 + βk∇q` k2L2 6 Ck∇m` k2L2 .
2
(30)
Let α > 0 be a constant to be chosen later and denote
h2` = km` k2L2 + κ̄k∇q` k2L2 + βkq` k2L2 + 2α
Z
m` · ∇q` dx.
Rd
As a result, from (30) and (27), we derive for some positive constant c0
2
1 d 2
h` + c0 k∇m` k2L2 + α ∇ 2 q` L2 + αk∇q` k2L2
2 dt
6 Cαk∇m` k2L2 .
(31)
Now choosing α suitably small, we deduce that
1 2
h 6 km` k2L2 + κ̄k∇q` k2L2 + βkq` k2L2 6 δh2` ,
δ `
(32)
for some positive δ. Thus, there exists a constant c > 0 such that
1 d 2
h + c22` h2` 6 0,
2 dt `
so that the proof of Lemma 2 is complete.
2
Proof of Proposition 4 (continued). – In view of Lemma 2 and formula (24), we have
k1` m(t)kL2 + k∇1` q(t)kL2 + k1` q(t)kL2
6 Ce−c2
2` t
k1` m0 kL2 + k∇1` q0 kL2 + k1` q0 kL2
109
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
+
Zt
e−c2
2` (t −τ )
k1` G(τ )kL2 + k∇1` F (τ )kL2 + k1` F (τ )kL2 dτ,
0
2
so routine computations yield Proposition 4.
4.2. Global existence and uniqueness
Let us first introduce functional spaces needed in the main global existence result. We
will prove existence in the space
e B d/2−1 ∩ B d/2 ∩ L1 B d/2+1 ∩ B d/2+2
E= C
e B d/2−1 ∩ L1 B d/2+1
C
d
,
and uniqueness in the larger space
e B d/2 × B d/2−1
Ee = C
d ∩ L2 B d/2+1 × B d/2
d .
We denote by k·kEe and k·kE the corresponding norms. Since B d/2 is a Banach space, it
is easy to verify that Ee and E are also Banach spaces. We now turn to our main global
existence theorem
T HEOREM 3. – Let ρ̄ > 0 be such that P 0 (ρ̄) > 0. Suppose that the initial density
fluctuation q0 belongs to B d/2 ∩ B d/2−1 , that the initial momentum m0 is in B d/2−1
and that the forcing term f is in L1 (R+ ; B d/2−1)d . Then there exists a constant η > 0
depending only on κ̄, µ̄, λ̄, ρ̄, P such that, if
kq0 kB d/2−1 ∩B d/2 + km0 kB d/2−1 + kfkL1 (B d/2−1 ) 6 η,
e In addition, (q, m) belongs to
then (21)–(23) has a unique global solution (q, m) in E.
E.
Proof. – Let us denote (qL , mL ) the “free” solution of the linearized system
qL (t)
mL (t)
q0
= W (t)
m0
+
Zt
W (t − s)
0
f(s)
ds.
0
We define the functional ΨqL ,mL in a neighborhood of 0 in E by
ΨqL ,mL (q̄, m̄) =
Zt
0
W (t − s)
0
G(qL + q̄, mL + m̄)(s)
ds.
(33)
To prove the existence part of the theorem, we just have to show that ΨqL ,mL has a fixed
point in E.
110 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
First step: stability of B(0, R).
We start by proving that the ball B(0, R) of E is stable under ΨqL ,mL provided R is
small enough. Denote q = qL + q̄ and m = mL + m̄. According to Proposition 4, we have
k(qL , mL )kE 6 Cη,
(34)
kΨqL ,mL (q̄, m̄)kE 6 CkG(q, m)kL1 (B d/2−1 ) .
(35)
Making the assumption
kqkL∞ (R+ ×Rd ) 6 1/2,
(H)
and using Proposition 3 and Lemma 1, we deduce the following estimates:
kG1 (q, m)kL1 (B d/2−1 ) 6 CkmkL1 (B d/2+1 ) kmkLe∞ (B d/2−1 ) 1 + kqkLe∞ (B d/2 ) ,
kG2 (q, m)kL1 (B d/2−1 ) 6 CkqkL1 (B d/2+1 ) kqkLe∞ (B d/2−1 ) ,
kG3 (q, m)kL1 (B d/2−1 ) 6 C kqkL1 (B d/2+2 ) kmkLe∞ (B d/2−1 ) + kqkLe∞ (B d/2 ) kmkL1 (B d/2+1 ) ,
kG4 (q, m)kL1 (B d/2−1 ) 6 CkqkL1 (B d/2+2 ) kqkLe∞ (B d/2 ) ,
kG5 (q, m)kL1 (B d/2−1 ) 6 CkfkL1 (B d/2−1 ) kqkLe∞ (B d/2 ) ,
since by interpolation, we have
kZk2L2 (B s ) 6 kZkLe∞ (B s−1 ) kZkL1 (B s+1 ) .
In the second inequality above, we also used that H (q) = q He (q) for a smooth function
He such that He (0) = 0. Therefore, assuming that R 6 1, we obtain
kΨqL ,mL (q̄, m̄)kE 6 Ck(qL + q̄, mL + m̄)kE k(qL + q̄, mL + m̄)kE + η
2
6 C (C + 1)η + R .
(36)
Let c be a constant such that k · kB d/2 6 c implies k·kL∞ 6 1/5. We choose (R, η) such
that
R 6 inf (5C)−1 , c, 1 and η 6 inf(R, c)/(C + 1), so that H, is satisfied.
(37)
From (36), we finally deduce that ΨqL ,mL (B(0, R)) ⊂ B(0, R).
Second step: Contraction properties.
Consider two elements (q̄1 , m̄1 ) and (q̄2 , m̄2 ) in B(0, R), and denote qi = qL + q̄i and
mi = mL + m̄i for i = 1, 2. According to (33) and to Proposition 4, we have
Ψq
L ,mL
(q̄2 , m̄2 ) − ΨqL ,mL (q̄1 , m̄1 )E 6 C G(q2 , m2 ) − G(q1 , m1 )L1 (B d/2−1 ) .
(38)
Under assumption (H) for q1 and q2 , we obtain estimates for G(q2 , m2 ) − G(q1 , m1 ).
Indeed, we just have to apply Proposition 3 and Lemma 1 to
G1 (q2 , m2 ) − G1 (q1 , m1 )
q2
q1
m2 ⊗ (m2 − m1 ) + (m2 − m1 ) ⊗ m1
= div m1 ⊗ m1
−
−
,
1 + q2 1 + q1
1 + q2
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
111
G2 (q2 , m2 ) − G2 (q1 , m1 ) = −∇ (q2 − q1 )He (q2 ) + q1 He (q2 ) − He (q1 ) ,
G3 (q2 , m2 ) − G3 (q1 , m1 )
q2
q2
q1
+ m1
−
1 + q2
1 + q2 1 + q1
G4 (q2 , m2 ) − G4 (q1 , m1 ) = κ(q2 − q1 )∇1q2 + κq1 ∇1(q2 − q1 ),
= − µ̄1 + (λ̄ + µ̄)∇div
(m2 − m1 )
,
G5 (q2 , m2 ) − G5 (q1 , m1 ) = f(q2 − q1 ).
This leads to the following inequality
Ψq
L ,mL
(q̄2 , m̄2 ) − ΨqL ,mL (q̄1 , m̄1 )E
6 Ck(q̄2 − q̄1 , m̄2 − m̄1 )kE
× k(q̄1 , m̄1 )kE + k(q̄2 , m̄2 )kE + 2k(qL , mL )kE + kfkL1 (B d/2−1 ) .
T
Now, if (R, η) satisfies (37) (for a greater constant C if needed), we deduce
Ψq
L ,mL
4
(q̄2 , m̄2 ) − ΨqL ,mL (q̄1 , m̄1 )E 6 (q̄2 − q̄1 , m̄2 − m̄1 )E
5
and the proof of the existence part of Theorem 3 is achieved.
Notice that in view of (34) and (37), (q, m) satisfies
kqkL∞ (R+ ×Rd ) 6 2/5 and
kqkLe∞ (B d/2 ) 6 2/(5C).
T
2
The solution (q, m) obviously belongs to the space Ee∞
defined in Section 6.2 (we
2
e
e
have E = E∞ ). Changing C into a greater constant if necessary, we therefore can apply
e
Lemma 4 to get uniqueness in E.
5. Local solutions away from vacuum
In this section, we want to show local well-posedness for the Korteweg system with
initial data (ρ0 , m0 ) such that (ρ0 − ρ̄, m0 ) in B d/2 × (B d/2−1 )d . Let us emphasize that
no smallness assumption is required: we just need the initial density to be bounded
away from zero. The pressure P may be any (possibly decreasing) smooth function
[d/2]+3,∞
of ρ (P ∈ Wloc
is enough).
It is convenient to rewrite (8) (9) in terms of q and m by using the same scaled
coefficients as in Section 4
∂t q + div m = 0,
∇m
div m
Dt m − µ̄ div
− (λ̄ + µ̄)∇
− κ̄∇ (1 + q)1q
1+q
1+q
= Γ (q, m) + f,
where Γ = Γ1 + Γ2 + Γ3 + Γ4 + Γ5 is defined by
Γ1 (q, m) = −∇ P ρ̄(1 + q) − P (ρ̄) /ρ̄,
(39)
(40)
112 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
Γ2 (q, m) = −div
m⊗m
,
1+q
1
Γ3 (q, m) = µ̄ div m ⊗ ∇
1+q
1
+ (λ̄ + µ̄)∇ m · ∇
1+q
,
κ̄
