INTERPLAY OF ELASTIC AND
INELASTIC SCATTERING OPERATORS IN
EXTENDED KINETIC MODELS AND
THEIR HYDRODYNAMIC LIMITS:
REFERENCE MANUAL
Jacek Banasiak,1 Giovanni Frosali,2 and
Giampiero Spigai
1
Department of Mathematics, University of Natal,
Durban 4041, South Africa
2 Dipartimento di Matematica "G. Sansone," Università
di Firenze-Via S. Marta 3, 1-50139 Firenze, Italy
E-mail: [email protected]
3
Dipartimento di Matematica, Università di
Parma-Via M. D'Azeglio 85,1-43100 Parma, Italy
ABSTRACT
In this paper we survey various hydrodynamic/diffusive limits
which can occur in the linear kinetic equation describing
elastic and inelastic collisions when the involved operators
act on different time-scales. The first part of the review is
devoted to the explanation to non-experts how to use the
compressed Chapman-Enskog procedure to analyse the
asymptotic behaviour of a kinetic equation when some
physical parameters are small with respect to the others. In
the second part is shown that all the formal steps of the
188
BANASIAK, FROSALI, AND SPIGA
procedure can be mathematically justified in a natural way
leading to a rigorous asymptotic theory.
KeY Words: Inelastic scattering; Lorentz gas; Asymptotic
analysis; Abstract Cauchy problems
AhiS Subject Classi cation:
82C40; 47D06; 35Q35
1. INTRODUCTION
Models in the kinetic theory can involve a large variety of different
phenomena, such as e.g., the elastic and inelastic collisions, thus it is natural
to investigate what happens when one (or more) of these phenomena is more
i mportane than the others. In such a case fit is customary to derive simpler,
approximate descriptions of the studied mode), introducing suitable new
continuum or hydrodynamic quantities. Such a continuum approximation
of the kinetic theory can be obtained mathematically by the asymptotic
analysis which, introducing suitable average quantities of the phase space
particle density, reduces the number of phase space independenYt variables
from seven to four.
The different importance of particular physical phenomena can be
accounted for in the mathematical mode) by introducing nondimensional
parameters reeated to them and investigating the limiting equation when
these parameters are very small or very large. The first analysis of this
type was carried out by Hilbert in his celebrated paper of 19121261 where
he expanded the solution of the Boltzmann equation in powers of a small
parameter (which in this case was the scaled mean free path) obtaining a
class of approximate hydrodynamic solutions, valid when the particle
collisions are dominant. The Hilbert theory has influenced much of the
later research in the kinetic theory yielding numerous papers on the detailed
description of the fluid approximation. However, a few years later there
appeared the Chapman-Enskog theory which treated the problem of
approximation of the Boltzmann equation by fluid equations in a much
more accurate way. Even if it is difficult to explain (without entering into
details) the differente between the Hilbert and Chapman-Enskog theories,
we can say that Hilbert expands the solution in the power series of the small
parameter (which yields the Euler equations at the first level of approximation), whereas Chaprnan and Enskog expand the equations obtaining
higher-order (e.g., Navier-Stokes) systems. For many years the ChapmanEnskog asymptotic procedure was used successfully in physics and in
SCATTERING OPERATORE IN EXTENDED KINETIC MODELS
f
189
The
practical app lications, even if it missed a rigorous foundations.
Hilbert and Chapman-Enskog theories are extensively discusseti in many
monographs devo 3dto kinetic equations; the reader can be referred to
1 and more recently Ref. [15].
the monographs,
In recent years, there bave appeared numerous papers attempting to
put the asymptotic theory of kinetic equations on a sound mathematic
basis. In this survey we shall focus on the compressed Chapman-Enskoi
expansion procedure, as adapted by J. R. Mika at the end of 1970s to the
asymptotic analysis of generai linear evolution equations. We apply this
method to provide a complete description of the hydrodynamic/diffusive
li mits which can occur in the linear kinetic equation modeling the interplay
between elastic and inelastic collisione when these collisione are allowed to
act on different time scalee.
The exposition is divided finto severa) sections. We present the
description of the physical mode) in Sec. 2 and we sketch the eneral
g
compresseti Chapman-Enskog procedure in Sec. 3. Sections 4 and 5 are
devoted to introducing the spectral projections which play the cruda) role
in the identification of the hydrodynamic subspaces of the equations, and to
derive formally the approximate evolution equations for the hydrodynamic
and kinetic quantities. Without entering into the details and referring instead
to the present literature, we give the main ideas for the rigorous proofs in
Secs. 6, 7, and 8. Finally we present a reference manual where the reader could
find the hydrodynamic/diffusive limits for different scaling parameters.
2. THE PHYSICAL MODEL
2.1. Description of the Model
In recent years there has occurred a considerable development in the
kinetic theory describing inelastic collisions. The interest in this field stems
mainly from the fact that such collisions are important for electron
transport even at low energies, such as in swarm propagation in gases and
slowing down electron beams in solids, and thus play a major role in the
semiconductor theory. However, they are also important in other branches
of the kinetic/transport theory, describing interactions of point particles
with composite systems, like the interaction of high-energy neutrons
with nuclei, or the interchange of kinetic energy by low-energy neutrons
propagating in gas media or solids.
From the physical point of view we consider a gas of test particles
having mass m, endowed only with translational degrees of freedom,
propagating through a three-dimensiona! host medium of particles having
190
BANASIAK, FROSALI, AND SPIGA
mass M. Such field molecules are usually much heavier than the test
particles, have a quite complicated structure, and thus non-negligible
internai degrees of freedom . As it is typical in the literature, also in this
paper such a structure is accounted for in a semiclassical way, that is, by
considering the molecules as point particles obeying the classical dynamics,
endowed with a set of quantum numbers which identify their internai
quantized state . Each of the several (infinite, in principle) discrete states
corresponds to a specific energy level, and thus the molecules in different
states must be considered as separate species .
In this paper wc shall stick to the simplest possible assumption, namely
that for the background particles only the first two energy levels are
significant (as it occurs al low temperatures), that is, the ground and the
first excited levels which are spaced by an energy gap 4E . In addition, we
assume the background to be al rest in thermodynamical equilibrium which
determines the distribution functions of the two background species, and we
consider the well-known Lorentz gas limit m/M -* 0 . In other words, the
test particles collide with something like a rigid net-they can be deflected
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
The number densities of the particles in the ground and in the excited
states, which are constant with respect to time t, are allowed to depend on
the spatial variable x ; we denote them by n l and n,, respectively . The condition of thermodynamical equilibrium relates them with each other through
AE/KT
< 1, where K is the Boltzmann
the Boltzmann factor b := n,/n t
= e
constant and T is the background temperature .
The time evolution of the distribution function f = f (x, v, t) of the test
2,341
particles is governed by the linear Boltzmann equation [2
ar
+v
.à =Cef
+ C`f,
(Cef)(x, vto, t)
=f
v[n t o t (v, w -o)')+ n2o2(v, w . w )]f (x, Ve)', t) dw
sz
f(x, vw, t) f~ v[n t o t (v, w
and energy of the interacting particles is conserved, but the kinetic energy is
conserved only in the elastic ones, since in the inelastic encounters the
is the elastic collision operator, and
the transport theory, due to the much larger density of the field particles, the
evolution of the test particle distribution function f is determined by the
collisions with them, since interactions of the test particles with themselves
are negligible . For the same reason, the test particles do not affect at ali the
evolution of the field particle distribution function . This assumption makes
the problem linear, al least with respect to the test particles . However, it also
implies that their momentum and kinetic energy are not conserved since
there occurs exchange with the external bath .
This subject is dealt with quite extensively in the literature . The
[13,ta,2o,2s~
and
essential features are given partly in standard textbooks,
1u,18,27,171
(we quote only some
partly in some pioneering pieces of work,
of them for the readers' convenience, without pretending to be exhaustive) .
A detailed analysis, based on the methods of the kinetic theory, with an
explicit derivation of the collision integrals in terms of the scattering crosssections, and under the standard assumptions for the validity of the integrodifferential Boltzmann equation, can be found in some more recent
.[22,3s,t,2a'
papers
We refer to these results as they are the starting point of
the present investigations .
(2 .1)
where
(elastic collisions), or exchange quanta of energy with the background
(inelastic collisions), but the classical continuous exchange of tl kinetic
energy is ruled out . In all collisions the total amount of mass, momentum,
quantity AE of the global impinging kinetic energy is transferred to or
from the internai energy of the field particle . Certainly, as it is implicit in
191
w ) + n2o2(v, w ' ai )] dci ,
(2 .2)
(C'f)(x, va), t)
=
f v[nt I2(v, w
• w)Î(x, v+w ,
t)
s2
+ n,I1 (v, to • ci)H(v - 8)f (x, v- tu', t)] d(o'
- )'(X, v(O, t) f Z v[n t it (v, u~ tti)H(v - S) + n,I2 (v, w w )] dw',
s
(2 .3)
is the inelastic collision operator . The standard five-fold integrai of kinetic
theory has collapsed to the two-fold one because field particles are "frozen"
as a consequence of the Lorentz gas assumption . Here, v = vaw is the velocity
variable, with modulus v and direction w, v. =,/v 2 ± 82 82 = 2AE/m and
H is the Heaviside function . Also, a l and o, are the elastic differential
collision cross-sections for the scattering of the test particles with the background molecules in the fundamental and excited state, respectively, and I l
and 12 are correspondingly the inelastic collision cross-sections (for the
endothermic and exothermic process, respectively) which obey the microreversibility conditions[271
v2, 1(v) = H(v - 8)vz I2 (v_)
v2 I2 (v) = v+II (v+) .
BANASIAK, FROSALI, AND SPIGA
192
When no confusion arises, the dependence of cross-sections on the angular
variable w • w is omitted in the notation, but implicitly understood, and the
same applies to the dependence of the background densities on the position x .
The Jacobian of the transformation between the precollisional and
postcollisional variables is not equal to one in the inelastic case, and this
fact has been used here, together with the microreversibility, in order to
interchange cross-sections and have ali of them evaluated at the same speed v .
The collision operators are more conveniently expressed by using the
elastic and inelastic collision frequencies, gí (v) = va 1 (v), gz = va2 (v), g' (v) _
vh(v) and g,(v) = vI2 (v) . In terms of the collision frequencies, the microreversibility conditions are
vg'(v)
= H(v - 8)v_g'(v-)
(2 .4)
vg'2(v) = v+gi(v+) •
(2 .5)
For v > 8, one of these two relationships is redundant since Eq . (2 .5) can be
obtained from Eq . (2 .4) by taking v + in piace of v . For v < 8, however, the
first relationship gives g'1 (v) = 0, an information that cannot be recovered
from the second relationship and that expresses the fact that the fxcitation
cross-section must vanish when the kinetic energy of an incoming particle is
below the inelastic threshold DE .
Using the collision frequencies (in the natural order as they appear)
instead of the cross-sections wc write the collision operators in the form :
(C ef)(x, voi, t)
f (x, vw, t) f
+
s
in,g' (v, w
[n t gi (v, w •
oti) + 11295 (V, w - w')] do'
w) + n2 g2 (v, w • w)] f(x, voti , t) dw',
(2 .6)
and
(Cf», ve),
S2
f (x, ve» f SZ,l(x, v, w • w) do'
v(o)
+
[n l gi (v, w • w)H(v - 8) + n 2g(v, w - w')] do'
+ fZ In igl(v+, w • w) -
f(x, 1'+w, t)
+ n,g2 (v_, w • w)H(v - 8) =f
v (x, v_w,
v
193
collision frequencies ?k = n kgti (elastic scattering), and vk = nkgk (inelastic
scattering), with k = 1, 2, where n 2 < n i . In the elastic process the test particle
speed remains unchanged, and the global effect of scattering is isotropization
in direction . In the inelastic collision operator the threshold effect described
above is accounted for by the Heaviside function H, and one may notice
scattering-in contribution at the speed v from test particles at speed v +
before collision (down-scattering), as well as from test particles at speed v_
before collision (up-scattering) . If these were the only interaction mechanisms
present in the system, a test particle with a given initial kinetic energy would
attain during its life only kinetic energies which differ from the initial one by
integer multiples of the energy DE . The test particles undergoing only inelastic scattering are thus partitioned into separate equivalence classes, modulo
AE with respect to kinetic energy, and such a quantity is essentially a mere
parameter with the range in the interval (0, DE) . Indeed, this is the actual
situation for our model, since the speed changes neither during free flight
(force fields have been neglected), nor under elastic scattering . Consequently,
the number of test particles in each class is bound to remain constant, and
neither of them feels the presene of the other classes .
At this point, the microreversibility conditions could be used again in
order to relate the two inelastic collision frequencies with each other and to
express C'f in terms of only one of them . Thus only (say) v 1 (x, v, (o • w) is a
free physical parameter in the model, since the other follows then by microreversibility . On the contrary, the elastic quantities X k (x, v, o • w) are both
free, since they are not correlated . They appear in the collision operator only
in the combination .l l + ), 2, which will be labeled as À in the sequel . For similar
reasons, the only inelastic parameter of interest, v 1 , will be renamed v .
