CERCIGNANI'S CONJECTURE IS SOMETIMES TRUE AND ALWAYS ALMOST TRUE

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Niveau: Supérieur, Doctorat, Bac+8
CERCIGNANI'S CONJECTURE IS SOMETIMES TRUE AND ALWAYS ALMOST TRUE CEDRIC VILLANI Abstract. We establish several new functional inequalities comparing Boltz- mann's entropy production functional with the relative H functional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at infinity. The proof rests on the method introduced in [39] combined with a novel use of the Blachman- Stam inequality. If the superquadraticity assumption is not satisfied, then it is known that Cercignani's conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from [39]. Finally, we consider the entropy-entropy production version of Kac's spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The first two results are sharp in some sense, and the third one is likely to be, too; they contain all previously known entropy estimates as particular cases. This gives a first coherent picture of the study of entropy production, according to a program started by Carlen and Carvalho [12] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation. Contents 1. Introduction 2 2. Superquadratic collision kernels 10 3. Nonvanishing collision kernels 17 4. General collision kernels 20 5. Further developments and open problems 30 6.

• boltzmann's entropy


• almost everywhere positive


• principle has


• kac's spectral


• collision kernels


• boltzmann equation


• functional inequalities comparing


• gap problem


• production functional

Sujets

• Boltzmann equation

Informations

• Publié par: mijec
• Nombre de lectures: 9
• Langue: English

Extrait

CERCIGNANI’S CONJECTURE IS SOMETIMES AND ALWAYS ALMOST TRUE

C EDRIC  VILLANI

TRUE

Abstract.We establish several new functional inequalities comparing Boltz- mann’s entropy production functional with the relativeHfunctional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at innit y. The proof rests on the method introduced in [39] combined with a novel use of the Blachman- Stam inequality. If the superquadraticity assumption is not satised, then it is known that Cercignani’s conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from [39]. Finally, we consider the entropy-entropy production version of Kac’s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The rst two results are sharp in some sense, and the third one is likely to be, too; they contain all previously knownentropyestimatesasparticularcases.Thisgivesarstcoherentpictureof the study of entropy production, according to a program started by Carlen and Carvalho [12] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation.

Contents

1. Introduction 2. Superquadratic collision kernels 3. Nonvanishing collision kernels 4. General collision kernels 5. Further developments and open problems 6. The entropy variant of Kac’s problem References

Date: January 31, 2005.

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C EDRIC  VILLANI

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Cercignani’s conjecture [19] asserts the domination of Boltzmann’s relativeH functional by a constant multiple of Boltzmann’s entropy production functional. Since its formulation twenty years ago, it has been disproved in greater and greater generality [8, 10, 47]. Nevertheless, we shall show in this paper that it is true in certain cases. In fact, if Grad’s angular cut-o is imposed, then it holds true in essentially all the cases which were not previously covered by counterexamples. We shall also show that a slightly weaker family of inequalities holds true in all physical cases, thus recovering and improving previous results in this direction [12, 13, 39]. These functional inequalities play a key role in our subsequent treatment of trend to equilibrium for the Boltzmann equation, both in a spatially homogeneous [34] and in a spatially inhomogeneous context [25]. We will not develop these issues here, in order to limit the size of the present paper; the reader who would like to consult a tentative global view on the subject is referred to [42, chapter C], or, better, to [46].

Beforegoingfurther,letusgiveprecisedenitionsofallthequantitieswhichwill be under study. Wheneverf=f(v) is a nonnegative integrable function onR N (N2), to be thought of as adensity in velocity spacenee,wed

1) the macroscopicdensity,velocityandtemperatureassociated tof, by the identities (1)= Z R N f(v)dv;u=  1 Z R N f(v)v dv;T= N 1  Z R N f(v)|v u| 2 dv;

2) theHfunctional, or negative of the entropy, by

(2)

H(f) = Z RN flogf;

3) thethermodynamical equilibrium, or maximum of the entropy under the constraints (1),

(3)

M f (v) =M ,u,T

(v)

|v u| 2  e   2T (2 T) N/2 ;

CERCIGNANI’S CONJECTURE IS SOMETIMES TRUE

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4) theH-dissipation, orentropy production 1 : (4)D(f=) 4 1 Z R N R N S N 1 (f 0 f 0   f f  ) lo f 0 f 0 B(v v  , )d  dv dv . g f f  HereS N 1 stands for the unit sphere inR N ; we have used the shorthandsf 0 =f(v 0 ), f  =f(v  ),f  0 =f(v 0 ), where (5)v 0 = v 2 +v  + |v  2 v  |  v ,  0 = v+ 2 v   |v  2 v  | (∈S N 1 ). Finally, the functionBappearing in (4) is Boltzmann’scollision kernel 2 .