Γ4 (q, m) = −κ̄ div(∇q ⊗ ∇q) − ∇|∇q|2 ,
2
Γ5 (q, m) = qf.
The proof of our local existence theorem relies upon the study of the linearized system
around (1 + q, 0) for a given q such that 1 + q is bounded away from zero, whereas the
first order linearized pressure term is dropped. The purpose of Section 5.1 is to derive
estimates for such a system. Well-posedness for (39) (40) is obtained in Section 5.2
through an iterative method.
5.1. Estimates for the linearized system
We now study the following linearized system
∂t q + div m = F,
∂t m − µ̄ div(a∇m) − (λ̄ + µ̄)∇(a div m) − κ̄∇(b1q) = G,
(LNSK2)
where m is a vector field in Rd , and a, b are scalar functions, bounded and bounded
away from zero
0 < c1 6 a 6 M1 < +∞,
0 < c2 6 b 6 M2 < +∞
on [0, T ].
(41)
Our purpose is to prove estimates for (LNSK2) in Besov spaces closely related to energy
spaces. We obtain
P ROPOSITION 5. – Let 1 6 r1 6 r 6 +∞, (q0 , m0 ) ∈ B d/2 × (B d/2−1 )d and (F, G) ∈
r
e 1 (B d/2−2+2/r1 × (B d/2−3+2/r1 )d ). Suppose (41), ∇b and ∇a belong to L
e 2 (B d/2 ),
L
T
T
1
∞
r
d/2+2/r
d/2−1+2/r d
2
d/2+1
e
e
∂t b ∈ L (0, T ; L ). Let (q, m) ∈ LT (B
× (B
) ) ∩ LT (B
× (B d/2 )d )
be a solution of the system (LNSK2). Then there exists a constant C depending only on
r, r1 , λ̄, µ̄, κ̄, c1 , c2 , M1 , and M2 such that the following inequality holds:
k(∇q, m)kLer (B d/2−1+2/r ) 1 − Ck∇bkL2 (L∞ )
T
T
6 C k(∇q0 , m0 )kB d/2−1 + k(∇F, G)kLr1 (B d/2−3+2/r1 ) + k∂t bkL1 (L∞ ) k∇qkLe∞ (B d/2−1 )
T
T
T
+ k(∇q, m)kLe2 (B d/2 ) k∇bkLe2 (B d/2 ) + k∇akLe2 (B d/2 ) .
T
T
T
Proof. – It is just a matter of showing appropriate estimates for 1` q and 1` m.
Denoting q` = 1` q, m` = 1` m, F` = 1` F , G` = 1` G, and applying 1` to (LNSK2),
we get
∂t q` + div m` = F` ,
(42)
∂t m` − µ̄ div(a∇m` ) − (λ̄ + µ̄)∇(a div m` ) − κ̄∇(b1q` ) = G` + R` ,
(43)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
113
where
r` = −µ̄ div([a, 1` ]∇m) − (λ̄ + µ̄)∇([a, 1` ]div m) − κ̄∇([b, 1` ]1q).
Using integrations by parts and Eq. (42), we obtain
−
Z
Rd
1 d
m` ∇(b1q` ) dx =
2 dt
−
Z
b|∇q` |2 dx
Rd
Z |∇q` |2
div m` (∇q` · ∇b) +
∂t b + b∇q` · ∇F` dx.
2
Rd
We now take the scalar product of (43) with m` and use the previous identity, which
yields
1 d
km` k2L2 + κ̄
2 dt
=
Z Z
b|∇q` | dx +
Z Rd
µ̄|∇m` | + (λ̄ + µ̄)|div m` |
2
2
2
a dx
Rd
(G` + R` ) · m` + κ̄ div m` (∇b · ∇q` ) +
Rd
|∇q` |2
∂t b + b∇q` · ∇F`
2
dx. (44)
In order to obtain a second estimate, we take the scalar product of the gradient of (42)
with m` , the scalar product of (43) with ∇q` and sum both inequalities. We obtain
Z
Z
d
∇q` · m` dx + κ̄ b (1q` )2 dx
dt
Rd
=
Z
Rd
(G` + R` ) · ∇q` + |div m` |2 + m` · ∇F` − µ̄ a∇m` : ∇ 2 q`
Rd
− (λ̄ + µ̄)a1q` div m` dx.
(45)
Let α > 0 be suitably small, and define
def
kl2 = km` k2L2 +
Z
κ̄b|∇q` |2 + 2α∇q` · m` dx and
ν̄ = inf(µ̄, λ̄ + 2µ̄).
Rd
Using (44), (45) and (41), we deduce that when
−1
ν̄ M1 (µ̄ + |λ̄ + µ̄|)2
α6
+1
2
2c2 κ̄
,
we have
Z
1 d 2 1
a ν̄k∇m` k2L2 + α κ̄b|1q` |2 dx
k` +
2 dt
2
Rd
6 kG` kL2 + kR` kL2 αk∇q` kL2 + km` kL2 + k∇F` kL2 αkm` kL2 + k∇q` kL2
1
+ k∂t bkL∞ k∇q` k2L2 + κ̄k∇bkL∞ k∇q` kL2 k∇m` kL2 .
(46)
2
114 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
Using (41), we clearly have for α > 0 small enough
1 2
k 6 km` k2L2 + κ̄
2 `
Z
Rd
3
b|∇q` |2 6 k`2 ,
2
(47)
so that using the above energy estimate, the spectral localization of (∇q` , m` ) and (41),
we get for some positive K
1 d 2
k + K22l k`2
2 dt `
6 C k` kG` kL2 + k∇F` kL2 + kR` kL2 + k∂t bkL∞ k∇q` k2L2 + 2` k`2 k∇bkL∞ .
(48)
Integrating with respect to time yields
k` (t) 6 e
−K22` t
k` (0) + C
Zt
e−K2
2` (t −τ )
k∂t b(τ )kL∞ k∇q` (τ )kL2
0
+ k∇F` (τ )kL2 + kG` (τ )kL2 + kR` (τ )kL2 + 2` k` (τ )k∇b(τ )kL∞ dτ.
(49)
Using convolution inequalities, we easily get
kk` kLr ([0,T ]) 6 C 2−2`/r k` (0) + 2−2`(1+1/r−1/r1) k(∇F` , G` )kLr1 (L2 )
+2
T
R` −2`/r L1T (L2 )
−2`/r
+2
k∇q` k
2
L∞
T (L )
k∂t bkL1 (L∞ )
T
+ k∇bkL2 (L∞ ) kk` kLr ([0,T ]) .
(50)
T
We first use (47) and (41) to infer that there exists a constant C > 0 such that
C −1 k` 6 k∇q` kL2 + km` kL2 6 Ck` ,
then multiply both sides of (50) by 2`(d/2−1+2/r) and sum over Z, which yields
k(∇q, m)kLer (B d/2−1+2/r ) 1 − Ck∇bkL2 (L∞ )
T
T
6 k(∇q0 , m0 )kB d/2−1 + k(∇F, G)kLer1 (B d/2−3+2/r1 )
T
+ k∇qkLe∞ (B d/2−1 ) k∂t bkL1t (L∞ ) +
T
X
2`(d/2−1)kR` kL1 (L2 ) .
T
q∈Z
We then obtain the desired inequality thanks to Lemma 5 in the appendix.
2
Remark 2. – When r1 = 1, estimates of Proposition 5 clearly enable us to prove the
existence and uniqueness of a solution (q, m) to (LNSK2) in the space L1T (B d/2+2 ×
eT (B d/2 × (B d/2−1)d ) as long as
(B d/2+1)d ) ∩ C
k∇akLe2 (B d/2 ) + k∇bkLe2 (B d/2 ) + k∂t bkL1 (L∞ ) 6 C −1 .
T
This stems from a basic duality method.
T
T
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
115
5.2. Local existence and uniqueness
First we define the functional spaces needed in our local existence and uniqueness
theorem. We will prove existence in the space
def
FT = L1T B d/2+2 × B d/2+1
d eT B d/2 × B d/2−1
∩C
d endowed with the norm
k(q, m)kFT = kqkL1 (B d/2+2 ) + kqkLe∞ (B d/2 ) + kmkL1 (B d/2+1 ) + kmkLe∞ (B d/2−1 ) ,
T
T
T
T
and uniqueness in the larger space
e 2 B d/2+1 × B d/2
FeT = L
T
def
d eT (B d/2 × B d/2−1
∩C
d endowed with the norm
k(q, m)kFeT = kqkLe2 (B d/2+1 ) + kqkLe∞ (B d/2 ) + kmkLe2 (B d/2 ) + kmkLe∞ (B d/2−1 ) .
T
T
T
T
T HEOREM 4. – Suppose that the exterior forcing term f belongs to (L1T (B d/2−1 ))d ,
that the initial momentum m0 belongs to (B d/2−1)d , and that the initial density ρ0
satisfies (ρ0 − ρ̄) ∈ B d/2 and ρ0 > c for a positive constant c. Then, there exists T > 0
such that the system (39) (40) with initial data ((ρ0 − ρ̄)/ρ̄, m0 ) has a unique solution
(q, m) in FeT . In addition, (q, m) belongs to FT .
Proof. – The existence part of the theorem is proved by an iterative method. We define
a sequence {(q n , mn )}n∈N as follows: the first term (q 0 , m0 ) is taken to be the solution of
the heat equation
∂t
q0
m0
−1
q0
m0
0
,
f
=
q0
m0
|t =0
=
q0
,
m0
(51)
with q0 = (ρ0 − ρ̄)/ρ̄. Assuming that (q n , mn ) belongs to FT , we then define q n+1 =
q 0 + q̄ n+1 and mn+1 = m0 + m̄n+1 with (q̄ n+1 , m̄n+1 ) solution of the following linear
system