For the readers' convenience, we rewrite the collision operators in
Eqs . (2 .6) and (2 .7) in the more concise form, where time is omitted because
it plays a role of a parameter :
(Cf»,
t)
f(x, vw, t)
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
fs~
À(x, v, w • w')f (x, vw) do',
(2 .8)
(C'f)(x, V(O)
t)] dw'
(2 .7)
Notice that the elastic collision operator is the same as for the
monoenergetic neutron transport, [131 and that there are two possible target
field particles, whose relative importance is determined by the macroscopic
= -f (x,
V(O)
[H(v-8)v(x,v,co .o)')+bL+v(x,v+,co-(o) do'
+ v±
v(x, v+ , w • w') f (x, v+ w) dai
v fSz
+ b
J s-
H(v - 8)v(x, v, w • (W) f (x, v_w) do' .
(2 .9)
BANASIAK, FROSALI, AND SPIGA
194
From a mathematical point of view, we assume that )l(x, v, w • co') is a
measurable function bounded almost everywhere :
0
<
a.min
< X(X,
v, -) <
vmin
< v(x, v, w . (ti) <
vmax <
+00 ,
for v
E
[S, oo[ .
(2 .10)
These assumptions are quite reasonable from the physical point of view .
Even in the highly idealized case of inverse power intermolecular potentials
without cutoff, they are valid for Maxwell molecules . However, they would
not be satisfied by the so called hard and soft potentials ; the latter could be
included to some extent into our analysis (see e .g ., Refs . [2,3,7,8]), but this
problem will not be considered here .
i
Kn e = ee
r
T
in front of the elastic and inelastic collision integrals, respectively . The
kinetic equation takes then the adimensionalized form
1 1/2
af
w
at + Sh~
ax
af
_
e
I
1
Kne C f + KniC
f
(2 .13)
where the collision operators are defined, with referente to the kinetic
variables only, by
Cf =-M,w)
f
s2
~,(~,w .w')dei' +f°X(~,w •(ti)f(~,ol)doti
s-
and
Cf
= M, w) [H(~ 1/2
1)
fs-
v(~, w •
(2 .14)
w) dei
f v(~+1 w w)
s'1/2 f
+ 1
z v(~+1,w cti)f( +l,w')dai
+(
+
2 .2 . Derivation of the Scaled Equations
+ b H(~ - 1) f v( , w . c)'f(~ - 1, ei) dai .
(2 .15)
The numbers Sh, Kne , and Kni measure the relative importance of the
streaming, elastic collisions, and inelastic collisions in the balance equation
for the test particle distribution function . We shall further simplify our
considerations by requiring that these three numbers are functions of a
single parameter e, which might represent the chosen smallness parameter .
Restricting our attention to regular functions of é we see that there is no
harm in assuming that the three numbers are power functions of e . Thus, we
shall consider Eq . (2 .13) in the form
at
EP Sf
+ E
Cef + Er Cif
(2 .16)
where p, q, r are integers, and the streaming operator S is defined by
3
M, (o) = 2 f
(2 .12)
Kni = Bi
+b(~+ 1)
Equation (2 .1) can be adimensionalized in terms of a typical length L,
a typical time -r, and typical values n*, gé and g; for density and collision
frequencies . Concerning the molecular speed, as a typical value for it wc will
take the quantity S corresponding to the inelastic transition . That introduces
spontaneously the Strouhal number [ ' 41 Sh = L/&c, and elastic and inelastic
mean collision-free times 0, = 1/n*gé and 9 = 1/n*g,* . We assume that the
elastic collision frequencies are of the saure order of magnitude, and that the
parameter b, smaller than unity, is 0(1), but other situations might be
analogously investigated . Scaled space and time variables will be denoted
by x and t again, and the saure applies for collision frequencies g with
k = 1, 2, and gi, as well as for background densities nk . The dimensionless
variable ~ = v 2 /8 2 will be used instead of adimensionalized speed, with the
jump in the inelastic transition equal to unity in the new scale . The new
distribution function, with split kinetic variables ~ and w, is labeled by f
again, and is given by
195
Easy manipulations single out the "Knudsen" numbers [141
Àmax < +00
for all (v, z) E [0, oc[ x [-1,1] .
Analogously, the following condition is required to hold true for the
independent inelastic collision frequency v = n i gi :
0 <
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(Vw)
v=(82
) 1/2
(2 .11)
S = -~ I / 2`w
a
. -
.
(2 .17)
196
BANASIAK, FROSALI, AND SPIGA
Sometimes, it proves convenient to resort to the quantity
2vf(vw)
Z
-~s )
(2 . 18)
,
(the scalar flux of the neutron transport) as a new dependent variable . In this
case the kinetic equation takes the same form as Eq . (2 .13), apart from a
slightly different expression of the inelastic scattering operator
C`lp = -cp(~, Q) CH(~ - 1)
+b(~
) 1/2
+i l
f
s=
v(~, w . w) d w'
)]+~ v( +l,cv cti)lp( +l,ai)dcti
sz
+bH(~-1)C
1)
s2
I v(~, o . w')q(~-1,cti)dcti .
1/2 ~
(2 .19)
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
197
where presene of the small parameter c indicates that the phenomenon
modeled by the operator A, is more relevant than that modeled by A 1 ,
which, in turn, is more relevant than that modeled by A o .
When in a kinetic equation the collision processes dominate over the
others, one is interested in finding a hydrodynamic evolution of the system,
and in the sequel we shall analyze different situations fitting into this
scheme .
In a mathematical framework, we can suppose to have ori the righthand side a family of operators {AE}E,o={Ao+(1/E)A1+(1/E2)A,}E,o
acting in a suitable Banach space X, and a given initial datum . The classical
asymptotic analysis consists in looking for a solution in the forra of a
truncated power series
f(n) (t) = f0(t)
+ Ef1(t) + E 2f2(t) + . . . +
l?'f (t),
and builds up an algorithm to determine the coefficiente fo , f1 ,f2 , . . .
Then f(" ) ( t) is an approximation of order n to the solution ff (t) of the
original equation in the sense that we should have
.
,
f
—
In any case, due to the negative powers of the variable ~ there appear
singular terms in the inelastic collision operator, the only differente being
its different location in Eqs . (2 .15) and (2 .19) . The singularity blisappears
from Eq . (2 .19) when the parameter b vanishes (no up-scattering) .
For a later reference, let us note that the test particle number density
follows from the different distribution functions as
n = f f(v) di, = f +~ f , ~ 1/2f(~, (9) d~ de) = +~ f , lp( , co) d dw .
(2 .20)
3 . THE COMPRESSED CHAPMAN-ENSKOG
PROCEDURE
fE(t)
-f(")( t)
Y = 0(é'),
for 0 < t < T, where T > 0 . Sometimes this approximation does not hold in
a neighbourhood of t = 0, because of the existence of an initial layer where
the estimate is not uniform with respect to t . For this reason it is necessary
to introduce an initial layer correction .
A first way to look at the problem from the point of view of the
approximation theory is to find, in a systematic way, a new (simpler)
family of operators, stili depending ori e, say BE , and a new evolution
problem
aY0,
The goal of this section is to give a concise overview of the asymptotic
analysis which is the basis of this paper, and which essentially stems from
the Chapman-Enskog procedure as revisited and modified by J . R . Mika in
the eighties . [3221
In order to introduce the reader into this asymptotic procedure,
let us consider a particular case of singularly perturbed abstract initial
value problem
Ms = AofE+ i A1f,+ i
A'f,
fE( ) = fo,
O
(3.1)
àt = BEwE,
supplemented possibly by an appropriate initial condition, such that the
solutions 10E (t) of the new evolution problem satisfy
11L(t) - 10<5(t)11X= o(E"),
(3 .2)
for 0 < t < T, where T > 0 . In this case wc say that BE is an operator
approximating A E to order n. This approach mathematically produces
weaker results than solving system Eq . (3 .1) for each é and eventually
taking the limit of the solutions as c -3 0 . But in real situation, c is small
but not zero, and it is interestìng to find simpler operators B E for modeling a
BANASIAK, FROSALI, AND SPIGA
198
particular regime of a physical system of interacting particles . For this type
of approach, we refer the reader to papers of the authors . [7 ' 81
A slightly different point of view consists in requiring that the limiting
equation for the approximate solution does not contain E . In other words,
tbc task is now to find a new (simpler) operator, say B, and a new evolution
problem
with an appropriate initial condition, such that tbc solutions cp(t) of the new
evolution problem satisfy
IfÎE(t) -
~
P(t) x - 0,
j1
as E
-->0,
(3 .3)
for 0 < t < T, where T > 0 . In this case we say that B is tbc hydrodynamic
limit of operators A E as E -i 0 . This approach can be treated as (and in fact
is) a particular version of the previous one as very often the operator B is
obtained as tbc first step in tbc procedure leading eventually to the family
{BE } E , o . For instance, for the nonlinear Boltzmann equation with the
originai Hilbert scaling, B would correspond to tbc Euler system, whereas
BE could correspond to the Navier-Stokes system with e-dependent
viscosity, or to Burnett equation on yet higher level .
In this review wc shall follow indeed this second point of view, looking
for suitable scalings of independent variables and physical parameters which
lead to tbc limitino equations not depending on c .
In any case the asymptotic analysis, should consist of two main points :
-
-
determining an algorithm which provides in a systematic way the
approximating family BE (or tbc limit operator B),
proving the convergence of fE in the sense of Eq . (3 .2) (or of
Eq . (3 .3)) .
Even if tbc formai part and the rigorous part of an asymptotic analysis
seem not to be related, the formai procedure can be of great help in proving
tbc convergence theorems .
The classical Chapman-Enskog procedure was adapted to a class of
linear evolution equations by J . R . Mika at the end of the 1970s . Later this
approach was extended to singularly perturbed evolution equations arising
in tbc kinetic theory . The reader interested in reviewing the applications of
the modified Chapman-Enskog procedure in tbc kinetic theory is referred to
the book by J . R . Mika and J . Banasiak . [311
The advantage of this procedure is that the projection of the solution
to the Boltzmann equation onto the null-space of the collision operator, that
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
199
is, the hydrodynamic part of tbc solution, is not expanded in E, and thus the
whole information carried by this part is kept together . This is in contrast to
the Hilbert type expansions, where, if applicable, only the zero order term
of tbc expansion of the hydrodynamic part is recovered from the limit
equation .
The main feature of the modified Chapman-Enskog procedure is that
the initial value problem is decomposed into two problems, for the kinetic
and hydrodynamic parts of the solution, respectively . This decomposition
consists in splitting the unknown function into tbc part belonging to
the null space V of the operator A,, which describes tbc dominant
phenomenon, whereas the remaining part belongs to the complementary
subspace W . Thus the first step of the asymptotic procedure is finding the
null-space of tbc dominant collision operator A, ; then tbc decomposition
is performed using the (spectral) projection P onto tbc null-space V by
applying P and tbc complementary projection Q = I - P to Eq . (3 .1) .
In this one obtains a system of evolution equations in tbc subspaces V and W .
At this point the kinetic part of the solution is expanded in series of E, but
the hydrodynamic part of the solution is left unexpanded . In other words,
we keep ali orders of approximation of the hydrodynamic part compressed
into a single function .
One of tbc main drawback of tbc classical approach is that tbc initial
layer contribution is neglected . To overcome this, two-time scaling is
introduced in order to obtain the necessary corrections . In generai, the
compressed asymptotic algorithm permits to derive in a natural way tbc
hydrodynamic equation, tbc initial condition to supplement it, and
tbc initial layer corrections . Hence, it is possible to give, under suitable
assumptions, an estimate for tbc error of tbc approximating solution,
uniformly in t > 0, in tbc sense specified later .
Summarizing, the originai Chapman-Enskog method is improved by
the introduction of two new ingredients :
the projection of the originai equation onto tbc hydrodynamic
subspace,
the analysis of tbc evolution equations in terms of the theory of
semigroups .
Taking these new ingrediente into account, we obtain the following
main advantages :
we can build an algorithm listing the steps of the procedure to be
followed,
we are able to establish ali the mathematical properties of tbc full
and limit solutions needed for the rigorous convergence proof .
200
BANASIAK, FROSALI, AND SPIGA
4 . THE HYDRODYNAMIC SUBSPACES
As we indicated in the previous section, the first step in the compressed
Chapman-Enskog method is to decompose the suitable Banach space
(usually L I ) into the hydrodynamic and kinetic subspaces . In this section
wc shali find the null-spaces of the elastic scattering and of the inelastic
scattering operators, respectively . It is remarkable that, in c ontrast . to the
standard kinetic theory, these spaces here are infinite dimensional . Next wc
shall study the relevant properties of the collision operators .
First let us consider the operator C e , in the form given by Eq . (2 .8)
made dimensionless by measuring v in units of 8 (unit spacing in the new
speed variable, labeled by v again) . The following theorem was . proved in
Refs . [3,10] . Since in all the considerations of this section x plays the role of
a parameter, we sha11 drop it from the notation .