Boltzmann’s entropy production functional describes the amount of entropy which is produced per unit of time by the collisions of particles in a dilute gas, at a given position in space. These collisions are assumed to be elastic and binary; one may think ofv 0 ,v 0 as the respective velocities of particles which are just about to  collide, and will have respective velocitiesv,v   Asas a result of this interaction. for Boltzmann’s collision kernel, it depends on the particular interaction between the particles, but it is always assumed to depend only on the two parameters|v v  | (modulus of the relative velocity) and cos=hk, i(cosine of the deviation angle), wherek= (v v  )/|v v  | examples are the hard-sphere models, in which. Typical B=|v v  |, and the inverse-power model, in whichB=|v v  | b(cos) for some exponent ∈Rand some complicated functionb, which is only known implicitly. Much more details can be found in [42] or in the classical references [20, 21, 22, 41]. Boltzmann’sHtheoremdeiesintDwith the entropy production and asserts that (i)D(f)0, (ii) if the collision kernelBis almost everywhere positive, then (6)D(f) = 0⇐⇒f=M f .

In other words, the entropy production is nonzero if the distribution function is not in thermodynamical equilibrium. This theorem is at the heart of most studies of the hydrodynamical approximation, or the long-time behavior of solutions of the Boltzmann equation. This gives strong motivation for establishingquantitative versions of this theorem. Accordingly, the following question has been studied by many authors (e.g. [24, 12, 13, 39]):

1 my own sign convention for theWith respect to previous papers, I have decided to change entropy (unphysical, albeit common in kinetic theory), and accept the idea that it should be nondecreasing with time. The in uence of Sasha Bobylev in this decision is acknowledged. 2 The kernelBis often improperly called the cross-section. Strictly speaking, the cross-section would rather beB(v v  , )/|v v  |.

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C EDRIC  VILLANI

Can one establish a lower bound on the entropy production, in terms of how much the distribution function departs from thermodynamical equilibrium ?

Thisquestionisveryrepresentativeofrecenttrendsinpartialdierentialequa- tions:inmanycasesofinterest,anentropyprinciplehasbeenidentied,anda detailed study of the entropy production principle provides a sharp insight into the problem of asymptotic behavior of the equation under consideration [4, 17, 18, 23, 35]. In the case of the Boltzmann equation, this problem is interesting not only for the sake of developing the theory of this particular equation, but also because it presents a number of interesting mathematical features, some of which are typical of kinetic equations and some of which are more specic. In our context, a natural way to measure the departure offtowards thermody- namical equilibrium is by means of therelativeHfunctional, (7)H(f|M flog. f f ) =H(f) H(M f ) = Z R N M f f This is nothing but the relative Kullback information offwith respect toM. After these preparations, we can state Cercignani’s conjecture: it consists in the validity of the functional inequality D(f)K(f)H(f|M f ),

(8)

whereK(fbe a positive constant depending on) would fonly via certain a priori estimates,suchassmoothness,decayatinnityorstrictpositivity. For the sequel of the discussion, let us introduce the functional norms which we shall use in our estimates. We dene the functional spacesL s 1 (weigthedL 1 ),L s 1 logL (weighted Orlicz space) andH k (Sobolev space) by the identities kfk L 1 s = Z R N f(v)(1 +|v| 2 ) s/2 dv, =f(v)(1 +|v| 2 ) s/2  1 +|logf(v)|  dv, kfk L 1 s logL Z R N and

kfk H k =   || X k Z R N |D  f(dv   2 v)| 2 , multi-index of length||, andD  f=∂  1 1 . . . ∂ N  N f course. Of have assumedf0.

in whichstands for a in these denitions we