∂t q̄ n+1 + div m̄n+1 = −1q 0 − div m0 ,






∇ m̄n+1
div m̄n+1

n+1

− (λ̄ + µ̄)∇
1 + qn
1 + qn
n
n+1
n
n
− κ̄∇((1 + q )1q̄ ) = Γ (q , m ) + H 0 (q n , mn ),
∂t m̄








− µ̄ div
q̄|tn+1
=0 = 0,
(52)
m̄n+1
|t =0 = 0,
where
H 0 q n , mn = −1m0 + µ̄ div
∇m0
1 + qn
+ (λ̄ + µ̄)∇
div m0
1 + qn
+ κ̄∇ 1 + q n 1q 0 .
116 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
First step: uniform bounds in FT
We want now to show that (q n , mn ) is uniformly bounded in FT . Denote E0 =
kq0 kB d/2 + km0 kB d/2−1 . Let ε ∈ (0, 1). In view of Proposition 2, we can choose T ∈ ]0, ε]
such that
kfkL1 (B d/2−1 ) 6 ε,
T
kq 0 kL1 (B d/2+2 ) + km0 kL1 (B d/2+1 ) 6 ε,
T
(Hε )
T
kq 0 kLe∞ (B d/2 ) + km0 kLe∞ (B d/2−1 ) 6 C(E0 + 1).
T
T
We are going to show that if ε is chosen suitably small, we have for all n ∈ N,
n n √
q̄ , m̄ 6 ε.
FT
(Pn )
√
Since q̄ 0 = 0 and m̄0 = 0, (P0 ) is true. Suppose that (Pn ) is fulfilled and that ε is less
than c/(4C1 ρ̄) (where
C1 is the norm of the embedding B d/2 ,→ L∞ ). From the fact that
Rt
n
q (t) − q0 = − 0 div mn (τ ) dτ and (Hε ), we gather
n
q − q0 L∞ ([0,T ]×Rd )
6 C1 div m0 L1 (B d/2 ) + div m̄n L1 (B d/2 ) ,
T
T
√
6 C1 (ε + ε),
6 c/2ρ̄.
We thus have
c
kρ0 kL∞
− 1 6 qn 6
2ρ̄
ρ̄
on [0, T ],
(53)
which entails, according to Lemma 1,
1
1 + q n − 1
1 ∇
1 + qn e2T (B d/2 )
L
d/2 )
e∞
L
T (B
6 Ckq n kLe∞ (B d/2 ) ,
(54)
T
q n = ∇
1 + qn e2T (B d/2 )
L
6 Ckq n kLe2 (B d/2+1 ) .
(55)
T
Apply Proposition 5 to (52), and use that ∂t q n = − div mn . This yields
n+1 n+1 q̄
∇q n 2 d/2 + ∇
, m̄
1
−
C
e (B ) FT
L
T
+ div mn L1 (B d/2 )
1
n
1 + q e
L2T (B d/2 )
T
6 C Γ q n , mn L1 (B d/2−1 ) + H 0 q n , mn L1 (B d/2−1 )
T
T
0
0
+ 1q 1 d/2 + div m 1 d/2 .
LT (B
)
LT (B
)
(56)
Next, we use Proposition 3, Lemma 1, (53), (54) and (55) to estimate the right-hand side
of (56). The following bounds hold:
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
117
Γ1 q n , mn 1 d/2−1 6 CT q n ∞ d/2 ,
eT (B )
LT (B
)
L
Γ2 q n , mn 1 d/2−1 6 C mn 2 2 d/2 1 + kq n k ∞ d/2 ,
eT (B )
eT (B )
LT (B
)
L
L
n
n
n
n
Γ3 q , m 1 d/2−1 6 C m 2 d/2 q 2 d/2+1 ,
eT (B )
eT (B
LT (B
)
L
L
)
Γ4 q n , mn 1 d/2−1 6 C ∇q n 2 2 d/2 ,
eT (B )
LT (B
)
L
n
n
n
Γ5 q , m 1 d/2−1 6 C q ∞ d/2 kfk 1 d/2−1 .
)
eT (B ) LT (B
LT (B
)
L
We also have
0 ∇ ∇m
n
1+q L1T (B d/2−1 )
6 Ck∇m0 kL1 (B d/2 ) 1 + q n Le∞ (B d/2 ) ,
T
T
∇ 1 + q n 1q 0 1 d/2−1 6 Ck1q 0 k 1 d/2 1 + q n ∞ d/2 .
L (B )
e (B )
L (B
)
L
T
T
T
Therefore, using (55), the above computations, (56) and (Pn ), we deduce that
p
n+1 n+1 q̄
1 − C ε(E0 + 1) 6 Cε(1 + E0 )2 .
, m̄
FT
−1
Choosing ε 6 (4C 2 (1 + E0 )) , this implies (Pn+1 ). The sequence {(q n , mn )}n∈N is
therefore bounded in the space FT . Moreover, (53) holds for all n ∈ N.
Second step: convergence of the sequence in FT .
Next, we are going to show that (q n , mn ) converges strongly in FT to a solution (q, m)
of (39) (40). We denote δq n = q n+1 − q n and δmn = mn+1 − mn . According to (51) and
(52), we have