Theorem 4 .1 . Let all the assumptions of Sec. 2 be satisfied. Then the operator
Ce is a bounded operator in X = L I (R 3 ) with the following properties
(i)
For any f
E
X and any non-decreasing firnction ic ive have
f 3 K(f)Cef dv
(ii)
(4 .1)
The
range of Ce
E
LI (R') ; f is independent of w} .
(4 .2)
is given by
R(C e ) = W =
{
f
E
L I (IIR 3 ) ;
f f do)
= 01 .
1s
(4 .3)
The spectral projection orto N(Ce) ( parallel to W) is given by
Pf
(iv)
For f
E
=
14~r
f
s2
f d w.
(4 .4)
W we have
f sign(f)Cef(v o)
dv < - 47rl,,,i„Il f 11X
and hence the spectral bound of
-47
rXmin
Ce,
(4 .5)
S(Ce), satisfies s(Ce)
<
201
Analogous properties hold in any weighted space L 2 (IIR 3 , w(v) dv) Aere iv is a
measurable strictly positive (a . e.) firnction .
Proof. Since À is symmetric in w and tu', for any g
veodw =
f52(gCef)()
-f s-, s=
f
_hence
fs
(gCef)(veo) da>
fS2
fL
E
L,<,( llR 3 ) wc have
X (v, w - ci)[g(vw)(f (vw) - f (veti))] dw dai
f52
X(v,w- (ó)[g(vw)(f(v(ti)-f(vw))]dwdcti,
= - f f .l(v, w •
s-2 s-
oi)[g(vw) - g(vw )]
.
x [ f(v(») - f (veti )] dw dai
Taking any bounded strictly increasing function
2
In particular, C e is dissipative .
The null-space of C e is given by
N(C e ) _ { f
(iii)
< 0.
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(K(
f)Cef)(vw) do - - f
f,
s2 s2
K : ll
-->
fIR wc obtain
),( v, w • w)[K( f(vw)) - K( f(vw ))]
x [ f (vw) - f (vcó)] do) dw'
< 0.
(4 .6)
Integrating over the remaining variable wc get the H-theorem . If, in
particular, we take K(t) = sign(t) wc obtain
f
3
sign(f )Cef d v< 0
which gives the dissipativity of C e . This proves (i) .
Let Cef = 0 ; then the left hand side of Eq . (4 .6) is zero, but due to
strict monotonicity of K this is possible only if
(4 .7)
AVO» = f (veti),
for almost all v, that is, to the kernel of C e may belong only functions
independent of w . On the other hand, such functions clearly belong to
N(C e ), therefore the kernel of Ce is given by Eq . (4 .2) .
Next we turn our attention to the solvability of
af(v)+f(v)
f
s2
À(v,w w)dcti -
f,
s-
X(v,w .(v)f(v(ó)dw =g(v)
(4 .8)
BANASIAK, FROSALI, AND SPIGA
202
for g
L 1 (R 3 ) . Denote
E
W= j f
l
For f
E
s
f
E
L I (R') ;
f
s~
d =0
Cf = -Nf + Kf .
defining
Z (slgn(f)C ef(vw) dw
2 fS2 S2
~.(v, w • w)[sign( f (veo)) - sign( f (ve )))]
+ f f sign( f(vw)f
s2 s2
-f
s- f,2
k
sign( f (vw)) f(vco) do) dw I
_ -47rÀmin lI f I
f,
s
v(v+ , w .
w) dw
(N- f)(v) = f(v)H(v2 - 1) f v(v, w •
sz
vo) dco dw
f ` f sign( f (vw))f (vw) dco do)'
where
v
(N+f)(i') =f (v)b i+
<
- - 1 >, min f f , sian( f (vw)) f (v(o) da> dw
2
( s- s-
L, (S`)
where, upon integration with respect to v, wc obtain that C' I li, - UI is
dissipative for a > -47rXmin . Since C'IYv is bounded, C'I w - aI must be
ni-dissipative, therefore if a > -47rXmin, then a E p(Ce l w) .
It is also clear that the spectral projection onto N(C e ) is given by
.
Eq (4.4) which ends the proof of (iii)-(iv) .
The statement for the L, space follows in the same way as all the
operation are first performed on the unit sphere and only later integrated
∎
with respect to v to get the estimates valid on R3 .
The analysis of the inelastic kernel is considerably more involved . The
case of purely inelastic collision operator C' has been thoroughly investigated in Ref. [1] . Analysis of the full operator C = C'+ C e is similar but wc
provide it for the sake of completeness . Note that since C e is a bounded
operator, the domain of C equals the domain of C', that is, D(C) = D(C') .
It is worthwhile to analyse the operator C in two spaces : X = L 1 (R )
and X2 = L,(l& 3 , b - °2 dv) where the Boltzmann factor b is defined in Sec . 2 .
Here again x plays the role of a parameter and thus will be suppressed in the
notation .
(4 .9)
Kf = K+f + K_f,
Nf = N+f + N-f,
x [ f(vw) - f (veti)] dw dw
-
203
To avoid confusion we shall write down explicitly the definitions of
the operatore which will appear in our considerations . For a continuous
function f we split the full collision operator C as
W we get
_ -1
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(K-f)(v)
vf ,
= 1
s
w) do)'
(4 .10)
v(v+, w • w)f(v+w) doti
(K+ f)(v) = bH(v2 - 1) f , v(v, w . wf(v_w)dw .
s
It follows that in X setting only the operatore K+ and N+ are unbounded
due to the singularity v -1 at 0 . On the contrary, in X 2 setting the operator
K_ is also unbounded with singularity \/v2 - 1 at v = 1 . Multiplication
operator N_ is bounded in both settings . 111
To characterize the domain of C, for an arbitrary positive measurable
function g we denote the weighted space L I (R3 ,g(v)dv) by X3 . It follows .
Ref. [1], that
D(C) = X 1+ ,,
.
Similar considerations can be carried out in the space X' = L2 (l83 , b- " dv) .
Then the domain will be denoted by D2 (C) . Using analogous notation,
wc have
D2 (C) = X' , n XH(v2_ 1)v _, n X' .
In the sequel wc shall find convenient to use the notation :
in Subsection 2 .2) and
Ow =f ~~),
=
v2
(introduced
(4 .11)
BANASIAK, FROSALI, AND SPIGA
204
so that f(vf(o)=0(x±1) . By Ref. [1] wc may put D(C) = D(K) and
D 2 (C)
=
D2 (K) .
The theorem below is an extension of Theorem 2 .2 of Ref . [1] to the
case that includes the elastic scattering term . Apart from one step, the proofs
of both theorems are similar but for the sake of completeness we provide
bere the proof for the present case .
Let f E D(C), then Cf = 0 if and only if 0, defined by Eq . (4 .11),
satisfies for ~ E [0, 1[ and n E N
+ n) = b"
0(~
(b)
Oo(~)
(4 .12)
where 00 E L 1 ([0, 1],d~) .
Let f ED 2 (C), then Cf = 0 if and only if Eq . (4 .12) holds with
0 0 E L2([0, 1], d~) .
Proof.
}
(a) Firstly, wc consider the case with bounded C' in X . This can be
done by regularizing v as follows
v(v, w • co)
v„ ((v,
v,0
w . w) =
for 1 + n -1 < v
for v < 1 +n - 1
(4 .13)
Following some ideas from Ref . [28] (see also Ref . [1]) wc obtain for
g
E
X* = L ro (f 3 )
(g, C+,f)
= fS2 fS2 f
+ce
v+v,r(v+ , w . w')[f(v+w)g(vw) + bf(vw)g(v+w)
- bg(v(9) f (vw) - g(v+oi) f (v+ (o)]v dv dw doi
1
fo
2 fs'- fs'-
X(v, w ()[g(vw) - g(vw')]
x [f (vw) - f (veti )]v 2 dv dw dai
R,
_ -
f
V
v,, (v+, w • w)U(v+w) - bf(va)]
(K(b-, ,2f), Cn
f)
f f
2 ~a' s'
À(v, w w )b °2 [g(vw) - g(vw )]
x [b - ` 2f (vw) - b -V f (veti)] dai dv,
v
S' L
+ v (v + (o , co)b ,2+1 [b _v2_lf(v+w) - b- '2f(vw)]
x [K(b-°--1f (v+w )) - K(b-°Zf (vw))]dw dv x [K(b-`2f(V(O)) - K(b-v2f(veti ))]
2
b-v 2
x [b -V f(va» f (ve)')] do)' dv,
fQ8fS2
a (V, w .
w)bv2
(4 .15)
hence for any f E LI (R3) wc have the H-theorem
(K(b-"f), C„f) < 0 .
(4 .16)
If we take as K any bounded strictly increasing function, then we see
that if for some f we have C„ f = 0, then
(K(b-v2f), C„f) = 0,
and since each term in Eq . (4 .15) is nonpositive, this is only possible when
wc simultaneously have
f(v+ (»') - bf (vw) = 0
f(v(O) - f (v(0) = 0
(4 .17)
for all w, cti E S2 and v E R+ . On the other hand, functions satisfying
Eq. (4.17) belong to the null-space of C,,, N(C„ ), by Eq . (4 .14), hence it is
described fully by Eq . (4 .17). To characterize these functions explicitly we
note that by Eq . (4 .7), the second equation in Eq . (4 .17) shows that the
elements of the kernel of C„ are independent of the angular variable . Hence,
using notation (4 .11) for Eq . (4 .17), by recurrence and Eq . (4.10) we get
0(~ + n) = b"Oo(~)
x [g(v+w) - g(vw)] doti dv
1
205
where ( •, .) denotes the duality pairing between L1(R3 ) and L,,, (l83 ), and C„
the collision operator corresponding to the regularized v,, . In particular, if K
is any bounded nondecreasing function, then we obtain
'3
Theorem 4.2 .
(a)
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(4 .14)
where ~ E [0, 1], which gives Eq . (4 .12), showing that N(C„) = N(C,,) .
Returning now to the original unbounded C, wc pass to the limit,
exactly as it was done in Ref . [1], Theorem 2.2, obtaining the same
statement .
206
BANASIAK, FROSALI, AND SPIGA
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(b) Let us consider now the problem in X2 . Again wc start with v„
defined by Eq . (4 .13) . For arbitrary g E X2 , similar calculations show that
by v„( ., w . a) = v„(, ai . w) we have
(g,
GA
(f,C1j)x'--=- f~
1
f v
l v,
f f
2 u3 s 2
E
+ n)
(v+,w •w )[f(v+w)-bf(vw)]2b-
X(v,w .w')[f(vw)-f(v(ti)] 2daidv .
(4 .18)
In the theory of hydrodynamic limits an important role is played by
the conditions of solvability of Cf = g . In this context we have the following
result . Again, this teorem is an extension of the result of Ref . [1] to cover
the case with C = C'+ C e .
Theorem 4 .3 .
Let C* be the adjoint to C in X and f
and only if
E
D(C* ) . Then C*f =
0
if
for all v e [0 . 1[, n E N and some fo E L~([0, 1]) .
The closure of the range R(C) is characterized by
g
E
R(C) if and only if
Ì=0
(c)
Jv +1
2
g(~v 2 + jw do) = 0
(4 .23)
Proof.
(a) Let g E N(C*) . Since then g E D(C*), in particular it is a bounded
function, thus b" g E X 2 . Moreover, if X is the weight function defining
D 2 (O), we obtain
f3
X(V)g2
1
(v)b 2v2
v"2 dv = f
+ro
o
f
s-
2
X(vw)g2 (vw)b v 2 di, dco < +oc,
where the integrability in neighbourhoods of v = 0 and v = 1 is due to the
i
boundedness of g and integrability of v i and v= over compact subsets of
118 3 . Hence b"2 g E D 2 (C) . Since X2 C X, if f E X 2 is bounded at v = 0, then
f E D(C) . Moreover, the set of such functions is dense in X 2 . Thus, by
self-adjointness of C in X 2 wc have
0 = (C* g,f)x = (g, Cf)x = (b g, Cf), - (C(b 'g),f)
0
(4 .20)
(4.24)
= f
ua
g(v)f(v) dv
= 1 f l Oo(~)
2
(~'A +jf , g(,/~ +j(,) dco) d~,
j=o
00
s
which, since 00 is arbitrary, gives
tiVe have the "spectral" decomposition
X = N(C) $ R(C)
w'+jf
rj
o
s2 Î(~/''-+yw)dw
=
47r
° o bJ \/v+j
and from the density wc see that g E D(C*) solves C*g = 0 if and only if
b" 2 g E N(C) in X2 . This shows that g is a 1-periodic function of the variable
independent of the angle .
