ON A NEW CLASS OF WEAK SOLUTIONS TO THE SPATIALLY HOMOGENEOUS BOLTZMANN AND

Description

ON A NEW CLASS OF WEAK SOLUTIONS TO THE SPATIALLY HOMOGENEOUS BOLTZMANN AND LANDAU EQUATIONS C. VILLANI Abstract. This paper deals with the spatially homogeneous Boltz- mann equation when grazing collisions are involved. We study in a unified frame the Boltzmann equation without cut-off, the Fokker- Planck-Landau equation, and the asymptotics of grazing collisions, for a very broad class of potentials; in particular, we are able to derive rigorously the spatially homogeneous Landau equation with a Coulomb potential. In order to do so, we introduce a new defi- nition of weak solutions, based on the entropy production. Contents 1. Introduction 1 2. Basic identities and formulation of the problem 6 3. Definitions and main results 12 4. Hard and moderately soft potentials 20 5. Lack of a priori estimates for very soft potentials 25 6. H-solutions 27 7. The Coulomb potential 33 Appendix A. Uniform gain of moments 35 References 36 1. Introduction We are concerned in this paper with equations arising from kinetic theory, describing the evolution of a gas (or a plasma) which is not at equilibrium. They can be written as (1) ∂f∂t = Q(f, f), t ≥ 0, v ? R 3 where the unknown function f(t, v) is assumed to be nonnegative, and stands for the density of particles at time t with velocity v; Q(f, f) is a quadratic non-local operator describing the collisions within the gas, 1

