SHARP ENTROPY DISSIPATION BOUNDS AND EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR

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SHARP ENTROPY DISSIPATION BOUNDS AND EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION G. TOSCANI AND C. VILLANI Abstract. We derive a new lower bound for the entropy dissipa- tion associated with the spatially homogeneous Boltzmann equa- tion. This bound is expressed in terms of the relative entropy with respect to the equilibrium, and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy, with an explicit rate. Our result gives a considerable refinement of the analogous estimate by Carlen and Carvalho [9, 10], under very little additional assumptions. Our proof takes advantage of the structure of Boltzmann's collision operator with respect to the tensor product, and its links with Fokker-Planck and Landau equa- tions. Several variants are discussed. Contents 1. Introduction 2 2. Preliminaries : Fokker-Planck and Landau equations 11 3. Symmetries for Boltzmann and Fokker-Planck equations 18 4. Integral representation of a lower bound for D 23 5. Main result 28 6. Extension to other kernels 35 7. The Kac model 37 8. Remarks about Fisher information and entropy dissipation 42 References 45 1

• kernels associated


• ?? ?


• another approach


• spatially homogeneous


• dv dv? d?


• boltzmann equa- tion


• constant kr


• kinetic equations


• boltzmann equation


• boltzmann's collision

Sujets

• Boltzmann equation

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 48
• Langue: English

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SHARP ENTROPY DISSIPATION BOUNDS AND
EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR
THE SPATIALLY HOMOGENEOUS BOLTZMANN
EQUATION
G. TOSCANI AND C. VILLANI
Abstract. We derive a new lower bound for the entropy dissipa-
tion associated with the spatially homogeneous Boltzmann equa-
tion. This bound is expressed in terms of the relative entropy with
respect to the equilibrium, and thus yields a differential inequality
which proves convergence towards equilibrium in relative entropy,
with an explicit rate. Our result gives a considerable refinement
of the analogous estimate by Carlen and Carvalho [9, 10], under
very little additional assumptions. Our proof takes advantage of
the structure of Boltzmann’s collision operator with respect to the
tensorproduct,anditslinkswithFokker-PlanckandLandauequa-
tions. Several variants are discussed.
Contents
1. Introduction 2
2. Preliminaries : Fokker-Planck and Landau equations 11
3. Symmetries for Boltzmann and Fokker-Planck 18
4. Integral representation of a lower bound for D 23
5. Main result 28
6. Extension to other kernels 35
7. The Kac model 37
8. Remarks about Fisher information and entropy dissipation 42
References 45
12 G. TOSCANI AND C. VILLANI
1. Introduction
This paper deals with the spatially homogeneous Boltzmann equa-
tion,
8
@f> N (t;v)=Q(f;f) t‚0; v2R (N ‚2);<
@t
(1)
>: 1 Nf(0;¢)=f ; f ‚0; f 2L (R ):0 0 0
The unknown f stands for the probability density of particles in the
velocity space. Q is the so-called Boltzmann collision operator,
Z Z ‡ ·
0 0(2) Q(f;f)= dv d!B(v¡v ;!) f f ¡ff⁄ ⁄ ⁄⁄
N N¡1S
N¡1 0 0where d! is the normalized measure on S , f = f(v), and so on,
and
8
0>v =v¡(v¡v ;!)!;⁄<
(3)
>: 0v =v ¡(v¡v ;!)!⁄ ⁄⁄
are the postcollisional velocities of two particles that collide with re-
spective velocities v and v , according to the laws of elastic collision⁄
8
0 0>v +v =v+v⁄< ⁄
(4)
>: 0 2 0 2 2 2jvj +jv j =jvj +jv j :⁄⁄
N(We denote by (a;b) = a¢b the scalar product in R .) On physical
grounds, it is assumed that the nonnegative kernel B(z;!) (the “cross
section”) depends only upon jzj and (z=jzj;!). Typical examples are
the (three-dimensional) hard spheres collision kernel
(5) B (z;!)=jz¢!j;HS
ormoregenerallythekernelsassociatedtotheso-calledhardpotentials
with cut-off,
?(6) B (z;!)=jzj b(fi)HP
where 0 < ? • 1, fi 2 [0;…] is the angle between z and !, and b 2
1L (0;…). For ? < 0, we speak of soft potentials with cut-off; for ? =0,
we speak of Maxwellian potential with cut-off. More generally, if B
depends only on (z=jzj;!), we speak of Maxwellian potential. We refer
to [42, 16] for a detailed discussion of other models.
TheBoltzmannequationisoneofthemostpopularmodelsinnonequi-
libriumstatisticalphysics. SoonafteritsintroductionbyMaxwell[33],
RSHARP ENTROPY DISSIPATION BOUNDS 3
BoltzmanndeducedfromitthecelebratedH-theorem,namelythatthe
entropy
Z
(7) H(f)= flogf
N
of any solution f to (1) is nonincreasing with time. From this fact
he gave plausible arguments for these solutions to converge towards a
definite equilibrium state as t goes to infinity (see [7] for instance). Let
us recall them briefly.
Let ’(v) be any function of the velocity variable. Multiplying (1) by
’ and integrating, we obtain
Z Z Z ‡ ·