(b) Since R(C) = N(C*) 1 , g e R(C) if and only if for any bounded
1-periodic function f of ~ = v 2
f(v2 +,1) =fo(v)
(b)
V5 0 (v2)
1 daidv
(4 .19)
= b"Oo(~),
(4 . 22 )
where
where ~ E [0, 1] and o e L,([0, 1], 4 d~) . The extension of this result to
unbounded v is performed as in X, case .
i
∎
(a)
2
b » Vfo(v 2)
(Pf)(v +n) =
X2 ; then
Equation (4 .18) shows that, as in X-setting, C„ f = 0 if and only if
O(() =f(,) satisfies
0(~
and the "spectral" projection onto N(C) along R(C) is given by
-2 = (C g,f)x•2
that is C, is self-adjoint in X2 . Let us take arbitrary f
207
(4 .21)
X
:/v22 +jf g(Jv 2 +jw) dw = 0 .
j=o
s-
(4.25)
BANASIAK, FROSALI, AND SPIGA
208
(c)
The
projection
(f - Pf, g) x = 0 for any g
onto N(C) along R(C)
N(C*) = R(C) 1 , that is
P
E
must satisfy
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
Aere, for v
=
f2 o
go(~)
rw
fio ( x 5 v ) =
~J +jf s-, f(/ +jw)dw
i=o
- 4nVo(~)Y -'bl J~
1=0
E [0, 1 [,
~
0 = f.3 (f - Pf)(v')g(v) dv
209
w
2 +lw) d(w
o,/l, +JfS2 f (x
47rr'
: bJ \/v 2 +j
(4 .32)
Thus
(4 .26)
+ f) d~,
and since go is arbitrary wc obtain Eqs . (4 .22) and (4 .23) .
From Eq . (4.25) wc obtain that N(C) rl R(C) = { 0}, and since
Pu E N(C) and by Eq . (4 .50), (I - P)u E N(C*)1 = R(C) hence wc have
(4 .27)
X, = N(C) ® R(C) .
N(C e ) D N(C) = N(C')'
and, in particular,
Ce P = 0 .
∎
We can summarize Theorems 4 .1, 4 .2, and 4.3 in the ~ul1 L I (lR 6 v )setting as follows.
Corollary 4.1 . Under the adopted assumptions
(a)
The null-space of C e is given by
N(C e ) _ { f
E L i (IIR6 ,
y ) ; f is independent on co} .
(4 .28)
and the spectral projection onto N(C e) is given by
(Pf)(x, v) =1
(b)
fsS
(4 .29)
, f (x, m) do) .
The null-spaces of C' and C = C'+ C e coincide and are given by
N(C) = N(C') _ { f
E
L I (R', v , ( 1 + v - ') dv dx) ;
f is independent of w and satisfies
f(x, v + ) = bf(x, v) for a .a . x
E
llRx, v
E QR
+}.
(4 .30)
The spectral projections onto N(C) and N(C') coincide and are
given by
(Ff», v 2 + n) = b",/i o (x, v2 ),
xEIlRY,vE[0,1[,1lENU{0},
(4 .31)
5 . DIFFUSIVE LIMITS-FORMAL DERIVATIONS
Let us return to the scaled Eq . (2 .16)
afE - ~ Sfl + ~ Cf, + 1 C fE
ót
E
E
E
Wc are looking for the diffusive/hydrodynamic limits of this equation .
According to the considerations of the previous section, there are two
possible hydrodynamic spaces : N(C`) and N(C) = N(C) = N(C' + C e ) .
We can expect evolution in N(Ce) if the elastic collisions are
dominant, and in N(C) if either inelastic collisions are dominant or
both elastic and inelastic collisions are much stronger than the
free-streaming .
To find the possible limiting equations wc use the compressed
Chapman-Enskog procedure, discussed in Sec . 2 . Hence the idea wc shall
pursue is to separate the hydrodynamic part of the solution to Boltzmann
equation by means of the appropriate spectral projection and then, by
expanding the remaining part into a series of c, to find and finally discard
terms of higher order in c, getting (at least formally) the limit
equation satisfied by the hydrodynamic part .
Accordingly, in the first case we will be looking for the situations when
the limit is the projection onto N(Ce ), and in the second when the limit is the
projection onto N(C) .
210
BANASIAK, FROSALI, AND SPIGA
5.1 . Evolution in N(Ce ) (Elastic Collision Dominante)
To find possible limiting evolutions in N(C e ) we shall use the
projection P defined by Eq . (4 .4) . It is important to note that since
N(Ce ) D N(C), PC' and C'P are not equal to zero . Denoting 0 = I - P,
we operate with these projections onto Eq . (5 .1), and denoting
and
vE = PfE
1v E =
Off,
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(limiting) equation for v E , wc obtain the limiting equations, independent of
E, in the form
and
ap =
- PSQ(QCe Q~) -1 Q~SPp + PC'Pp,
p
a = -PSQ(QCe Q) 1 QSPp,
at
we obtain
I
ir
PSQw 6 +
PC ' PV E
at
Ep
E
ai_vE _
Q~sQ~wE
I QSPV E
at
Ep
EP
aV E
+ 1
if r = 0,
if r < 0 .
(5 .3)
(5 .4)
If p < k, then the power of the coefficient multiplying PSQ(QC e o) -1 x
+1
,. PC ,. 0112,
E
+ Er
1 QC'Pv E +
211
OSLPp is positive and therefore this term is negligible when e tends to zero .
i Q~CQwE
Er
(5 .2)
Then, the possible limiting equations are
+ i Q1Ce Q~u' E .
E
where we already used the fact that PSP = 0 (see Ref . [3]) . Sirice we assumed
that the elastic collisione are dominant, we must assume that q > max{ p, r} .
Since we are looking for the limiting equations, the equation for the
approximation of v E cannot contain c . This yields r < 0 and shows that p
must be less or equal to the index k of the first nonzero term in the expansion of 1vE = wo + ew 1 + E2 1v 2 + • • • . Let us consider first the case when
p = k . Inserting this expansion into the second equation in Eq . (5 .2) we obtain
lVO
Eq-poSpl,E + E9-p . SQ~(w0 + Ew 1 + . . .)
E 9 (aa + E aatl-+ . . . =
+ E9-rQC'pVE + E a-rQC ' Q(wo + EW1 + . • .)
+ QCeQ(wo + Ew1 + . . .)
Since q > r and q > p, wc obtain
QCzQpwo = 0
which yields w 0 = 0, because Q is the complement to the spectral projection .
Clearly, the first nonzero term in the expansion of 1v will be Wk with k
satisfying k = min{q - p, q - r} . However, if q - p > q - r, then r > p, but
r < 0 yields p < 0 which contradicts the assumption that p = k . Thus
k = q - p and q = 2p . In any case we obtain W k = -PCQ) -1 QSw,
(provided the inverse exists) . Changing now the notation from v E into p to
emphasize the fact that the forthcoming equation is an approximating
and
ap
= PC' Pp,
ap =
at
0,
if r = 0,
if r < 0 .
(5.5)
(5 .6)
To summarize, wc consider the generic (simplest) combinations of
powers of E giving particular limiting equations . Thus, wc see that Eq . (5 .3) is
(formally) the limiting equation for the scaling
afE
at
-
e
SfE + CefE + C 'JE,
E
and Eq . (5 .4) for the scaling
aE
= E SfE + É CefE + ECfE .
Further, Eq . (5 .5) is the limiting equation for the scaling
afE - Sf +
E C efE + C'fE,
and Eq . (5 .6) for the scaling
E
= SfE +C efE + EC`fE .
BANASIAK, FROSALI, AND SPIGA
212
5 .2 . Evolution in N(C) (Inelastic Collision Dominance)
t
As wc have seen in Sec . 4, the cases when either C', or C' + C e dominate have the same hydrodynamic subspace N(C) and the same
projectors onto it . Since N(Ce ) D N(C'), wc have CeP = PC' = 0, where
P is the projector defined by Eqs . (4 .22) and (4 .23) . Operating with P and
Q = I - P onto Eq . (5 .1) and denoting
vE
and
= PfE
PSQwE
= 1
at -
1
QSPv.
Ep
+ Ep
1 QSQw E
+ lr
E
QC`QW E +
E
(5 .7)
QCeQIVE,
where wc used PSP = 0.J`? As before, we observe that p must~be less or equal
to the index k of the first non-zero term w k in the expansion
lv = w0 + Ewl + • • . . However, if it is strictly smaller, then the equation for
p (the approximation to VE ) will be trivially reduced to
at
independently of what is happening in the second equation (though as wc
shall sec in Subsection 8 .3, the initial layer corrector complementing the
hydrodynamic equation changes slightly depending ori whether C' and C'
are of the same or different magnitude) . Hence, wc can safely assume that
p = k > 0 . Let us then consider the expanded version of the second equation
in (5 .7)
(IV O + CIV I +
1
= E QSPV
+
Er
a
at
> q.
Multiplying the last equation by
Er
(Wp + Elvl + . . .)
= Er-PQSPV E + E r-POSO(w 0 +
+ QCQ(lv 0
+ CIVI +
. . .) +
CIVI
Er 9 QCe Q(wo + CIVI + . . .) .
OSPQ = -QCQwk
when r - p = k, that is r = 2p . The index q seems to have no significance
whatsoever on the final result . Consequently, the limiting equation for the
approximation p of v E is of the forra
ap =
- PSQ(QC'Q) I QSPp,
(5 .10)
provided the inverse exists .
The case r = q is similar . Multiplying Eq . (5 .10) by Er we obtain
ap - 0,
at
Let us consider first the case r
wc obtain
213
Due to the definition of Q we have, as before, w 0 = 0 . Let lvk be the
first non-zero term of the expansion of w . Because r - q > 0, wk at QCeQ is
multiplied by Er- q+k which is always of higher order than Ek . The term w k at
OSO is multiplied by Er -P+k = Er , and since QSP is multiplied by Er-k with
k > 0, we see that Wk is to be determined from
lv E = QJ,
wc obtain
v
alt,
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
1
r
E+
)
1
a
a (wo +
Ew l + . . .) = E r-PQSPVE + Er-POSQ(WO + Ew
+ QCQ(wo
+ Ewl + • • •)
+ QCe Q(wo +
l
+ . . .)
CIV I + • • •) .
Due to the definition of Q wc have as before lv 0 = 0 . Let Wk be the first
non-zero term of the expansion of lv . Here r - p > 0, and therefore the term
wk at QSO is multiplied by E r- P +k = e' which is of higher order than Er -P
since k > 0, and consequently W k is to be determined from
QSPQ = -Q(C' + C e )Qw k
E QSQ(wo
QC . Q(w o
Er
+ Ewl + . . .)
1
+ Ewl + . . .) + E9
QC e Q(wo +
Here we have to distinguish two cases : r
we must have of course r > p) .
> q
Ewl + . . .),
and r
= q
(5 .9)
(in both cases
when r - p = k, that is r = q = 2p. Consequently, the limiting equation is of
the forra
~p =
- PSQ(Q(C' + C e )Q) l QSPp,
provided the inverse exists .
(5 .11)
214
BANASIAK, FROSALI, AND SPIGA
Thus typical cases will be
(i)
àE = S.ff, +
CefE + C'fE,
(5 .12)
with q < 1, giving the hydrodynamic limit
ap
ar - 0,
afE _ 1
1
e
1
E SfE+~~CfE+E? CfE,
ar
(5 .13)
with q < 2, giving the limit
8p
_ - PSQ(QC' t Q) -1 QSPp
and
tE
a = ~ SfE + É CefE +
Cf,
ar
-PSQ(Q(C` + Ce)Q) - `QSPp .
6 . MATHEMATICAL ANALYSIS : PRELIMINARY
REMARKS AND DEFINITIONS
6 .1 . The Relevance of Rigorous Results
In this section we shall give a survey of rigorous results concerning the
hydrodynamic limits discussed in the previous section . By "rigorous results"
wc mean the results providing the estimate of the error between the solution to
the kinetic equation (2 .16) and the solution to the respective limiting equations
(5 .3)-(5 .6) or (5 .8)-(5 .11) (supplemented, if necessary, by an appropriate
215
initial layer corrector) . Such an estimate should be provided, if possible, in the
space L1(llG v) which is a natural space from the physical point of view .
Such an analysis consists of two steps . At the first one we have to
determine all the terms of the asymptotic expansion which would give the
desired estimates of the error, provided they are well-defined (that is,
the corresponding equations are solvable), and bave sufficient regularity .
The second step is to prove that all the equations obtained via the asymptotic
procedure are solvable, and that their solutions bave sufficient regularity for
the error estimates to be available .
It is to be noted that the last step seems to be purely of mathematical
interest-a practitioner would be probably satisfied with the statement that
the solution to the limiting equation approximates the solution to the original equations provided all the terms are sufficiently regular, thus from his
point of view the asymptotic analysis would have been completed after the
first step . However, even if such an argument leads very often to satisfactory
results, one must remember that in many cases the equations defining the
terms of the asymptotic expansion are rather artificial and don't reflect any
physical reality ; thus their solvability cannot be judged from the fact that
"the Universe exists" .