∂ δq n + div δmn = 0, 
 t
n

div δmn

 ∂ δmn − µ̄ div ∇δm
− (λ̄ + µ̄)∇
− κ̄∇ (1 + q n )1δq n
t
n
n
1+q
1 + q n−1 n−1 
n
n

=
Γ
q
,
m
−
Γ
q
,
m
+
H
q n−1 , mn−1 , q n , mn ,


 n
n
δq
|t =0
= 0,
δm
|t =0
with
(57)
= 0,
δq n−1 ∇mn
H q , m , q , m = −µ̄ div
(1 + q n )(1 + q n−1 )
δq n−1 div mn
n−1
n
− (λ̄ + µ̄)∇
+
κ̄∇
δq
1q
.
(1 + q n )(1 + q n−1 )
We keep the same assumptions on ε as in the first step. According to Proposition 5, (53),
(Pn ) and (55), we thus get
n−1
n−1
n
n
δq n , δmn 6 C Γ q n , mn − Γ q n−1 , mn−1 1 d/2−1
FT
LT (B
)
n−1
n−1
n
n
+ H q , m , q , m 1 d/2−1 .
LT (B
def
)
(58)
Denote δΓin = Γi (q n , mn ) − Γi (q n−1 , mn−1 ). Thanks to Proposition 3, to Lemma 1, (53),
(54) and (55), we obtain the following estimates:
118 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
n
δΓ 1
L1T (B d/2−1 )
. T 1 + q n Le∞ (B d/2 ) + q n−1 Le∞ (B d/2 ) δq n−1 Le∞ (B d/2 ) ,
T
T
T
n
δΓ n
∞ d/2 mn−1 2 d/2
eT (B )
eT (B )
L1T (B d/2−1 ) . 1 + q L
L
n
n−1 + m Le2 (B d/2 ) δm Le2 (B d/2 )
T
T
n−1 2
n−1 + m Le2 (B d/2 ) 1 + q Le∞ (B d/2 )
T
T
n
n−1 × 1 + q Le∞ (B d/2 ) δq
d/2 ) ,
e∞
L
T
T (B
2
n
δΓ 3
L1T (B d/2−1 )
T
T
T
+ mn−1 Le2 (B d/2 ) q n−1 Le2 (B d/2+1 )
T
T
+ q n Le2 (B d/2+1 ) δq n−1 Le∞ (B d/2 ) ,
T
T
n
δΓ 4
L1T (B d/2−1 )
5
L1T (B d/2−1 )
n
δΓ . q n Le2 (B d/2+1 ) δmn−1 Le2 (B d/2 ) + mn−1 Le2 (B d/2 ) δq n−1 Le2 (B d/2+1 )
T
. ∇δq n−1 Le2 (B d/2 ) ∇q n−1 Le2 (B d/2 ) + ∇q n Le2 (B d/2 ) ,
. f T
L1T (B d/2−1 )
T
n−1 δq
d/2 )
e∞
L
T (B
T
,
δq n−1 ∇mn
(1 + q n )(1 + q n−1 ) L1T (B d/2 )
. 1 + q n−1 Le∞ (B d/2 ) 1 + q n Le∞ (B d/2 ) ∇mn L1 (B d/2 ) kδq n−1 kLe∞ (B d/2 ) ,
T
T
T
T
n−1
δq
1q n L1T (B d/2 )
. 1q n L1 (B d/2 ) δq n−1 Le∞ (B d/2 ) .
T
T
Using (Hε ) and (Pn ) in the above computations, then (58), we thus get
√
δq n , δmn ) 6 C ε(E0 + 1)3 .
FT
Now, if we choose an ε such that ε 6 (4C 2 )−1 (E0 + 1)−6 holds, (qn , mn )n∈N is clearly
a Cauchy sequence and thus converges in FT to a limit (q, m) which satisfies (53).
The verification that the limit is solution of (39) (40) in the sense of distributions is a
straightforward application of Proposition 3. 2
Third step: uniqueness in FeT
Consider the solution (q, m) built in the previous part and suppose that (q 0 , m0 ) ∈
2
e
eT (B d/2 × (B d/2−1 )d ) also solves (39) (40) with initial data
LT (B d/2+1 × (B d/2 )d ) ∩ C
0
(q0 , m0 ). Denote δq = q − q and δm = m0 − m. We have



∂t δq + div δm = 0,






∇δm
div δm


− (λ̄ + µ̄)∇
− κ̄∇ (1 + q)1δq
 ∂t δm − µ̄ div
1+q
1+q



= Γ (q 0 , m0 ) − Γ (q, m) − H (q 0 , m0 , q, m),







 δq
= 0, δm
= 0.
|t =0
|t =0
(59)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
119
Let T ∗ ∈ [0, T ] be the greatest time such that (53) is satisfied by q 0 on [0, T ∗ ]. Continuity
for q 0 in C([0, T ]; B d/2 ) implies that 0 < T ∗ 6 T .
We now apply Proposition 5 to (59) using bounds of step 1 and (53) for q. Unlike
in the second step, our assumptions on (q 0 , m0 ) only provide us with bounds for H in
e 2 ∗ (B d/2−2). This leads to the following estimate:
L
T
k(δq, δm)kFe ∗ . kΓ (q 0 , m0 ) − Γ (q, m)kL1 ∗ (B d/2−1 ) + kH (q 0 , m0 , q, m)ke
L2
T
T∗
T
(B d/2−2 )
.
Using the same estimates as in step 2 for kΓ (q 0 , m0 ) − Γ (q, m)kL1 ∗ (B d/2−1 ) , and the fact
T
that
kH (q 0 , m0 , q, m)ke
L2
T∗
(B d/2−2 )
. 1 + kqke
1 + kq 0 kLe∞ (B d/2 ) km0 kLe2
L∞ (B d/2 )
T∗
(B
T∗
T∗
d/2 )
+ k∇q 0 kLe2
(B
T∗
d/2 )
,
we finally gather
k(δq, δm)kFe ∗ . 1 + kqkLe∞ (B d/2 ) 1 + kq 0 kLe∞ (B d/2 )
T∗
T
T∗
∗
× T + kf kL1 ∗ (B d/2−1 ) + K kmkLe2 (B d/2 )
T
T∗
+ k∇qke
L2
T∗
+ k∇q 0 kLe2
(B d/2 )
T∗
(B d/2 )
+ km0 kLe2
T∗
(B d/2 )
k(δq, δm)kFe ∗ ,
T
with K(z) = z + z . This obviously entails δq ≡ 0 and δm ≡ 0 on a suitably small
interval [0, T 0 ] with T 0 > 0.
Using the same arguments as for the proof of Lemma 4, we can now conclude that the
two solutions coincide on the whole interval [0, T ]. 2
2
6. Local strong solutions near equilibrium
In this section, we want to show that local well-posedness for Korteweg system when
the density is close to a constant also holds in spaces of type Bps with p 6= 2, that is, in
spaces which are not related to physical energy spaces. Recall that this approach was
extensively used for the study of incompressible Navier–Stokes equations (see [7] and
the references enclosed).
This viewpoint enables us to get well-posedness even if the initial velocity belongs to
a space Bps such that the regularity index s is negative, which in particular is relevant for
oscillating initial velocities.
To avoid tedious discussions about the definition of qf, we suppose from now on, that
f ≡ 0 (see Remark 3 at the end of the section). Using the same notations as in Section 4,
the Korteweg system rewrites as follows:
∂t q + div m = 0,
(60)
∂t m − µ̄1m − (λ̄ + µ̄)∇div m − κ̄∇1q = G(q, m),
(61)
(q, m)|t =0 = (q0 , m0 ),
(62)
120 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
where we define G = G1 + G2 + G3 + G4 by
G1 (q, m) = −div
m⊗m
,
1+q
G2 (q, m) = −∇ P (ρ̄(1 + q)) − P (ρ̄) /ρ̄,
q
q
G3 (q, m) = −µ̄ div ∇
⊗m+
∇m
1+q
1+q
q
q
− (λ̄ + µ̄)∇ ∇
·m+
div m ,
1+q
1+q
and
κ̄
G4 (q, m) = − ∇ |∇q|2 − κ̄ div q∇ 2 q .
2
Let us emphasize that no stability assumption on the pressure law is required: we just
[d/p]+3,∞
have to assume that P is suitably smooth (P ∈ Wloc
is enough).
In Section 6.1 we study the linearized system around (0, 0) where we drop the first
order linearized pressure term. Local well-posedness for (q0 , m0 ) is then proved in
Section 6.2 through a fixed-point argument.
6.1. Estimates for the linearized pressure-less system
This section is devoted to the proof of estimates for the following linear system
∂t q + div m = F,
∂t m − µ̄1m − (λ̄ + µ̄)∇div m − κ̄∇1q = G.
(LNSK3)
The main result of this section is the following proposition:
P ROPOSITION 6. – Let s ∈ R, p ∈ [1, +∞], 1 6 ρ1 6 +∞ and T ∈ ]0, +∞ ]. If
e ρ1 (B s−2+2/ρ1 × (B s−3+2/ρ1 )d ), then the linear
(q0 , m0 ) ∈ Bps × (Bps−1 )d and (F, G) ∈ L
T
p
p
eT (B s × (B s−1 )d ) ∩ L
e ρ1 (B s+2/ρ1 ×
system (LNSK3) has a unique solution (q, m) ∈ C
T
p
p
p
(Bps−1+2/ρ1 )d ). Moreover, for all ρ ∈ [ρ1 , +∞], there exists a constant C depending only
on µ̄, λ̄, κ̄, ρ, ρ1 and d such that the following inequality holds:
kqke
s+2/ρ + kmk ρ
s−1+2/ρ
ρ
e
)
)
L (B
L (B
T
p
T
p
6 C kq0 kBps + km0 kBps−1 + kF kLeρ1 (B s−2+2/ρ1 ) + kGke
ρ
s−3+2/ρ1 .
)
L 1 (B
T
p
p
T
Proof. – Apply operator 1 to the first equation and operators div and curl (with
curl g := ∂j gi − ∂i gj ) to the second one. Denoting ν̄ = λ̄ + 2µ̄, we obtain