0 0(8) Q(f;f)’= dvdv d!B f f ¡ff ’;⁄ ⁄⁄
2N N¡1S
where for simplicity we omit the arguments of B. For fixed !, the
transformation
0 0(9) T :(v;v )7¡!(v;v )! ⁄ ⁄
isinvolutiveandhasunitJacobian. Usingthischangeofvariables, and
also the transformation
(10) R :(v;v )7¡!(v ;v);⁄ ⁄
we easily obtain
Z Z ‡ ·‡ ·1 0 0 0 0(11) ¡ Q(f;f)’= dvdv d!B f f ¡ff ’ +’ ¡’¡’ :⁄ ⁄ ⁄⁄ ⁄
4
2Choosing ’ = 1;v (1 • i • N);jvj , by (4) we deduce that (11)i
Nvanishes, and hence that the quantities ‰ > 0 (mass), u2R (mean
velocity), T >0 (temperature) defined by
(12) Z Z Z
2 2‰= f(v)dv; ‰u= f(v)vdv; ‰(juj +NT)= f(v)jvj dv
are preserved with time. Now, choosing ’(v)=logf(v) in (11), we get
Z
(13) ¡ Q(f;f)logf =D(f)
where D(f) is the entropy dissipation,
Z ‡ · 0 01 f f0 0 ⁄(14) D(f)= dvdv d!B(v¡v ;!) f f ¡ff log :⁄ ⁄ ⁄⁄4 ff⁄
RR4 G. TOSCANI AND C. VILLANI
Since (x;y)7!(x¡y)log(x=y) is a nonnegative function, so is D(f).
Moreover, at least if B > 0 a.e., D(f) vanishes if and only if for almost
all v;v ;!,⁄
0 0(15) f f =ff :⁄⁄
Boltzmann proved that if f is smooth, the equality (15) implies (with
the notations of (12))
2jv¡uj‰ ¡ f
2T(16) f(v)= e ·M (v):
N=2(2…T)
Such distributions are called Maxwellian.
The assumption of smoothness was later proved to be unessential
(see [17] and the references therein; see also the proof of Perthame [36]
relying on Fourier transform). Lions also gave a direct proof [31]
1 N 1that (15) implies f 2C (R ) as soon as f 2L .
As a conclusion, if f(t) is a solution of (1), then H(f(t)) is strictly
fdecreasing with time unless f =M , and this strongly suggests that
f(17) f(t)¡¡!M
t!1
fin some sense. By the way, the fact that M is the only minimizer of
H intheclassoffunctionssatisfying (12)isadirectconsequenceofthe
identity
Z ‡ ·f f ff fH(f)¡H(M )= M 1¡ + log
f f f
N M M M
and the positivity of x7!1¡x+xlogx.
Once these formal arguments have been cast, it is very difficult to
go further and to prove that (17) actually holds, say in the sense of
1the topology induced by the L norm. Results have been obtained by
several authors, in particular Arkeryd [3] and Wennberg [49], for hard
(or Maxwellian) potentials with cut-off. More precisely, they prove
f ¡‚t(18) kf(t)¡M k•C(f )e0
pfor various weighted L norms (p‚1), with a constant C(f ) depend-0
ing on the initial datum. The rate ‚ is obtained by a compactness
argument, and therefore completely unknown. These results rely on
the study of the linearized Boltzmann equation and the rate is not
given by the entropy dissipation. But, as pointed out by Carlen and
Carvalho[9], suchquantitiesasthisrate ‚, thatarenotexplicitlycom-
putable, may be completely irrelevant from the physical point of view
(as is Poincar´e’s recurrence theorem for statistical physics).
Another approach is to try to transform the assertion
fD(f)=0()f =M
RSHARP ENTROPY DISSIPATION BOUNDS 5
into a quantitative result of the form
fD(f)‚d(f;M )
where d is some suitable metric on a space of functions which is sta-
1 Nble under the action of the Boltzmann equation (for instance, L (R ),2
1the space of all L functions with finite moments of order 2). Desvil-
lettes [19] was the first to obtain a result in this direction. His lower
bound reads
Z
8R >0 D(f)‚K inf dvjlogf(v)¡logM(v)jR
M2M jvj•R
where M is the set of all Maxwellian distributions, and f is assumed
to be bounded below by a fixed Maxwellian distribution. His proof
gives no indication on the way the constant K (obtained by the useR
of the open mapping theorem) depends on R. This result, proven for
a certain class of kernels, was extended by Wennberg [48] to cover the
physically realistic cases. Variants have also been derived for other
kinetic equations.
Another bound below was obtained by Gabetta and Toscani [24]
in the case of Kac’s model, which is a one-dimensional caricature of
Boltzmann’s equation (see section 7). Their bound reads
‡ ·
fD(f)‚? I(f)¡I(M )f
where ? is a constant depending on f in a complicated way, andf
Z Z
2 pjrfj 2(19) I(f)= =4 jr fj
f
istheso-calledFisherinformationoff. Again,theproofdoesnotyield
anyindicationontheway? varieswithf,andhencetheresultcannotf
be exploited for the trend towards equilibrium.
Someyearsago, Carlen andCarvalho[9] wereableto derivethe first
inequality of the form
¡ ¢
f(20) D(f)‚Φ H(fjM ) ;
where
f f(21) H(fjM )=H(f)¡H(M )
fis the relative entropy of f with respect to M , and Φ is a nonnegative
function, strictly increasing from 0 (but very slowly). Φ depends on
a few qualitative properties of f (essentially, the existence of a finite0
moment of order higher than 2). Though Φ is implicitly defined, its
construction is entirely explicit, and in particular for all ">0 one can6 G. TOSCANI AND C. VILLANI
compute · > 0 such that Φ(·)‚ ". The result holds for kernels that
are bounded below, in the sense
B(z;!)‚” >0:
In a second paper [10], Carlen and Carvalho adapted their analysis to
1the case of the hard s