To present the rigorous asymptotic results in a unified framework wc
shall limit ourselves to the homogeneous case when the collision frequency is
independent of the spatial variable and to the simplest case of the isotropic
scattering . Thus wc shall consider the following scattering operators :
(5 .14)
giving the limit
ap =
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(Cef)(x, v) = - 4nX(v)f(x, v) + X(v)f f(x, v(i) dai,
s,
(6 .1)
with
0 < Xmin < ?( v) <- Xmax < + ce
for all v
E
[0, ce[, and
(Cf)(x, v) = f (x, v)47r(H(v 2 - 1)v(v) + b v+ v(v+ ))
v
vv
+
')(V+) Z f(x, v+ (0) de
s
f
+ bv(v)H(v2 - 1) f f (x, v- o)') dai,
sZ
(6 .2)
where we recall that H denotes the Heaviside function, v ± = _/V2 + 1
and 0 < vmin < v(v) < Vmax < + oo for v E [1, ce[ . We note that the
extension of the forthcoming results to the case of x-dependent coefficients
BANASIAK, FROSALI, AND SPIGA
216
is straightforward even though computationally unpleasant . To allow more
generai (but stili isotropie in v) scattering cross-sections is more demanding
but can be accomplished in several cases (see e .g ., Refs . [2,8]) . The extension
to to dependent scattering cross-sections seems to present serious difficulties
due to the adopted L 1 -setting and the employed techniques .
In this subsection wc introduce the function spaces relevant to the
further considerations . Since from now on the dependence on the spatial
variable will become important, in contrast to the previous sections wc must
introduce notation which will distinguish L 1 spaces in x and v variables .
The basic space is
X, = L 1 (Rr, Xv) = L 1 ([183 > X~) = L1(I6, ,),
(R3) for a = x, v . Most considerations will be carried out in
where X,, = Li
X, with fixed x . Typically, if A C is an operator in X v (possibhv depending on
x as a parameter), then by A we will denote the extension of this operator to
X . If A C is unbounded in X, with domain D(A r ), then A is considered on the
natural domain
®R
D(A) = {f
E
X ; f (x, .)
E
D(A x) for a .a . x, x -> (AXf)(x)
E
X} .
If A i does not depend on x, and it is clear from the context in which
space it acts, we will omit the subscript x . The same convention will be
applied to operators A v acting in YC .
Occasionally, if the above procedure can be reverted, for acting in X
operator A wc shall write Ar or A,, to denote this operator acting with x or v,
respectively, fixed as a parameter (that is, e .g ., A Cf = A(f ® 1) where
f E X, if the latter defines an element of Xv ) .
The asymptotic analysis requires some additional regularity of
the data . Typically, the required regularity in v variable is related to the
integrability with respect to a certain weight function and the required
regularity in x variable is related to the differentiability . Accordingly, wc
introduce
X,, = L I (R, lv(v)dv)
and for the most typical "moment" weight w(v) = tiv(v) = 1 + vk , k e Z,
wc denote
X,,, k = L 1 (R'V , ( 1 + Vk ) dv) .
217
Consequently, wc denote
X,r = L1 (R x,
Xv, w
and
Xk =
6 .2 . Notation for Operators and Function Spaces
X = Xr
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
3
LI (Rx, Xv, k)
If A is an operator in X with domain D(A), then the domain of its part
in Xk will be denoted by Dk (A) .
Combining the spatial and velocity regularity, for a given operator A
wc introduce
Xlkm,A = {f
E
Xk ; 8 f
E
Dk (A m ), I8I < l},
(6 .3)
where for the multi-index ,B = (PI, 8,,,8 3 ) with I$I = $ 1 + $2 + $3 we
This space can be normed by the natural graph
denoted 8 = 8~
x
z ~3 .
norm.
Note that the space XlkO,A is independent of the operator, thus it is
sensible to denote it by X lk ; we have then
XlkO,A - X1k - {f
E
Xk ;
8
f
E Xk, I/3I
1} .
In many cases the operator under considerations is independent of x .
The most important example is rendered by the collision operator
and to illustrate the notation discussed above we specify it for this
particular case .
It foliows from Sec . 4 that the domain of the operator C = C' + C e =
C+ + C_ + C e in X, satisfies
D(C) = D(Ci) = X,, ,- 1,
thus, if treated as an operator in X, C has the domain D(C) = X_ 1 .
Since the moment weight 1 + v k and the weight defining the domain of
C don't affect each other, it is easy to show that the domain of C in X,,, k is
given by
Dk(C) =
Xv,,k+,-1
(6 .4)
(the term 1 can be omitted as any function integrable with respect to v k and
v-1 is necessarily integrable) . Considering again C as an operator acting in
BANASIAK, FROSALI, AND SPIGA
218
=
wí(R', Dk(C"')) .
(6 .5)
Very often we shall use the space
XOkm, C
=
Xkm, C
3
= L i (Rx , Dk(C
m
)) .
In particular, using the equivalent definition of D k (C), we get
XOkl,C
=
XkI,C
=
«x X,,'
L1,
+v ) - X
Note that in accordance with the compressed Chapman-Enskog procedure,
discussed in Sec . 3, the hydrodynamic term of the bulk part of the expansion, p, is not expanded .
The number of terms in each expansion and the value of n in the
definition of r are determined in some sense a posteriori after having written
the formai equations for the error, so that the error could be conjectured to
be of the required order .
k+v
7 . RIGOROUS ANALYSIS : DOMINANT
ELASTIC SCATTERING
.
Summarizing, f E X lk,,,, C if 8s f E D k (C") for IsI < l and the norm x -
R3 . In particular, if m = 1, then it is
is integrable over
sufficient that all the derivatives be integrable over IRi ~ with weight function
defined in Eq . (6 .4) .
II
8 f (x, •) II Dk(c»)
6 .3 . Full Asymptotic Expansion-Preliminary Comments
The preliminary considerations leading to the hydrodynamic
equations have been carried out in Sec. 5 . To be able to obtain the desired
error estimates, in most cases wc have to supplement these equations
with bulk and initial layer correctors . This is done by an extension of the
compressed Chapman-Enskog asymptotic procedure of Secs . 3 and 5, which
is sketched below .
The asymptotic solution is sought in the form
J (t, r) =f(t) +f(v) = p(t) + lwo(t) + El v l (t)
+ . . .+ o(z)+Epl(i)+ . . .+tivo(r)+Ewl(r)+ . . .,
where r = t/E" for some n
p, po, pl . . .
E
N(Ce )
E
(6.6)
fil . The terms
(resp . N(C)),
are called the hydrodynamic part of the expansion, whereas
ivo, tiv l , . . . . lvo , fv l , . . .
E
N(C e )1
219
initial lager ; they are to be determined independently of each other .
the full space X k , wc can write Eq . (6 .3) in the simpler form :
Xlkn',C
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(resp . N(C) 1 ),
and are called the kinetic part of the expansion .
Moreover, the terms depending on t are referred to as the bulk part of
the asymptotic expansion and the terms depending on r are known as the
In this section wc shall provide a mathematical analysis of the
procedures leading to the limiting equations (5 .3)-(5 .6) in the hydrodynamic
space N(C e ) . WC start with the kinetic-diffusion equation (5 .3) which is the
most interesting mathematically case ; some of the presented results for this
model can be found in Refs . [3,5] .
Next we shall briefly discuss the case (5 .5) which has been recently
solved in Ref. [4] . Equation (5 .5) can be viewed as a simplified (spatially
homogeneous) version of Eq . (5 .3) and no surprisingly basic ideas of Ref . [4]
play the important role in the analysis of the latter equation .
Two other equations of this section, Eqs . (5 .4) and (5 .6) are much
simpler and though they haven't been analysed before, we shall limit ourselves to some generai comments .
7 .1 . The Kinetic-Diffusion Equation
Let us consider the Boltzmann equation
t
= 1E Sf
+ z Cef + Cf,
E
(7 .1)
The preliminary considerations leading to the limiting equation (5 .3)
have been carried out in Subsection 5 .1 . Here wc have to supplement the
hydrodynamic equations with bulk and initial layer correctors which will
enable us to obtain the desired error estimates .
WC start with the system (5 .2) where wc put p = 1, q = 2, r = 0 . The
assumed isotropie form of the scattering operators allows us to simplify
Eq . (5 .2) even further . To do this wc note that by Corollary 4 .1, the operator
220
BANASIAK, FROSALI, AND SPIGA
C' reduces N(Ce) and since W = N(Ce ) 1 ', we get DC'O = QC'P = 0 .
System (5 .2) takes the forni
at -
E PSQ~wE +
.
PC'
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
-i co . Thus we obtain wo = po = 0 and
1
01 = -(QCe4~) c Spp,
O Ceo)-1 QSOw 1 ( QCQ Qu) -1 QpSQ(QCe Q) -I QSpp (7
Iv2 = _(
=
.6)
w0 = e t c ce Q w .
zero as
PV' ,
ai,,
E Q~SwE
+
(~
SQ~wE
+
Q~C
Q~IVE
E
QC' O'V E ,
fE =
with the initial conditions
VI(0) =
w,(0) = Of
= YV
Inserting Eq . (6 .6) finto Eq . (7 .2) and equating the terms at the same
powers of c we find that wc have to take r = t/E2 so that wc obtain the
following system
~t =
-C~SQ~(QTlC e Q) -1 Q~SPp + PCPp,
(7 .3)
OC, Q111V° = 0,
QCe Qw1 + QSP,0 = O,
O Ce WV2 + SQ1V1 = 0,
Q
ap0 - 0
ar
,
=1' ,
=
VE
-
P'
Z, = w E -
l1'° -
Ew1 - E2 VV2 .
(7 .7)
Assuming that the solution and the terms of the asymptotic expansion
are regular enough (we require that they belong to the domains of ali the
operators involved in Eq . (7 .2)) and taking into account Eqs . (7 .3)-(7 .5)
wc obtain the following system of equations for the error
aYE
at
-
1
1
PSQZE - PC'PY E = EPSQ1V2 + PSl
E
E
o,
(7 .8)
a,_1
1
1
Q S PY E - E OSO -' - QC r OZE - C2 OCe QZ E
at
e
1
= EQ SQ1h'2 + e2 Q C'Q11' 2 + QISQwo + EQZ1C' QJlw1
E
aw 1 - E2 a1v2
+ QC'ow
E
at '
al
with the initial conditions
_ Q Ce W1, o ,
y, (0) = 0,
which, as wc shall see, defines enough terms of the asymptotic expansion to
obtain, at least formaily, the convergente of the differente E(t) - f (t, r) to
zero as e - 0 .
In fact, let us assume for a time being that ali the equations
above can be solved and that the solutions are sufficiently regular to
make the manipulations to follow available . It can be proved that on
this level of approximation the correct initial values for Eqs . (7 .3) and
(7 .5) are
p(0)
Hence, we take the pair (p, îv o + E1V1 + E2 d'2) as the approximation of
(ve , VE ) ; the error of this approximation is given by
° -
ar
avo
r
YE
P f = v,
221
0
ll'o (0) = w .
Note that the equations for w 1 and w, do not require any side conditions, and the solution to Eq . (7 .4) is determined by the stipulated decay to
J E(O )
=
E(Q~C e Q~) -1 QSE
V -
E 2 (Q~C eQ) -1 a sQ(OC e Q) -1 Q SP V .
Keeping in mind our assumption that ali the terms of the asymptotic
equation are sufficiently regular, we see that the error eE = Y E + zE is a classical
solution of the problem
aee~ -
l Se, - C'e, -
EZ
Ce e,
a t
SWV2 +Q~C i OiV 1 +EQQC'(iV2 - a tivl -E E(
at
w
1
+ SQ wo +QC'Qivo ,
E
-1 O SO ( Q C e Q) QSE V .
)
e E(O = E(QZlC e Q) -1 QSEP v -E2 (QC e Q)
-1
(7 .9)
BANASIAK, FROSALI, AND SPIGA
222
The semigroup solving this equation is contractive in X,
the Duhamel formula wc obtain the estimate
II eE(t) II x
<
-
o -1
E (UCeO)-1 USp v - E(OCe ) QSO(OCe U)
I
'Pr
SQ1V2 (S) +
+ E
J (r''
+
EJ
1
II
Q
2
C Q1V1(S) + EQC Q~w2(S) -
SU11Vo(S/E ) +
eoC
' QVO(s/E2 ) x
II
ds .
-1 QS((D v
a
alv,
X
(S) - E
11
thus using
2
a
a1~~
(S)
}
3
2
Ir,, Qf,
Lemma 7 .1 . If f E D(S) n D(C), then
Eqs . (5 .1) and (7 .2) are equivalent .
E D(S) n D(C')
and therefore
The explicit form and the solvability of Eq . (5 .3) have been investigated in Ref. [3] for much more generai models . In the theorem below we
shall summarize the main results of Ref . [3], specified to the case in hand .
Theorem 7 .1 .
(a)
-1
Let us denote D = - pSU(UCeQ) OSP . Then deflning d =
d(v) = 4n/(3À(v)) > 47r/(3X nmx ) we have
(Dp)(x, v) = d v2 (Ap)(x, v)
Aere A is the Laplace operator in x-variable .
D(D) = {f
(7 .10)
(7 .11)
(d)
223
The operatore D,, = d v 22 A (v fixed) defined on the domains
D(D,) = L 1, 2(R3) C Xi (Bessel potential space, see e .g . Ref. [25])
for v > 0 and D(D 0 ) = X,, generate positive semigroups of
contractions in X c , denoted hereafter by (G D (t)) r>o .