 ∂t 1q + 1div m = 1F,

∂t div m − ν̄1div m − κ̄12 q = div G,
∂t curl m − µ̄1curl m = curl G.
(63)
Proposition 2 gives the following estimates for the third equation, which decouples from
the first two equations
kcurl mkLeρ (B s−2+2/ρ ) 6 C kcurl m0 kBps−2 + kcurl GkLeρ1 (B s−4+2/ρ1 ) .
T
p
T
p
(64)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
121
The following lemma points out a smoothing effect for the first two equations:
L EMMA 3. – Let s ∈ R, (p, ρ1) ∈ [1, +∞ ]2 and T ∈ ]0, +∞]. Suppose that
e ρ1 (B s−2+2/ρ1 ))2 . Then the system
(c0 , v0 ) ∈ (Bps )2 and (h, k) ∈ (L
T
p

 ∂t c + 1v = h,

∂t v − ν̄1v − κ̄1c = k,
(c, v)|t =0 = (c0 , v0 ),
(65)
eT (B s ) ∩ L
e ρ1 (B s+2/ρ1 ))2 . Moreover, for all ρ ∈
has a unique solution (c, v) in (C
T
p
p
[ρ1 , +∞], there exists C > 0 depending only on ν̄, κ̄, ρ, ρ1 such that
k(c, v)ke
s+2/ρ 6 C k(c0 , v0 )kB s + k(h, k)k ρ1
ρ
e (B s−2+2/ρ1 ) .
)
p
L (B
L
T
p
T
p
Using (64), Lemma 3 with c0 = 1q0 , v0 = div m0 , h = 1f , k = div g and noticing
that 1m = ∇div m + divcurl m, Proposition 6 is now obvious. 2
Proof of Lemma 3. – Denoting by U (t) the semi-group associated to (65), we deduce
from Duhamel’s formula that
c(t)
v(t)
c
= U (t) 0
v0
+
Zt
h(s)
U (t − s)
k(s)
ds,
0
with U (t) = e−t A(D) and
A(ξ ) =
0
κ̄|ξ |2
−|ξ |2
.
ν̄|ξ |2
Straightforward computations show that
e
−t A(ξ )
=e
2
− t ν̄|ξ|
2
h1 (t, ξ ) + ν̄2 h2 (t, ξ )
−κ̄h2 (t, ξ )
h2 (t, ξ )
h1 (t, ξ ) − ν̄2 h2 (t, ξ )
with
sin(ν 0 |ξ |2 t)
, if ν̄ 2 < 4κ̄,
ν0
h1 (t, ξ ) = 1, h2 (t, ξ ) = t|ξ |2 , if ν̄ 2 = 4κ̄,
sinh(ν 0 |ξ |2 t)
h1 (t, ξ ) = cosh(ν 0 |ξ |2 t), h2 (t, ξ ) =
, if ν̄ 2 > 4κ̄
ν0
h1 (t, ξ ) = cos(ν 0 |ξ |2 t),
h2 (t, ξ ) =
p
and ν 0 = |κ̄ − ν̄ 2 /4|.
Let ϕe be a smooth function supported in {ξ ∈ Rd | |ξ |±1 6 2} and such that ϕe ≡ 1 on
Supp ϕ. Denote aij (t, ξ ) the coefficients of the matrix e−t A(ξ ) and
q
Hij (t, x)
= (2π )
−d
Z
e −q ξ ) dξ.
eix·ξ aij (t, ξ )ϕ(2
122 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
We assume that the following inequality holds:
kHij kL1 6 Ce−c min(1,4κ̄/ν̄
q
2 )22q ν̄t
,
(66)
where C depends only on ν̄, d and κ̄, and c is a universal constant. Since
U (t)
1q c
(x) =
1q v
q
q
(H11 (t, ·) ? 1q c)(x) + (H12 (t, ·) ? 1q v)(x)
,
q
q
(H21 (t, ·) ? 1q c)(x) + (H22 (t, ·) ? 1q v)(x)
estimate (66) yields
kU (t)(1q c, 1q v)kLp 6 Ce−c min(1,4κ̄/ν̄
2 )22q ν̄t
k1q ckLp + k1q vkLp .
Now, we complete the proof of Lemma 3 coming back to the definition of Besov spaces,
and using convolution inequalities.
q
q
In order to prove (66), we first remark that kHij kL1 = khij kL1 with
hij (t, y) = (2π )−d
q
Z
e
eiy·η aij t, 2q η ϕ(η)
dη.
q
All the functions hij are of the type
hq (t, x) =
Z
e ) dξ,
eix·ξ f 22q |ξ |2 t ϕ(ξ
(67)
for a function f ∈ C ∞ (R+ ). Using integrations by parts and Leibniz’ formula, we get
for all α ∈ Nd ,
(−ix) h (x) =
α q
X α Z
β
β6α
e ) dξ.
eix·ξ ∂ β f 22q |ξ |2 t ∂ α−β ϕ(ξ
(68)
Next, from Faà-di-Bruno’s formula, we deduce that
∂ f 2 |ξ | t =
β
2q
2
X
f
(m)
2 |ξ | t 2 t
2q
2
2q
m
m
Y
!
∂ (|ξ | ) .
γj
2
(69)
j =1
γ1 +···+γm =β
|γi |>1
Let us suppose first that ν̄ 2 < 4κ̄. Then, we just have to prove that
khq kL1 6 Ce−c2
0
for f (u) = eiν u e−ν̄u/2 and ν 0 =
2q ν̄t
p
κ̄ − ν̄ 2 /4. We have
f (m) (u) = iν 0 −
ν̄
2
m
0
eiν u e−ν̄u/2 ,
(70)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
123
so that |f (m) (u)| 6 (ν 0 + ν̄/2)m e−ν̄u/2 . Using (68), (69) and that Supp ϕe ⊂ {ξ ∈ Rd |
|ξ |±1 6 2}, we prove the existence of constants Cα,β,m such that
|β|
α q XX
m
2q
x h (x) 6
Cα,β,m 22q t e−ν̄t 2 /8 .
β6α m=1
For any constant c < 1 and m ∈ N, there exists Cm such that um e−u 6 Cm e−cu . This
clearly yields (70).
When ν̄ 2 = 4κ̄ , we must verify (70) for f (u) = ue−ν̄u/2 and f (u) = e−ν̄u/2 . This is
obvious in view of (69) and Leibniz’ formula. When ν̄ 2 > 4κ̄, we must verify (70) for
s
ν̄
f (u) = exp − 1 ±
2
! !
4κ̄
1− 2 u .
ν̄
Using again (69) we thus get
√
α q x h (x) 6 Ce−cν̄t 22q (1± 1−4κ̄/ν̄ 2 ) 6 Ce−c0 (κ̄/ν̄)22q t
2
and we conclude to (66).
6.2. Local well-posedness for an initial density close to a constant
In this section, we agree that B s stands for Bps . Let us introduce the functional spaces
needed in the local existence theorem. We will prove existence in
eT B d/p × B d/p−1 )d ) ∩ L1 B d/p+2 × B d/p+1
ET = C
T
p
d and uniqueness in
eT B d/p × B d/p−1
EeT = C
p
e 2 B d/p+1 × (B d/p
∩L
T
d .
We have the following result:
T HEOREM 5. – Let p ∈ [1, +∞[. Then there exists η > 0 such that if q0 ∈ B d/p ,
m0 ∈ (B d/p−1 )d and
kq0 kB d/p 6 η,
then there exists T > 0 such that system (60)–(62) has a unique solution (q, m) in EeT .
p
In addition, (q, m) belongs to ET .
p
Proof. – For T > 0 and p ∈ [1, +∞[, we denote
k(q, m)kF p = kqkLe∞ (B d/p ) + kqkL1 (B d/p+2 ) + kmkLe2 (B d/p ) + kmkL1 (B d/p+1 ) .
T
T
T
T
T
124 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
Let (q0 , m0 ) be as in Theorem 5 and denote by (qL , mL ) the solution of the linearized
pressure-less system on the interval [0, T ]:

 ∂t q + div m = 0,

∂t m − µ̄1m − (λ̄ + µ̄)∇div m − κ̄∇1q = 0,
(q, m)|t =0 = (q0 , m0 ).
Denoting by V (t) the semi-group generated by the above system, we have
(qL , mL )(t) = V (t)(q0 , m0 ).
Let us define
def
ΦqL ,mL (q̄, m̄) =
Zt
V (t − s) 0, G(qL + q̄, mL + m̄)(s) ds.
0
In order to prove the existence part of the theorem, we just have to show that ΦqL ,mL has a
p
p
fixed point in ET . Since ET is a Banach space, we are going to prove that ΦqL ,mL satisfies
p
the hypotheses of Picard’s theorem in a ball B(0, R) of ET for sufficiently small R.
1st step: Stability of B(0, R)
Denote q = qL + q̄ and m = mL + m̄. According to Proposition 6, we have
kΦqL ,mL (q̄, m̄)kEp 6 CkG(q, m)kL1 (B d/p−1 ) .
T
(71)
T
Under the assumption
kqkL∞ ([0,T ]×Rd ) 6 1/2,
(H)
and using Proposition 3 and Lemma 1, we state the following estimates:
kG1 (q, m)kL1 (B d/p−1 ) . (1 + kqkLe∞ (B d/p ) )kmk2Le2 (B d/p ) ,
(72)
kG2 (q, m)kL1 (B d/p−1 ) . T kqkLe∞ (B d/p ) ,
(73)
T
T
T
T
T
kG3 (q, m)kLe1 (B d/p−1 ) . kqkLe∞ (B d/p ) k∇mkLe1 (B d/p ) + kqkLe2 (B d/p+1 ) kmkLe2 (B d/p ) ,
(74)
kG4 (q, m)kL1 (B d/p−1 ) . k∇qk2Le2 (B d/p ) + kqkLe∞ (B d/p ) k∇ 2 qkL1 (B d/p ) .
(75)
T
T
T
T
T
T
T
T
T
This leads to the following inequality:
kΦqL ,mL (q̄, m̄)kEp 6 C 1 + kqL kLe∞ (B d/2 ) k(qL , mL )kF p + k(q̄, m̄)kF p
T
T
T
× T + k(qL , mL )k
p
FT
+ k(q̄, m̄)k
p
FT
T
.
Let c be a constant such that k · kB d/2 6 c implies k·kL∞ 6 1/5. We choose
R = min (10C)−1 , c, 1 ,
(76)
and suppose that kq0 kB d/p 6 R/2. Then, since qL ∈ C([0, T ]; B d/p ), we have
k(qL , mL )kF p 6 R for T small enough so that (H) is fulfilled. We also suppose that
T
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
T 6 R/4 and we get
ΦqL ,mL B(0, R) ⊂ B(0, 9R/10).
125
(77)
2nd step: Contraction properties
p
Suppose that (q̄1 , m̄1 ) and (q̄2 , m̄2 ) belong to the ball B(0, R) of ET , and denote q1 =
qL + q̄1 , m1 = mL + m̄1 , q2 = qL + q̄2 and m2 = mL + m̄2 . Using again Proposition 6,
we get
Φq
L ,mL
(q̄2 , m̄2 ) − ΦqL ,mL (q̄1 , m̄1 )Ep 6 C G(q2 , m2 ) − G(q1 , m1 )L1 (B d/p−1 ) .
T
T
Under assumption (H) for q1 and q2 , we can derive estimates for Gi (q2 , m2 ) −
Gi (q1 , m1 ). We apply Proposition 3 and Lemma 1 to the following identities:
G1 (q2 , m2 ) − G1 (q1 , m1 )
= div (m1 ⊗ m1 )(qe2 − qe1 ) − qe2 m2 ⊗ (m2 − m1 ) + (m2 − m1 ) ⊗ m1 ,
G2 (q2 , m2 ) − G2 (q1 , m1 ) = −∇(P (ρ̄(1 + q2 )) − (P (ρ̄(1 + q1 ))/ρ̄,
G3 (q2 , m2 ) − G3 (q1 , m1 )
= −µ̄ div ∇ qe1 ⊗ (m2 − m1 ) + qe1 ∇(m2 − m1 )
+ ∇(qe2 − qe1 ) ⊗ m2 + qe2 − qe1 )∇m2 − (λ̄ + µ̄)∇(· · ·),
κ̄
G4 (q2 , m2 ) − G4 (q1 , m1 ) = − ∇ ∇(q2 − q1 ) · ∇(q2 + q1 )
2
− κ̄ div (q2 − q1 )∇ 2 q2 + q1 ∇ 2 (q2 − q1 ) ,
def
where qei = qi /(1 + qi ). We finally get a constant C such that
Φq
L ,mL
(q̄2 , m̄2 ) − ΦqL ,mL (q̄1 , m̄1 )Ep
T
6 CT kq̄2 − q̄1 kL∞
d/p ) + Ck(q̄2 − q̄1 , m̄2 − m̄1 )kF p
T (B
T
× k(q1 , m1 )kF p + k(q2 , m2 )kF p .
T
(78)
T
We make the same assumption on R and T as in the first step (replacing C with a larger
constant if necessary) and get
Φq
L ,mL
(q̄2 , m̄2 ) − ΦqL ,mL (q̄1 , m̄1 )Ep 6
T
9
k(q2 − q1 , m2 − m1 )kF p .
T
10
p
This completes the proof of the existence of a solution (q, m) for (60)–(62) in ET , which
in addition satisfies
kqkL∞ ([0,T ]×Rd ) 6
2
5
and
kqkLe∞ (B d/2 ) 6
T
1
.
5C
(79)
126 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
3rd step: Uniqueness
Uniqueness is a straightforward application of the following lemma, which yields also
the global uniqueness result of Section 4.
L EMMA 4. – There exists a constant C depending only on d and p such that if
T ∈ ]0, +∞] and (qi , mi ) (i = 1, 2) are two solutions of (60)–(62) belonging to EepT
and (q1 , m1 ) satisfies (79) with the constant C, then (q2 , m2 ) ≡ (q1 , m1 ) on [0, T ].
Proof. – Let T ∗ be the largest time such that (H) is satisfied by q2 . As kq2 (0)kL∞ 6
1/5 and q2 ∈ C([0, T ] × Rd ), we have T ∗ > 0. Let [0, Tm ] ⊂ [0, T ∗ ] be the biggest
interval such that the two solutions coincide on [0, Tm ]. Suppose that Tm < T . Let
def
e i (t) = qi (t − Tm ), mi (t − Tm ) .
qei (t), m
Continuity in time for qi implies that (H) is satisfied by qe1 and qe2 on an interval [0, ε]
ep × E
ep .
e i )}16i62 belongs to E
for ε > 0 small enough. Moreover {(qei , m
ε
ε
We now use the decomposition
G(qi , mi ) = G1 (qi , mi ) + G2 (qi , mi ) + G03 (qi , mi ) + G04 (qi , mi ) + G05 (qi , mi )
with
G03 (qi , mi ) = −µ̄1
qi mi
1 + qi
qi mi
− (λ̄ + µ̄)∇div
,
1 + qi
κ̄
G04 (qi , mi ) = − ∇ |∇qi |2 + κ̄ div ∇qi ⊗ ∇qi ,
2
0
G5 (qi , mi ) = −κ̄1(qi ∇qi ).
Proposition 3 provides us with estimates for G1 (qi , mi ), G2 (qi , mi ) and G03 (qi , mi ) in
e 2 (B d/p−2 ). Thanks to Proposition 6,
L1T (B d/p−1), and for G04 (qi , mi ) and G05 (qi , mi ) in L