• grazing


• weak solution


• collision


• differences between hard


• soft potentials


• cut


• fokker- planck-landau equation


• without cut


• boltzmann equation

Sujets

• Landau in der Pfalz
• Grazing
• Weak solution
• Collision
• Boltzmann equation

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 43
• Langue: English

Extrait

ON A NEW CLASS OF WEAK SOLUTIONS TO THE
SPATIALLY HOMOGENEOUS BOLTZMANN AND
LANDAU EQUATIONS
C. VILLANI
Abstract. ThispaperdealswiththespatiallyhomogeneousBoltz-
mannequationwhengrazingcollisionsareinvolved. Westudyina
unified frame the Boltzmann equation without cut-off, the Fokker-
Planck-Landauequation,andtheasymptoticsofgrazingcollisions,
for a very broad class of potentials; in particular, we are able to
derive rigorously the spatially homogeneous Landau equation with
a Coulomb potential. In order to do so, we introduce a new defi-
nition of weak solutions, based on the entropy production.
Contents
1. Introduction 1
2. Basic identities and formulation of the problem 6
3. Definitions and main results 12
4. Hard and moderately soft potentials 20
5. Lack of a priori estimates for very soft potentials 25
6. H-solutions 27
7. The Coulomb potential 33
Appendix A. Uniform gain of moments 35
References 36
1. Introduction
We are concerned in this paper with equations arising from kinetic
theory, describing the evolution of a gas (or a plasma) which is not at
equilibrium. They can be written as
@f 3(1) =Q(f;f); t‚0; v2R
@t
where the unknown function f(t;v) is assumed to be nonnegative, and
stands for the density of particles at time t with velocity v; Q(f;f) is
a quadratic non-local operator describing the collisions within the gas,
12 C. VILLANI
whose form we shall make explicit below. These equations are called
homogeneous since f is assumed to be independent of the position.
In most of what follows, the velocity space can be assumed to be
N 3R rather than R , but this implies to change many constants and
exponents, so that we shall restrict ourselves to the physically realistic
case N =3.
ThefirstmodelthatweshallconsideristheBoltzmannequation[10,
11, 33],
Z Z
@f 0 0(2) =Q (f;f)= dv d!B(v¡v ;!)(f f ¡ff )B ⁄ ⁄ ⁄⁄@t 3 2S
2where S is the 2-dimensional unit sphere,
‰
0v =v¡(v¡v ;!)!⁄(3) 0v =v +(v¡v ;!)!⁄ ⁄⁄
0 0and we adopt the usual convention f =f(v), and so on.
0 0We recall the usual interpretation of (3) : v, v stand for the ve-⁄
locities of two particles that are about to collide, and v, v for the⁄
velocities of these particles after collision. In particular, they satisfy
the conservation of mass, momentum, energy
‰
0 0v +v =v+v⁄⁄
0 2 0 2 2 2jvj +jv j =jvj +jv j :⁄⁄
B is a weight function for all possible values of the parameter !. Its
particular form depends on the interaction between particles, but on
physical grounds it is always assumed that B(z;!) depends only onjzj
andj(z=jzj;!)j.
Themeaningofequation(2)isclear: ifweformallysplitthecollision
operator,
+ ¡(4) Q (f;f)=Q (f;f)¡Q (f;f)B
it is easily seen that the second term (loss term) “counts” all collisions
in which particles of velocity v acquire a different velocity, while the
first one counts the particles that the velocity v.
In most of what follows, we shall assume that the particles inter-
act through repulsive forces following an inverse power law : that is,
any two particles apart from a distance r exert on each other a force
sproportional to 1=r . In this case [10],
? ¶\s¡5 z …
?(5) B(z;!)=jzj b(fi); ? = ; fi = ;! ; 0•fi• ;
s¡1 jzj 2
wherefiistheangleformedbyz and!,andbisanonnegativefunction,
defined implicitly, locally bounded, with one nonintegrable singularity
RH-SOLUTIONS 3
…for fi» ; more precisely,
2
s+1‡ ·¡… …s¡1
(6) b(fi)» C ¡fi as fi! :
2 2
This singularity reflects the importance, for such potentials, of the so-
0called “grazing collisions” (that is, the collisions such that v ’ v).
These are a source of great mathematical difficulties because the split-
ting (4) is impossible, and therefore most works on the subject ([1, 18],
etc.) make the assumption of Grad’s angular cut-off [10]
‡ ·…1b2L 0; :
2
With or without the cut-off assumption, the properties of the equa-
tion depend heavily on ?. It is customary to speak of hard poten-
tials for s > 5 (i.e. ? > 0), and soft potentials for 2 • s < 5 (i.e.
¡3 • ? < 0). The special cases, s = 5 and s=2, are called respec-
tively the Maxwellian and Coulomb potentials. Soft potentials often
appear to be much more difficult to study, as regards both the ho-
mogeneous theory and the near-equilibrium inhomogeneous setting [8].
However, this does not seem true for renormalized solutions [18], and
we shall come back to this.
The theory of the homogeneous Boltzmann equation for hard poten-
tials with cut-off is by now fairly complete. After the pioneering works
by Carleman [9] and Povzner [29], Arkeryd [1] proved existence and
uniqueness of a solution to the Cauchy problem under the assumption
of boundedness of some moments of the initial data, and the conver-
gence to equilibrium if the initial entropy is finite. These results have
been improved by several authors [19, 22, 14, 38]), and finally the re-
cent results [28] are almost optimal as regards the existence and the
uniqueness of solutions.
On the other hand, there is very little literature dealing either with
soft potentials, or with potentials without cut-off. The first significant
result in this field was given by Arkeryd [2], who proved existence of
weaksolutionsforpotentialswithoutcut-offand“nottoosoft”,thatis,
for? >¡1. Then Arkeryd [3] and Elmroth [19] studied these solutions
in the hard potential case.
We shall first show in this paper how Arkeryd’s argument can be
adapted to yield weak solutions for ¡2 • ? • ¡1. We mention that
this extension has also been treated independently by Goudon [21].
However, for ? < ¡2, this method does not work any more, and we
shall explain why. This will lead us to introduce a new concept of
weak solution, that can be handled for very soft potentials (essentially,4 C. VILLANI
¡4<? <¡2). WecallthemH-solutionsbecausetheyrelyuponafor-
mulation of the Boltzmann equation involving the entropy production.
The basic idea behind the definition is very simple : roughly speaking,
the finiteness of the entropy production yields a partial “regularity-
type” a priori bound on the tensor product function f(v)f(v ), which⁄
is enough to give a meaning to the collision operator even for very
singular cross-sections. Precise statements are given in section 3.
For a Coulomb potential (proportional to 1=r in dimension 3), the
Boltzmannequationbecomesinadequatebecausegrazingcollisionsbe-
come preponderant over all other collisions. In 1936, Landau [24] de-
rived from the Boltzmann equation another equation in which only
grazing collisions are taken into account :
‰ ? ¶ Z
@f @ @f @f⁄
(7) = dv a (v¡v ) f ¡f ;⁄ ij ⁄ ⁄
@t @v 3 @v @vi j ⁄j
where a (z) is some nonnegative symmetric matrix function whoseij
formwillbedetailedbelow, andweusetheconventionofimplicitsum-
mation over repeated indices. This equation is also called the Fokker-
Planck equation and has considerable importance in plasma physics.
The limiting process involved, called the “asymptotics of grazing
collisions”, wasstudiedbyDegondandLucquin[12], whileDesvillettes
[13] did an analogous work for potentials other than the Coulomb one
(the nature of the asymptotics is in fact quite different in those two
cases). These works were only concerned with formal results, proving
that the Boltzmann collision operator reduced to the Landau one for
a smooth enough density f, when all collisions become grazing.
According to [13], for a given inverse power potential, we have
?+2(8) a (z)=Λjzj Π (z)ij ij
?where Π(z) is the orthogonal projection on z ,
zzi j
(9) Π (z)=– ¡ ; z =0;ij ij 2jzj
andΛissomeconstantdependingontheparticularformoftheasympto-
tic process.
On the other hand, Arsen’ev and Buryak [4] have shown how a rig-
orous theorem could be obtained, proving that solutions of the Boltz-
mann equation converge towards solutions of the Landau equation,
when grazing collisions prevail. There, the limiting process was carried
for weak solutions, using essentially compactness tools developed by
Arkeryd [2]. However, this theorem was proven only under very strong
assumptions on the kernel (smooth, truncated forjzj near 0 and1).
6RH-SOLUTIONS 5
We shall show how to treat the asymptotics of grazing collisions
in a frame consistent with the existence theorems on the Boltzmann
equation without cut-off and much weaker assumptions than in [4].
Under very weak hypotheses we shall recover (8). We are also able to
treat the Coulomb case, thus giving a first rigorous basis to the work
of Degond and Lucquin.
We emphasize the importance of the study of these asymptotics, as
one of the basic justi&