The operator D with the dornain
dS
/,2
o
2
(b)
(c)
From the above inequality we see that if ali the expressions in the first two
terms exist and are bounded in t on [0, to ], 0 < t o < oc, then the contribution
of this integrai is of order of c on this interval . As far as the second integrai
is concerned, the initial layer is assumed to be exponentially decaying with
for some w > 0 . If this property is
r - . oc, that is, to be of order of e`
preserved after having operated on 1v with the operatore SU and QC'(,
then upon integration we obtain that also the contribution of this term is of
order of e, thus IIeEI x = O(e) and the convergente is proved . Hence we see
that to complete the analysis we must prove that all the tern3s exist and have
the desired regularity . For instante, for QSci v2 to be well-defined we need
the existence of S p or, in other words, the solvability of Eq . (5 .3) in the
moment space X together with three-fold differentiability with respect to x ;
wc also need certain regularity of the moments with respect to the operator
C' : the existence of QC'WV requires that S p be in D(C), etc .
The first step is to establish the equivalente of the forms (7 .2) and (5 .1)
(with relevant scaling) . This requires the projections of the solution to Eq . (5 .1)
to belong to the domain of S . In this respect wc have the following lemma [41
which also applies to any other case with dominant elastic scattering .
3
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
E
X ; f(., v) E D(D,,), (x, v) - (D,, f)(x, v) E X}
generates a semigroup of contractions (G D (t)) r>o in X . This semigroup is conservative for nonnegative initial data .
The operator T = D + PC'P defined on the domain D(T) _
D(D) fl D(C) generates a positive semigroup of contractions in
X, denoted by (GT(t))r>o, which is conservative for nonnegative
initial data .
It turns out be more convenient in further considerations to replace the
speed v by the energy related variable according to v = ~, as it was done in
Sec. 4 . To avoid introducing yet another set of definitions, we shall not
change the symbols of the functions appearing in the problem . With this
convention, the Cauchy problem for the limit hydrodynamic equation (5 .3)
takes the form
2
ap
= ~ d Ap +~
i
H(~ - l)rn(~) + b
m( +
1)p( + 1)+bH(~- 1)lnOp(s~ - 1),
(7 .12)
p(0) = p' ,
where m(~) = 4mrv(~) .
To carry on the rigorous analysis, we are interested in proving the
existence of the moments of the solution to Eq . (5 .3), thus the problem
(7 .12) has to be studied in X . .
Under our simplifying assumptions on the homogeneity and isotropy
of scattering (see Subsection 6 .1), in the velocity space X, ali the terms can be
averaged with respect to co and this space is reduced to X, = L
dv) _
t
Li (R+ , ~ d~)
I (R+, v2
The crucial result is summarized in the foliowing theorem, whose
proof is quite lengthy and requires some preliminary lemmas . For a
complete proof we refer the reader to Ref. [5], where the full problem related
to Eq . (5 .3) is completely investigated . Here we limit ourselves to an outline
of the main steps of the proof.
224
BANASIAK, FROSALI, AND SPIGA
Theorem 7 .2. Fo,• any k > 0, the operator T = D + PC`P generates a
positive semigroup in X k , denoted by (GT(t))r>o . Thus if p E Dk (T), then the
corresponding Cauchy problem (7 .12) :
(7 .13)
p
a
T
at
p
with the initial condition
(7 .14)
p(0) = p
has a unique solution
p E
C ([0, cc[, X k ) .
For the proof of this theorem it is convenient to split the operator T
into the following sum
T=D+ff C`lP=D-N+K_+K+,
where N, K_ and K+ are defined by Eqs . (4 .9) and (4 .10) . First it is possible
to prove that D - N generates a positive semigroup, using the fact that N is
a multiplication by a non-negative measurable and almost tverywhere finite
function .
It can be proved (Ref. [5]) that the resolvent of R(À, D - N + K+ )
exists and is a positive operator in X k for any ? > 0, thus by Desch's
theorem (see e .g . Ref. [33], Theorem 8 .1) it follows that D -N+ K+
generates a positive semigroup . Finally, because K_ is a bounded positive
operator in X, Theorem 7 .2 is completely proved using the Bounded
Perturbation Theorem e .g . Refs . [9,19] .
∎
Now we are ready to provide the error estimates . Let us recall that
according to Eq . (6 .5) we have
Xlkm,T =
{f
E
Xk ;
8~Xs f E
(7 .15)
D(Tm ), I,Bj < 1} .
We have the following lemma, whose proof is given in Ref . [5] .
Lemma 7.2 . Let v = P f
Then for each interval [0, t o ], 0 < to < +oo,
there exists a constant M such that
E X331, T .
o
II(QCe Q) -1 QSP v-E(UC e Q) -1 QSQ(QaC e ) -1 QSP v IIx
+ rmax
[ool
< M.
SQw 2 (t) + QC'QpW1(t)
+
EQIICQJi 2(t) -
a8
t1 (t) - E aaf, (t)
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
225
The regularity of the initial layer is dealt with in the next lemma .
Lemma 7 .3 . Let w = l f
such that for any t > 0
E X111,
ci . Then there exist a positive constant L
, mi
IISQtii'o(t/E2 )IIx+EIIQC'Q o(t/E 2 )IIx < Le . t1 E2 .
( 7 .17)
Proof. Since the terms of the initial layer expansion are exactly the same as
in Sec . 7 .3, this lemma coincider with Lemma 4 .3 of Ref. [4] .
Thus we have the theorem .
E X331, T gnd Q f E X 111 , ci . Let fE be the
solution of Eq . (7 .1) witk the initial datura f, and p be the solution to Eq . (5 .3)
with the initial value Pf . Then for each interval [0, t o ], 0 < to < +oo, there
exists a constant K depending only on the initial data, the coefficients of the
equation and t o , such that
Theorem 7 .3 . Assume that P f
fE(t) - p(t) - e ar/EZ Qf
x
< KE
(7 .18)
uniformly on [0, t o ] .
Proof. For the proof wc note that the assumptions on the initial data
adopted here are not weaker than that of any lemma (in particular,
D(T) c D(S) n D(D)) so that ali the steps of this subsection are justified .
Hence using Lemmas 7 .2 and 7 .3 ; we have by Eq . (7 .10)
IleE(t)Ilx
<
E (OCe o) -1 QSPv-E(QC e Q) -1 (QSQ(QCe Q) -1 (QSPv
+
E
J0
+ 1E
r
r
fo
EMt0
S4~W2 (s)
+ QC`Qpw 1 (s)
IIS(QíV0(S/E 2)
+
EL
%lIE`
J0
+
EQ~C'Q)
+
EQ~C`Q~11' 2 (S) -
o(S/e2)IjX
V
x
7t2
aasl (s) - E 8s
(s) x ds
ds
e_~1min'dr < KE .
The only difference now is that in Eqs . (7 .7) and (7 .10) we had e E =f€ - pw o - EW 1 - E 2 vv2i whereas in Eq . (7 .18) the last two terms are missing .
However, the estimates of Lemma 7 .2 can be carried also for 1v 1 and w,
BANASIAK, FROSALI, AND SPIGA
226
alone, showing that they are bounded on [0, to] . Since they are multiplied by E
and c2 respectively, they can be moved to the right-hand side of the inequality
(7 .18) without changing it .
∎
7 .2 . Purely Diffusive Hydrodynamic Limit
from
In this subsection we shall describe the steps leading asymptotically
8f
Clearly the estimates are analogous with the only difference that this
time p is a solution of the diffusion equation in x multiplied by v 2 , as seen
from Eq. (7 .20), and the solution to this equation must have the regularity
required in Lemma 7 .2 . Since the operatore of differentiation and
multiplication by vk commute with D, and D generates a C0 -semigroup,
the assumPtions will be much milder here and the proof of the counterpart
of this lemma is much easier, hence we shall only sketch it . Recalling that
Xlk = WI(R',Xv,k),
_ E sfE + É c ife + EC,ff
(7 .19)
to Eq . (5 .4) which by Theorem 7 .1 is given by
(7 .20)
ap = v2 dOp
at
Since the limiting equation is simpler than in the previous case, wc shall skip
most technical details .
Using the isotropy of the scattering wc arrive aY the following
counterpart to Eq . (7 .2)
av,
at
aaE
-1 PSQW E + EIPC`PV E ,
E
_ 1 4~1SlvE + i Q~SQQ1V, + E~Q1 C ' 0WE +
QpCe Q~W E .
(7 .21)
Apart from the hydrodynamic equation (5 .4), all the other terms of the
asymptotic expansion coincide with those given by Eq . (7 .6) . Defining the
approximation and the error as in Eq . (7 .7), we obtain the error equations in
the form
a
227
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
EE - PSQZ E - EPCPYE = ePSQWZ + EPC`rp+ ! PSWWO ,
a z, _ 1 Qspy,
- 1 QSWE - EOCQZE - E,, QC e Qz = EQSwi',)
ar e
E
+E 3 4pCQW2
+ 1 Q sWv0 +E2 Q~C`Q7w l
+ EQ~1CO1vO - E
e
aat`
a
t2
- EZ a ,
with the initial conditions
YE(0) = 0,
z E (0)=E(QC e Q) -I QSPv-E2 (Qa1Ce
(7 .22)
wc have the following lemma .
Lemma 7 .4 . Let v = Pf
X44 . Then for each interval [0, t 0 ], 0 < t0
there exists a constant M such that
E
II(OCe Qp) -I QSPV - e(0Ce
o
) -1 c Sc (4Ce 1l)
<
+
oo,
-I QpSPvIX
a (t)_E aa
t2 (t)
+ max ; SeW2(t)+EQCQi 1(t)+e 2 0Ci 11w2(t)- at
1E [O, to]
i
(7 .23)
< M.
Proof. Following the approach of Lemma 7 .2 in the same order we see
that the estimate (7 .23) is satisfied if: v3 as~ v E X for 1f1 = 3, (v + l)a~e v E X
for 1,81 = 1, (v'` + v)as~ v E X for If I = 2, and v k a~, v E D(D) for I13I = k,
k = 1, 2 . Recalling the definition of D(D) we see that if v E X 44 , then all
∎
the above requirements are satisfied .
The initial layer terms are the same so that for the estimates wc can use
Lemma 7 .3 .
To make the statement of the final theorem more clear, wc recall that
the space XI I I, C used in Lemma 7 .3 is given by
I
3
X111, C - WI (Rr, X,,,,,+,-~).
o
To ensure that the fE is the classical solution we require that P f E
D(C) in addition to the assumptions of Lemma 7 .4 . With these we have the
following counterpart of Theorem 7 .3 :
a
(3) 1 QSQ(QCeQ) 1 Q3Spv .
Theorem 7 .4 . Assume that P f E X 44 ,, Co and Qf E X I 11, C . Let fE be the
solution of Eq . ( 7 .19) with the initial datum f , and p be the solution to Eq . ( 7 .20)
with the initial val ue Pf . Then for each interval [0, t 0 ], 0 < t0 < +oc, there
228
BANASIAK, FROSALI, AND SPIGA
exists a constant K depending only on the initial data and the coe cients of
the equation and t0 , such that
fE(t) - p(t) - e
w
E2
of
x
< Ke
(7 .24)
=
Now we shall present the counterpart of Theorems 7 .3 and 7 .4 for
the scaling
aE
Sf +
i CefE + Cf
at
E
(7 .25)
- Qspy E - QsQz E - QC'Q ; - QCe ` zE
( 7 .26)
!
where, as before,
+1 m( + l)p( + 1) +bH( - 1)m()p( - 1),
with m() = 47rv() .
This problem was thoroughly investigateti in Ref . [4] so that wc
mention here only that the solvability of Eq . (7 .26) in the moment spaces
Xk presents similar (though technically less involved) difficulties to those
encountered for Eq . (5 .3) .
It follows that in this case there is no need to go to
1v2
as the terms
0
w o = e'QC Q w
where r = t/E, and p is the solution to Eq . (7 .26) with the initial condition
p(0) = v, suffice to obtain the desired estimates . This follows as the error of
-11'0
- EYV1,
uv i
at ,
zE(0) = - Ew1(0) = e(OCe Q) -1 Q5P v .
YE( 0 ) = 0>
The conditions under which the error is of order of s are given in the
following theorem, which was originally proved in Ref. [4] .
E X221 C and 0Q~f E X111
ci • Let f, be the
solution of Eq . (7 .25) with the o initial datura f , and p be the solution to
Eq . (5 .5) with the initial value f . Then for each interval [0, t0], 0 < t 0 < + co,
there exists a constant K depending only orl the initial data, the coe cients of
the equation, and t 0 , such that
fE(t) - p(t) - e _" `
~+1
PC'Pp=-CH(~- 1)nz(:)+b
m(~+ 1)'p
+
=wE
EQC'(QYV1 + (QC'Qw0 - E
(7 .27)
o
ap = PC'Ip,
ar
zE
+
Theorem 7 .5 . Assume that Pf
resulting in the hydrodynamic limit (5 .5)
the approximation
EDSQtiv1 + PSWV 0 ,
and the initial conditions
7 .3 . Purely Kinetic Hydrodynamic Limit
yE - VE - p,
8yE
- PSQz E - PC'Pyf
al
= e SjT'1 + QSQVp
iv l = -(QC e Q) 1 QSPp,
229
formally satisfies
at
uniformly on [0, t 0 ] .
p,
SCATTERING OPERATORE IN EXTENDED KINETIC MODELS
I E(Q
o
f
x
1
<_ Ke
(7 .28)
uniformly, on [0, t0 ] .