T
we get
e2 −m
e 1 )kEep 6 Z(ε)k(qe2 − qe1 , m
e2 −m
e 1 )kEep ,
k(qe2 − qe1 , m
ε
ε
with
Z(ε) = Cε + C sup kqi kLe∞ (B d/p ) + kqi kLe2 (B d/p+1 ) + kmi kLe2 (B d/p ) .
T
T
T
i∈{1,2}
eT (B d/p ), we infer that
From (79) for q1 and the definition of the space C
lim Z(ε) = 2Ckq1 (Tm )kB d/p 6 4/5,
ε→0
e 2 ) = (qe1 , m
e 1 ) on [0, ε]. This
thus Z(ε) < 1 for ε small enough. We therefore get (qe2 , m
achieves the proof of Lemma 4. 2
Remark 3. – In Theorem 5, we supposed that the external forcing term f vanishes.
We can easily show that the local existence and uniqueness still holds if f belongs to
L1T (B d/p−1) where p < 2d. Indeed, the usual product maps B d/p × B d/p−1 into B d/p−1 ,
provided p < 2d.
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
127
7. Further remarks
7.1. Weak solutions in dimension 2
We now focus on the 2-dimensional problem and study existence results of weak
solutions. A weak solution (ρ, u) of (8)–(10) in D 0 (R+ × R2 ) is required to satisfy the
finite energy a priori bounds (16) (17) for initial data verifying (15). More precisely, we
require that for all test functions (ψ, φ) ∈ C0∞ ([0, ∞) × R2 ) × C0∞ ([0, ∞) × R2 )2 ,
Z
ρψ(t) dx =
R2
Z
Z
ρ0 ψ(0) dx −
R2
Z
ρu(t) · φ(t) dx =
R2
Zt
Z
ds
0
ρ0 u0 · φ(0) dx +
R2
Z Zt
ρu ⊗ u : D(φ) − µ D(u) : D(φ)
ds
0
ρu · ∇ψ dx,
R2
R2
− (λ + µ)div u div φ + p(ρ)div φ
|∇ρ|2
ρ2
+κ
div φ + ∇ρ ⊗ ∇ρ : D(φ) − 1div φ dx.
2
2
As pointed out in the introduction, we are not able to prove a global existence theorem
like in [17] when κ 6= 0 because of the quadratic terms ∇ρ ⊗ ∇ρ, even though ∇ρ is a
priori bounded in L∞ ((0, T ); L2 (Rd ))d . We focus now on weak solutions in dimension
d = 2 near a stable equilibrium, i.e. solutions (ρ, u) close to (ρ̄, 0), where ρ̄ > 0 satisfies
P 0 (ρ̄) > 0. Let us introduce
ρ(t) − ρ̄ δ(t) = ρ̄
L∞
,
(80)
which measures the density fluctuation in L∞ . Unfortunately, such an a priori bound
does not seem to be available for finite energy initial data. However, in view of the
results of the preceding sections, it is valid for suitably smooth initial data and small
enough time, which motivates the following result
P ROPOSITION 7. – There exists η > 0 such that as long as
E0 + sup δ(t) 6 η,
t ∈[0,T ]
there exists a weak solution (ρ, u) on (0, T ) of (NSK) such that ρ − ρ̄ ∈ L2 (0, T ;
Ḣ 1 (R2 ) ∩ Ḣ 2 (R2 )), u ∈ L2 (0, T ; Ḣ 1 (R2 ))2 ∩ L∞ (0, T ; L2 (R2 ))2 .
Proof. – First, multiplying the equation of momentum conservation by ∇ρ 2 and
integrating by part, we easily infer
d
dt
Z
R2
m · ∇ρ 2 dx +
κ
2
Z
2
1ρ 2 dx
R2
128 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
=
Z
2ρ|div m|2 + 2κ|∇ρ|4 + ∇ 2 ρ 2 : (ρu ⊗ u)
R2
− (λ + 2µ)1ρ 2 div u − 2ρP 0 (ρ)|∇ρ|2 dx,
(81)
0
hence, denoting Z(s) = 2sP (s), we obtain
Z
Z
2 d
κ
m · ∇ρ 2 dx +
Z(ρ̄)|∇ρ|2 + 1ρ 2 dx
dt
4
R2
6
R2
3 C 1 + kρkL∞ k∇uk2L2 + CkρkL∞ ku · ∇ρk2L2 + Ck∇ρk4L4
+ Ckρk2L∞ kuk4L4 + CkZ(ρ) − Z(ρ̄)kL∞ k∇ρk2L2 .
Let us now recall Gagliardo–Nirenberg’s inequality
kf k2L4 6 Ckf kL2 k∇f kL2 .
(82)
Thus, we obtain assuming that δ(t) < 1/2
kuk4L4 6 Ckuk2L2 k∇uk2L2 ,
2
2
k∇ρk2L4 6 C ρ −1 L∞ ∇ρ 2 L4 6 C 1 + δ(t)2 ∇ρ 2 L2 1ρ 2 L2
6 C 1 + δ(t)3 k∇ρkL2 1ρ 2 L2 ,
ku · ∇ρk2L2 6 kuk2L4 k∇ρk2L4 6 C 1 + δ(t)3 kukL2 k∇ukL2 k∇ρkL2 1ρ 2 L2 ,
√
kukL2 6 C(1 − δ(t))−1/2 k ρukL2 .
Hence, for δ(t) < η (where η > 0 depends on Z 0 ), we have for some C0 > 0
Z
d
m · ∇ρ 2 dx + C0 k∇ρk2L2 + k1ρ 2 k2L2
dt
R2
6 Ck∇uk2L2 + CkukL2 k∇ukL2 k∇ρkL2 1ρ 2 L2
2
+ Ck∇ρk2L2 1ρ 2 L2 + Ckuk2L2 k∇uk2L2 ,
2
6 Ck∇uk2L2 1 + kuk2L2 + Ck∇ρk2L2 1ρ 2 L2
2
√
6 Ck∇uk2L2 1 + k ρuk2L2 + k∇ρk2L2 1ρ 2 L2 ,
2 6 CE0 k∇uk2L2 + 1ρ 2 L2 + Ck∇uk2L2 .
Therefore, as soon as E0 is small enough, we have for some constant C1 > 0
d
dt
Z
2 m · ∇ρ 2 dx + C1 k∇ρk2L2 + 1ρ 2 L2 6 Ck∇uk2L2 .
R2
Let α > 0 and define wα by
1 √
κ
wα (t) = k ρuk2L2 + π(ρ) − π(ρ̄)L1 + k∇ρk2L2 + α
2
2
Z
Rd
m · ∇ρ 2 dx.
(83)
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
129
Then, choosing α small enough, we deduce that
1
3
wα (t) 6 w0 (t) 6 wα (t),
2
2
and from the energy bounds and (83) that
2 d
wα (t) + C2 k∇uk2L2 + k∇ρk2L2 + 1ρ 2 L2 6 0,
dt
(84)
so that we have the claimed a priori bounds. Now considering a suitable approximate
problem, we can easily pass to the limit in quadratic terms like ∇ρ ⊗ ∇ρ, since ∇ρ
is bounded in L2 (0, T ; H 1 (Rd )), and also in C([0, T ]; H −1 (Rd )), as can be seen by
writing a linear transport equation on ∇ρ. 2
7.2. Blow-up of solutions with compactly supported density
Let us finally make a few remarks based upon the work of Z. Xin [22] in the noncapillary case. We consider the full Navier–Stokes Korteweg system (1)–(3) (with energy
equation), for which Hattori and Li [14] proved global existence of H s solutions (for