7 .4 . Continuity Equation as the Hydrodynamic Limit
The last case of the limit evolution in the hydrodynamic space N(Ce ) is
given by the scaling
aa
EE - SfE
+
EC'fE
+
E C efE ,
( 7 .29)
which produces formally the trivial hydrodynamic limit
ap
0.
ar =
For the sake of completeness we note that the standard asymptotic
procedure (with the initial layer time r = t/E) gives the saure terms of the
230
BANASIAK, FROSALI, AND SPIGA
expansion as in tbc case of purely kinetic hydrodynamic limit . The error
equation takes the form
ayE
- PSQzE - EDC'Py E = epsWY i + PS©tiv o + e C'ip,
at
z - tSPyE - QSQ?E - E6.
= et SQw,
+
COZE - 1 t3 C, Or,
E Z QC'Uw 1 + USU1v o
+
a
(7 .30)
EvVC'Q-Vwo - E a
`t 1
with the initial conditions
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
231
simplified version of these models (without C e) was thoroughly investigated in Refs . [2,8], and from the numerical point of view in Refs . [6,16,17] .
It follows that taking C e into account does not change the overall
picture-the differences are rather cosmetic as was already noted in Sec . 4 .
It is useful to remark that the null-spaces N(C) and N(C + Ce ) coincide
and tbc inclusion of N(C) into N(Ce ) simplifies the asymptotic analysis .
Thus in this section we shall present only formai considerations to show that
tbc presente of Ce does not necessitate the introduction of any assumptions
other than those in the papers cited above . Recali also that by Corollary 4 .1
tbc null-spaces and the projection operators P and Q coincide in all the cases
discussed in this section .
yE(0) = 0,
z E (0)
=
E(QCe Q) -1 OSQD v,
8 .1 . The Case of C` and C e Being of the Same Magnitude
and wc see that tbc only difference with Eq . (7 .27) (apart from possibly
higher powers of E in some places) is the presence of the term EPC'EPp
in tbc first equation of Eq . (7 .30) . However, the solution to tbc limiting
equation is constant in time
p(t) = p(0)
= v,
(7 .31)
so that ali tbc regularity requirements for p will be satisfied provided they
are imposed on tbc initial value v .
Thus we can state tbc theorem
o
o
X221, ci and O f E Mi 11, c ; . Let f be the
solution of Eq . (7 .29) with the initialdatum f . Then for each interval [0, to],
0 < to < +oo, there exists a constant K depending only on the initial data,
the coe cients of the equation, and t o such that
Theorem 7 .6 . Assume that P f
E(t) - Pf - e x`!EQf
uniformly on [0, to] .
x
E
< Ke
(7 .32)
First we shall consider the Boltzmann equation with tbc scaling
1
aE
at
= 1e g + e1 CefE + E
In this section we shall discuss the hydrodynamic limits in the cases
with dominant either inelastic, or both inelastic and elastic scattering . A
(8 .1)
which as wc know has the hydrodynamic limit
p
aa = - PSQ[Q(C' + C e )Q]-1 QSPp .
Let us first note that the question of the invertibility of Q(C + C e )Q is
much more complicated than in the cases with dominant elastic scattering .
The case without the elastic collision operator Ce was investigated in
Refs . [2,8] . Here wc shall show that the addition of C e , as defined in
Eq . (6 .1), introduces only insignificant changes .
Due to tbc translational character of tbc operator C' it is convenient to
introduce the reduced energy ~ E [0, 1[ and to re-define all the functions as
functional sequences in the following way : for n = 0, 1, 2 . . . and fixed
~E[0,1[
P" (x,
0 = p(x, ~ +n),
d„(~) = d(~ + n),
8 . RIGOROUS ANALYSIS : DOMINANT
INELASTIC SCATTERING
Cf',
v,,(~) = v(~ + n),
~» = '/~ + li,
P,A)
.
_ ' ~' v»+1(0
232
BANASIAK, FROSALI, AND SPIGA
It follows that using this notation wc can introduce the following
equivalent norm in Xk
Ilfllk=
f
0
I
~olifo( •, ~)Ilx,d~+
j-1
k~2
f
0
i
~j11f( •, ~)IIY,d~<oc,
(8 .2)
and the domain Dk (C') can be identified by the finiteness of the following
expression
f
o
1
Il fo(,
y, d4+
1
j=1
' f s~jllf(4,w)IIY,d: < cc .
(8 .3)
0
To solve the equation (C + C') f = g we use the previous notation to
obtain
go(x,
44 , w) = -4m[X0(4) + bpo(4)]fo(x, 4, w) +po f ,
s
+ Xo(4) f
gn(x,
fi
(x, 4, w) do
` fo(x, 4, m') dai ,
+P"(0 f f, +t (x, 4, o)) doi + bv„(4) f
s
, f,-1 (x, s:, co) dai
s-
+ x„(4)f s~,f,(x,4,w)do),
g„0(x,
(Pg,)(x,
1
4) =
47r
f
s2
g, (x,
where
4, w) dw,
and g„ 1 = g„ - g„o , and introduce the similar notation for the unknown
function f, we see that due to Eq . (6 .1) wc must have
goo(x,
g, 1 (x, S,
+ bp» (4-) +
w) =
1(x,
4, w),
so that we see that the presence of the elastic scattering operator C e affects
only the easy diagonal part of the equation, wheres the troublesome part is
the same as in the case C e = 0, and we can repeat the proof of Theorem 2 .1
of Ref. [2] .
Thus, if wc denote by X ° the subspace of X of functions satisfying the
compatibility condition of Theorem 4 .3 :
= {f E
L, (Rb, v) ;
E
j=0
J4
[O, 1],x
E
j
f S-
f (x, s/~ +
fw)
de) = 0
IIRx ~,
and, as above, we decompose g = go
then the following theorem is valid .
+ g1,
where go
= Pg
and g 1
= g - go,
If g E X° is such that g0 E Xk+2, then there exists a
unique sohztion f e Dk (C) fl X° to the equation (C' + Ce)f = g, and there
exists a constant M such that for any such g
Theorem 8 .1 . Let k > 0 .
(8 .4)
g„ = g„o + g„1,
go1(x, 4, .w) = -4 r[À0(4) + bp0(4)]f01(x, 4, (9),
for a .a . 4-
4, w) _ -4z[? n + bp„(4) + v, («f, (x, 4, (9)
233
and in the same time
,
S
where n > 0 . If we decompose
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
4) = - 4irbpo(4)foo(x, 4) + 4zpo(4)f10(x, s:),
g»o (x, 4) = - 47r[bp„ (4) + 1 ]f,o(x, 4) + 4zp„ (4)f,+1, o (x, 4) + 47rbf,-1, 0(x, 4-),
Ilfllxo,,, <
M(Ilgollx '+ , + 1191 11x,1
Remark 8.1 . It is worthwhile to note that Theorem 8 .1 gives only sufficient
conditions for the solvability of (C'+ C e )f = g, and the condition (8 .7) is
not necessary, as can be checked by considerino , the sequence
with fo defined so that the compatibility
(fo)n>1 = ((-1)"%, l n Z )„>1
.
It
is
straightforward
to check that g = Cf does not
condition is satisfied
satisfy Eq . (8 .7) .
On the other hand it can be proved that if g is of constant sign at least
for large velocities, then Eq . (8 .7) gives also the necessary condition for it to
belong to the range of C, hence C(D(C) n X° ) ~ X ° . The details of these
considerations can be found in Ref . [8] .
BANASIAK, FROSALI, AND SPIGA
234
Denote
1N = f
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
235
Let us recall that by Theorem 4 .3 the spectral projeetion onto
E
X; f f(x, s~, co)dw = 0 for a .a . x
s
e R
~ E
[0, oo[ } .
J-
Let us observe that W c X° , that is, any function annihilating constants on
S2 automatically satisfies the compatibility condition (4 .43) . This makes W
a very useful space as follows from the following corollary to Theorem 8 .1 .
Corollary 8 .1 . The operator C' + C e is continuously invertible on W with the
inverse given by
I
fo = - (fio + bpo) go>
.f;, = -(X„ + bp„ + v„) -l g, ,
Moreover C' + Ce is coercive in W and generates there a semigroup of
negative tvpe -y = -(Xmin + bLmin) .
Proof. The first statement follows immediately from Eq . (8 .6) . The proof
of tbc second statement is similar to the proof of Proposition 4 .2
of Ref. [2] .
∎
This corollary allows the construction of the diffusion operator . Before
doing this we should however note that the structure of the diffusion
operator bere is completely different than that of for the dominant elastic
scattering . In the latter case the diffusion reflected the feature of the elastic
scattering that the particles could be only deflected without changing their
velocity, thus the kinetic energy ~ E [0, ce[ entered the diffusion only as a
parameter labeling particles coming from classes of different energy . Here
the inelastic collisions could change the energy of particles, however, for a
given particle this change could only occur among countably many states . In
other words, the particles are divided into non-overlapping classes
with energies taking values ~ + N, where ~ E [0, 1 [, and these classes
remain separate throughout the evolution . As we shall see, this feature
is to some extent reflected in the diffusion operator, where the energy
parameter is free only in the interval [0, 1 [ and for higher energies the
values of the solution are reproduced in a periodical manner .
N(C' + Ce ) is given by
00
(Pf)„(x, ) = b"G -1 (oy~~j(pf)j (x>0,
j=0
(8 .9)
1
where b = n ) /n 1 < 1, (Pf)„ = ,,,f, and G(~) = F_j_ 0 bl~j . Here ~
x E ll8;. . It can be proved, Ref. [8], that
Q(f)
E
[0, 1[ and
ce
= ref
j=0
is the proper hydrodynamical quantity, that is, it remains constant
throughout the evolution . Thus, our limiting diffusion equation will determine the approximation of o . This will be defined only for ~ E [0, 1 [, and the
extension to the approximation valid for all ~, according to Eq . (8 .9), is
given by p = (p„),0 where
(8 .10)
p„ = b"G-1 Q .
To find Q we have the following result which is a straightforward
generalization of Proposition 3 .1 of Ref. [2] .
Proposition 8.1 . The diffusion equation (5 .11) is of the forni
aQ
B(~)
AQ,
at - 3G(~)
(8 .11)
where
Xo(~)~o + be l vi (~)
bj ~ja
j=1 Xj(S)Sj + b~j+I vj+I
(S) + Sjvj(S)
The existence of the initial layer is also not automatic as the operator
Q(C` + C')Q is not continuously invertible on QX . Therefore even if
Q(C' + Ce )Q generates a semigroup, this semigroup cannot be of negative
type and wc will not have the exponentially decaying initial layer . To
circumvent this difficulty we use Corollary 8 .1 and restrict the initial
values to such that
o
0
f =v+w=Pf+uf,
(8 .12)
BANASIAK, FROSALI, AND SPIGA
236
that is, we assume that tiv e W . It follows from Eq . (8 .9)o that fQ = 0,
hence for the initial values satisfying Eq . (8 .12) we have Q f =Q f .
Let us write down formai expressions for the remaining terms of the
asymptotic expansion and for the error . The projected system takes the form
avE -
e
at
Eu
aWE - 1 OSQwE
wE(0)
= tiv
:=
o
+
(Z E, YE) =
E
-
YE
~ PSQZ E _ PSQwO + EPS_Oiv2,
at e
_
Z
Qf,
Q(C' + Ce )Qtiv2 = }QSQwI,
(8 .13)
where p is defined by Eqs . (8 .10) and (8 .11) . Fortunately, QSPp is linear in m
and thus belongs to W, hence the equation for w l is solvable, defining
- ( Q(C' + Ce)Q) -1 QSPp .
Moreover, due to the presence of the exponential factor in Eq . (8 .10),
QSQw I E X 0 fl Xk for any k (at least for sufficiently smooth in x function o) .
Thus by Theorem 8 .1 the term w, is also well-defined .
The initial layer time is given by -c = t/E 2 and the standard procedure
produces po = 0 and
ai-VO
(8 .14)
a-V = Q(C' + Ce)QtivO .
Due to the assumption (8 .12) wc restrict this equation to the subspace W
where, due to Corollary 8 .1, wc have the generation of a semigroup of a
negative type.
Finally, we obtain that the proper initial values for Eqs . (8 .11) and
(8 .14) are given by, respectively
p(O) = v = Pf ,
o
tii'o(0) = tiv =1f
y,(0) = 0,
E QSPYE - E QSQZE - É Q(C` + Ce )Q~E
at -
Pf ,
TC , + C e )Qtivl = - QSPp,
=
Combining these results we see that the error
p, w E - IV O - Ewl - E2 w2) is (formally) the solution to
(V
1
QSPYE + 2 Q(C' + C e )Q19E ,
E
E
where we have taken into account that PC = CF = 0 and that PSP = 0
which follows from Eq . (8 .9) (see also Ref. [2]) . As before wc obtain that
for the bulk part 0 0 = 0 and
wl
237
PSQwE ,
at
V,(0) =
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
1
QSQ1"vO + eQSQw 2 -
E (O) = -Ew l (0) - e2 w,(0) .
ativ l
e at -
Ez
aw2
(8 .15)
at ,
The rigorous analysis of this system is not different from that provided
in Ref. [2] . However, due to a simpler form of the diffusion operator (the
Laplacian commutes with the differentiation with respect to x) we can case
the assumptions on the spatial regularity of the initial value . It follows that
in this case we have the following theorem .
Theorem 8 .2. Let us assume that v = Pf
E X411 , c i and Qf E X11 c fl W .
For any t0 < + oc there exists a constant M (depending on t o in an a ne way)
such that
II fE(t) - p(t) - 0 0(t/e2 )I1x < EM .
(8 .16)
8 .2 . Strictly Dominant Inelastic Scattering
Here we shall consider the Eq . (5 .13) with q = 1 which presents the
most involved case . That is, we have
t
=CSfE +ECefE
+e
CE,
which, as wc know, has the hydrodynamic limit
ap _
- PSQ(QC'Q) - 'QSPp .
(8 .18)
It is clear that due to Theorems 4 .2, 4 .3, 8 .1 and Corollary 4 .1 the
asymptotic analysis in this case will not differ much from that for Eq . (8 .1) .
The hydrodynamic part p is again given by Eq . (8 .10) with the only
difference that the diffusion coefficient B in Eq . (8 .11) should be calculated
with À,, = 0 for any n .
238
BANASIAK, FROSALI, AND Sp IG
It follows that w 0 , lv l , w0 are the same and lb'2 is the solution of
QSQw J + QCe Qiv l = QC'Qtiv2 .
(8 .19)
Note that the operation with QCe Q on w 1
not change tbc properties of
iv i which are relevant to the solvability does
of Eq . (8 .19) (integrability
weight 1 + v k) so that the terra lv2 is well-defined as in Eq
with
. (8 .13) .
Defining the approximation as in the previous case we obtain the error
equations in the form
aYE
8t
1
_ 1
_
E PSQze ! PSQwO
azE
1
1
=
QSQ }t'o +
QwO + EQCe Qw2
. QCe
(8 .20)
v = Pf E X411, c, and Qf
+ oc there exists a constant M such that
E
Xi 11, ci n W .
(8 .21)
8 .3. Continuity Equation as the Hydrodynamic Limit
The last case is when wc have the scaling
C
E= Sff +E Ceff
+6 f,
with q = 1 or q = 0, both giving the hydrodynamic limit
(8 .23)
1}'0(0) = W .
at
PSQz6 = PSQivO
+
yE (0) = 0,
EPSQIVI,
QCeQZE
1
7E (0) _ -EYV,(0)
J jfE(t) :5
p(t) - iVO(t/e2)IIX
EM.
ap
at = 0 .
ax'° = QC'Qw0,
at - QSQZ e + E QC'QZE
= QSQIVO + QCe Qii, o + EQCe QI1'1 + EQSQ_lvt,
We see that but for the terms É QCe QwO and EQC e QW2,
Eq . (8 .20) is
the same as Eq . (8 .15) . However, since the operator
QC e Q is bounded, the
addition of these terms will not affect the error estimates and we see
that
also the main theorem will not change here .
<
and
It follows that the proper initial layer time is i = t/E ; apart from this the
initial layer corrector is defined as above by
QSPy.
awl
8w2
+ EQSQ }v2 - E ó- e 2
ót t ,
zE(0) _ _Ci,(0) - E2 w2 (0) .
For any t0
Ff
li, l = _(QC'Q) -1 QSP p .
a t -
_
Theorem 8.3 . Let us assume that
Clearly, then p = p =
lowing equation
1
1
q = 0.
Defining the error by y = v E - p, z = w - Elv l - w 0 wc obtain the fol-
y€ (0) = 0,
at - E QSPYE - E QSQzE - E QC e QzE - E- QC 1 QzE
_ 1
Let us consider first the case with
àt
_
EPSQlv2 ,
+
239
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
(8 .22)
(8 .24)
where we used the fact that alv i /at = 0 .
If wc take q = 1, then the only change will be that
lvl = - ( Q(Ce + C`)Q) -1 QSPP,
and, with the same r and the initial value w 0 (0)
aW0 = Q(C e + C )Q 1N0 .
az
(8 .25)
Wc know that the addition of C e doesn't affect relevant properties of C', so
the above equations are solvable as in the previous cases . The only differente
will be that in the error system (8 .24) wc won't have terms involving QCe Q
on the right hand side of the limiting equation but this doesn't affect the
error estimates . Thus, both cases can be treated as one .
The regularity assumptions can be further relaxed here . Wc note
that for the estimates we require the existence of only two spatial
derivatives of p and the regularity with respect to v is o ensured by the
exponential factor appearing in the definition of p = Pf . All the other
assumptions on Ff come from the requirement that the error be the
classical solution of Eq . (8 .24) . The assumptions on Qf are the same as
in both previous cases, and are dictated by the solvability of Eqs . (8 .23)
240
BANASIAK, FROSALI, AND SPIGA
and (8 .25) .
theorem .
SCATTERING OPERATORS IN EXTENDED KINETIC 1VIODELS
Thus wc can summarize this discussion by stating the
i and 1v = Qf e
+ oe there exists a constant M such that
Theorem 8.4. Let us assume that v = Pf
W . For any t 0
<
E
X, i1
c
9 .1 . Dominant Elastie Scattering
We define
X111,cin
IIfE(t) - Pf -}1'0(0E)II x < EM .
(8 .26)
lPf = 1 ffdw .
47r '-
(9 .2)
and0=I-P
9 .1 .1 . The Case p = 1, q = 2, r = 0
9 . REFERENCE MANUAL
In this section we have collected the main formular of this paper for
easy reference . Wc avoid here making any regularity assumptions-it is
enouah to state that all the results are valid if the initial data are smooth
with respect to x and decay sufficiently fast to zero as v -+ oo . The only
additional assumption (adopted for technical reasons) is that the kinetic part
of the initial datum annihilates constants over S 2 in the clses of dominant
inelastic, or elastic and inelastic scattering . Wc are dealing with the
following Boltzmann equation
i
The details are in Subsection 7 .1 .
Hydrodynamic limit
àt
=~dAp- (H(~- 1)m(~)+bi m(~+
1)1p
+~+l m( +1)p(~+1)+bH(~-1)m(~)p(i - 1),
p(o) = Pfo ,
where m(4) = 4TrvO and the diffusion coefficient d is given by
af E - _ sfE + Cf, + ~P CIfE
at EP
eq
E
vco ax + i (-47rkf +x
d
J
, f dco /
s-
+- (-4,r(Hv+b i+v+) f + l
l + v+
f
s
f+ dw +bvHf f_dw),
s-
f€(0) =f
where v+ = w' ± 1, for any function g we denoted g f = g(wz ± 1) and
H = H(v z - 1) is the Heaviside function . Functions 2, and v are functions of
v variable only and v + = v(v + ) . In what follows we shall mainly use the
energy variable 4 = v'` . In all the cases the O(E) approximation in
L 1 (R6, v ), uniform on finite intervals [0, t0 ], is given by
f (t, x, v) = p(t, x, v) + W0(t/E k , x, v) + 0(c),
E
where k is equal to the highest power of
or k = 2 .
é
_
4Tr
3)(v)
2
241
in Eq . (9 .1), that is, either k = 1
Initial layer corrector
tiv 0 (t/E', x, v) = e-j(Vkl
62
~f(x, n
) .
9 .1 .2 . The Case p = 1, q = 2, r = The details are in Subsection 7 .2 .
Hydrodynaric limit
ap
ar
p(o) = P.f ,
BANASIAK, FROSALI, AND SPIGA
242
where the diffusion coefficient d is given by
d
p(t) = Pf
Initial layer corrector
Initial layer corrector
2
a(i)t/EZ
ivp(t/E , x, v) = e
Q
Wo(t/E, x, v) = e-X(v)tl'O f (x, v) .
(V, x) .
9 .2 . Dominant Inelastic Scattering
9 .1 .3 . The Case p=0, q= 1,r=0
In this subsection it will be convenient to use the sequential
notation for functions : f = (f„),,, o , where f (~ + n) = f„(4) for ~ E [0, 1 [
and n = 0, 1, . . . .
The spectral projection on either N(C) or N(C` + Ce) is given by
The details are in Subsection 7 .3 .
Hydrodynamic limit
- H(~ - 1)rn(~) + b
+,/
+
~ o
p(O) = Pf
1 m(~ + 1)
I p
1 rn( + 1)p(~ + 1) + bH(~ - 1)m(~)p(~ - 1),
(Pf)n(x,
o
-x(V)`IE(
lv o (t/E, x, v) = e
af(x, v) .
where b = n,/n 1 < 1,
by Eq . (9 .2) .
j=0
~j(Pf)j(x, 0,
j
+j, GO _ ~j o b'~j and P is defined
Hvdrodvnamic limit
Wc have p = (p„),,, o where
p,, =
9 .1 .4 . The Case p-0, q= l,r=-1
b n G -1
Q,
and Q is the solution to
The details are in Subsection 7 .4 .
p(o) = Pf ,
= b 77 G -1 (~)
The details are in Subsection 8 .1 .
Initial layer corrector
Hydrodvnamic limit
)
9 .2 .1 . The Casep= 1,q=r=2
where m(~) = 4nv(~) .
at = 0,
at
243
thus
4rr
3),(v)
~
a _
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
where
I
aQ
at
B(~)
3G(~)
0
A
cQ '
Q(0) = Pf ,
B(4) _
0
X0(5) 0+belv1()
bj a
j=1 À1( ) j+bj+1vj+1( )+ jvj( )
BANASIAK, FROSALI, AND SPIGA
244
Initial layer
9 .2 .3 . The Case p = 0, q = r = 1
The details are in Subsection 8 .3 .
In all cases below we require that w satisfy Pw = 0, and we use the
standard notation
SCATTERING OPERATORE IN EXTENDED KINETIC MODELS
p ( ) _ +n+1
Hydrodynamic limit
We have
v(~+n+ 1) .
With these
0
wo, ,, ( t/Ez , x, w) =
p(o) = Ff ,
0
e-(a0(~)+bp0($»t1E2 tivo(x, , w),
e_(xn( )+bn„(g)+v (gpr/EZ
tivn (x, ~, w)
i
for n > 1 .
thus
9 .2 .2 . The Case p= 1,q= 1,r=2
p(t, x, v) = Ff .
The details are in Subsection 8 .2 .
Initial layer
Hydrodynamic limit
We have p =
e (fio($)+bpo())tlEH,°(x,
(p, t ),,, o
Yt0,,,(t/E, x, (0) _
where
e(4(0+bv„A)+v,AMlEy°pn(x, ~, w)
pi, = b"G -1 Q,
and e is the solution to
9 .2 .4 . The Case p = 0, q = 0, r = 1
aQ _ B(~)
at
3G(O
The details are also in Subsection 8 .3 .
Q(o) = Pfo ,
Hydrodynamic limit
where
B()
Wc have
_
0
E
b, a
bitvi( ) +
b j+tvj+1 ()+ j vj ( )
p(o) = Pf ,
Initial layer
wo, (t/E2 , x, eco) =
e -b)0( )r/E2 11, 0(x, , w),
e -(bp .( )+0„( ))VEZlv7(x
thus
w)
for n > 1 .
p(t, x, v) = Pf .
,w),
for n > 1 .
245
246
BANASIAK, FROSALI, AND SPIGA
Initial lager
1b'o,l ( t/e, x, ~(0) =
SCATTERING OPERATORS IN EXTENDED KINETIC MODELS
6.
e bna ()t/E tiv o (x, ~, (o),
e
(bP .
($)+v„(
W,, (x, ~, o»
))t~E °
for ti > 1 .
ACKNOWLEDGMENTS
A significant part of this paper was prepared when one of the authors
(J . Banasiak) visited Dipartimento di Matematica Applicata "G . Sansone"
at the Università di Firenze . The support received for this visit from the
National Group for MathematicalPhysics of the Istituto Nazionale di Alta
Matematica (INdAM-GNFM) is highly appreciated .
The work of J . Banasiak was also partly supported by National
Reseach Foundation of South Africa .
This work was also partly supported by the Italian Ministery of
University (MURST National Project "Problemi matematici delle teorie
cinetiche"), by the CNR Special Project "Metodi Dhatematici in
Fluidodinamica Molecolare," and by the European TMR Network
"Asymptotic Methods in Kinetic Theory."
The authors would like to express their sincere thanks to all the
colleagues who, on various occasiona, contributed to this paper through
stimulating discussions, suggestiona and criticism .
7.
8.
9.
10 .
11 .
12 .
13 .
14 .
15 .
16 .
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Received September 5, 2000
Accepted March 23, 2002