s > 0 large) close enough to constant states |(ρ0 , m0 , θ0 ) − (ρ̄, 0, θ̄)| 1 when the heat
conduction parameter α is positive. We expect that the preceding results, namely global
well-posedness of the Korteweg system, still hold in scale invariant Besov spaces for
solutions close to constant states.
In contrast to the preceding approach where ρ is close to a constant, we assume now
that the initial data satisfy
def
A(0) =
Z Rd
|u0 − x|2
ρ0
+ ρ0 e0 dx < +∞,
2
which is the case for instance if ρ0 has compact support. For the sake of simplicity,
we consider a perfect gas law p = ρRθ , e = cv θ , and set γ = 1 + R/cv . When
γ ∈ (1, 1 + 2/d], we define σ (t) = (1 + t), whereas when γ ∈ (1 + 2/d, ∞), we take
σ (t) = t. Then, we deduce from easy computations (see [22]) that
d
A(t) = σ (t)
dt
Z
(2ρe − dp) dx,
Rd
where
A(t) =
Z Rd
|uσ (t) − x|2
(d + 2)
ρ
+ σ (t)2 ρe (t) dx + κ
2
2
Zt
Z
σ (s)
0
|∇ρ(s, x)|2 ds dx.
Rd
Using the fact that p = (γ − 1)ρe and denoting δ = 2 − d(γ − 1), we obtain
Ȧ(t) 6 δ
A(t)
,
σ (t)
hence A(t) 6 A(0)σ (t)max(δ,0).
130 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
Therefore, we have
σ (t)min(2,d(γ −1))
Z
p dx 6 (γ − 1)A(0).
(85)
Rd
As a consequence, we obtain in the isothermal case θ = θ̄
RM θ̄σ (t)min(2,d(γ −1)) 6 (γ − 1)A(0),
denoting by M the total mass, which proves that solutions blow up after some critical
time T0 as soon as A(0) is finite.
Let us remark that similar observations can be done in the isentropic case, as well as
in the non-isentropic case when the thermal diffusion is neglected (i.e. α = 0). In order
to get blow up estimates, we have to consider initial densities compactly supported in
Rd (see [22]), and observe that the support of the density does not grow as time evolves.
Then, blow up estimates stem from estimate (85) and Hölder’s inequality.
Appendix
This section is devoted to a commutation lemma that we used to prove Proposition 5.
e 2 (B d/2+1 ) and B ∈ L
e 2 (B d/2 ). Then the following
L EMMA 5. – Suppose A ∈ L
T
T
estimate holds on [0, T ]:
k∂k [A, 1` ]BkL1 (L2 ) 6 Cc` 2−`(d/2−1)kAkLe2 (B d/2+1 ) kBkLe2 (B d/2−1 ) ,
T
where C depends only on d, and
T
P
`∈Z c`
T
6 1.
Proof. – The proof of the above lemma requires some paradifferential calculus. We
have to recall here that paradifferential calculus enables to define a generalized product
between distributions, which is continuous in many functional spaces where the usual
product does not make sense (see the pioneering work of J.-M. Bony in [4]). The
paraproduct between u and v is defined by
def X
Tu v =
S`−1 u1` v.
`∈Z
We thus have the following formal decomposition (modulo a polynomial):
uv = Tu v + Tv0 u,
def X
with Tv0 u =
S`+2 v1` u.
`∈Z
Coming back to the proof of Lemma 5, we split ∂k [A, 1` ]B into
∂k [A, 1` ]B = ∂k T10 ` B A − ∂k 1` TB0 A + [TA , 1` ]∂k B + T∂k A 1` B − 1` T∂k A B.
R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
From now on, we agree that (c` )`∈Z denotes a positive sequence such that
According to (20) and to the definition of T 0 , we have
X
∂k T10 ` B A =
P
`∈Z c`
131
6 1.
∂k (Sm+2 1` B1m A).
m>`−2
e ρ (B s ) spaces,
Thus, in view of Bernstein’s lemma and the definition of L
T
X
k∂k T10 ` B AkL1 (L2 ) .
T
2m 2`d/2 k1` BkL2 (L2 ) k1m AkL2 (L2 ) ,
T
m>`−2
X
. 2`d/2k1` BkL2 (L2 )
T
T
2−md/2 2m(d/2+1) k1m AkL2 (L2 ) ,
T
m>`−2
. 2−`(d/2−1) 2`(d/2−1) k1` BkL2 (L2 )
X
T
2m(d/2+1) k1m AkL2 (L2 ) ,
T
m>`−2
. c` 2−`(d/2−1) kBkLe2 (B d/2−1 ) kAkLe2 (B d/2+1 ) .
T
T
We use classical estimates for the paraproduct to bound the second term of the right-hand
side (see [7] and [8]). We get
kTB0 AkL1 (B d/2 ) . kBkL2 (B d/2−1 ) kAkL2 (B d/2+1 ) .
T
T
T
e ρ (B s ) spaces, we get
Using spectral localization of 1` and definition of L
T
k∂k 1` TB0 AkL1 (L2 ) . c` 2−`(d/2−1) kBkLe2 (B d/2−1 ) kAkLe2 (B d/2+1 ) .
T
T
T
According to (20), the third term reads
[TA , 1` ]∂k B =
X
[Sm−1 A, 1` ]1m ∂k B.
|m−`|64
Applying first order Taylor’s formula, we get for x ∈ Rd ,
[Sm−1 A, 1` ]1m ∂k B(x)
−`
=2
Z Z1
h(y) y · Sm−1 ∇A x − 2−` τ y 1m ∂k B x − 2−` y dτ dy.
Rd 0
Convolution inequality thus yields
[Sm−1 A, 1` ]1m ∂k B L2
hence
[TA , 1` ]∂k B L1T (L2 )
. 2−` k∇AkL∞ k1m ∂k BkL2 ,
. c` 2−`(d/2−1) k∇AkL2 (L∞ ) kBkLe2 (B d/2−1 ) .
T
Finally, thanks to (20), we have
T∂k A 1` B =
X
|`−m|61
Sm−1 ∂k A 1` 1m B
T
132 R. DANCHIN, B. DESJARDINS / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 97–133
so that
kT∂k A 1` BkL1 (L2 ) 6 k∂k AkL2 (L∞ ) k1` BkL2 (L2 ) .
T
T
T
Classical estimates for the paraproduct yield
kT∂k A BkL1 (B d/2−1 ) . kBkLe2 (B d/2−1 ) k∂k AkLe2 (B d/2 )
T
T
so that the proof of Lemma 5 is achieved.
T
2
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