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- into account
- constructive arguments fail
- boltzmann's argument
- time reversal
- specular reflection
- transport operator
- interaction between
- pre-collisional velocities

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EQUATIONS

C. VILLANI

We report here on a recent trend about the mathematical study of

convergence to equilibrium for kinetic equations involving collisions,

the archetype of which is the Boltzmann equation

@f

(1) +v¢r f =Q(f;f)x

@t

Z Z

0 0(2) Q(f;f)= dv d B (v¡v ;?)(f f ¡ff ):⁄ ⁄ ⁄⁄

N N¡1S

Along with explaining the notations in (1)–(2), we shall point out the

important features of this equation for our analysis.

The unknown in (1) is the density function of the gas, f(t;x;v)‚0.

Here t stands for the time variable, x for the position variable and v

for the velocity variable. We shall assume that x varies in a smooth,

Nconnected, bounded domain Ω of R , then v varies in the tangent

Nspace to Ω at x, which can be identiﬁed to R . At each time t, one

can think of f(t;¢;¢) as the probability distribution of the particles in

Nphase space Ω£R . The main distinctive feature of kinetic theory lies

inthekeepingtrackof thedependence of thedensity upon the velocity

variable.

There are two integro-diﬀerential operators appearing in (1). One

is the transport operator, v¢r . Its presence reﬂects the fact thatx

particleswouldtravelalongstraightlinesiftheydidnotinteract(New-

ton’s law), or, to be more precise, broken lines which end at, and start

from the boundary of Ω. The second one is the Boltzmann opera-

tor, Q, which models the eﬀect ofinteractions among particles. This

operator is quadratic, as a consequence of the modelling assumption

that the gas is dilute enough that only binary collisions count. It is

non-local with respect to the velocity variable, but local with respect

to the position variable, which reﬂects the modelling assumptions that

interactions between particles are localized in space (occur only if the

particles are very close to each other), but may involve particles with

very diﬀerent velocities.

1

R2 C. VILLANI

To (1) must be added some boundary conditions which take into

account some model for the interaction between the boundary and the

particles. This interaction is often very complex and poorly under-

stood; however, here we shall just consider the simplest case ofspecu-

lar reﬂection, i.e. particles bounce on the boundary according to the

Snell-Descartes laws. This can be translated as

(3) f(x;R v)=f(x;v) (x2@Ω)x

where R v = v¡ 2hv;n(x)in(x), n(x) standing for the normal unitx

vector to @Ω at point x.

Thepositionvariablexappearsonlyasaparameterin(2),sothatQ

isactuallyanoperatorwhichtoanyfunctionf(v)associatesafunction

Q(f;f)(v).

It is only in (2) that the details of the collisions are taken into ac-

0 0count. Here we have used the following standard notations : v;v⁄

are the pre-collisional velocities, v;v are the post-collisional veloci-⁄

0 0 0 0ties, and f, f , f , f respectively stand for f(v), f(v ), f(v), f(v ).⁄ ⁄⁄ ⁄

0 0Sincecollisionsareassumedtobeelastic, oneimposesv +v =v+v ,⁄⁄

0 2 0 2 2 2jvj +jv j =jvj +jv j . ThisleavesroomforN¡1parameters, which⁄⁄

N¡1aretakenintoaccountbythevariable? 2S . Ourparameterization

will be

8

v+v jv¡v j⁄ ⁄> 0v = + ?>< 2 2

(4)

>> v+v jv¡v j⁄ ⁄0:v = ¡ ?:⁄ 2 2

Underlyingthederivationof (2)istheimportantchaosassumption

which states, roughly speaking, that the velocities of particles which

are about to collide are not correlated; and the microreversibility

assumption, a version of which states that the microscopic laws of

collision are unchanged under time reversal.

Finally, the cross-section B(v¡ v ;?) depends on the details of⁄

the interaction between particles, and recipes for its computation have

been given by Maxwell at the very beginning of kinetic theory.

Among the main features of the Boltzmann equation, we insist on

the geometric input at the level of (4), the monotonicity feature which

will be induced by the nonnegativity of the cross-section, and the fact

that the collision operator only act on the velocity dependence, not on

the position dependence.

We stop here for this short presentation of the Boltzmann equation

and refer to [22] and the references therein for much more.ON THE TREND TO EQUILIBRIUM FOR KINETIC EQUATIONS 3

1. The H theorem and its implications

ThemostfamousdiscoveryofBoltzmanniscertainlytheH theorem.

Let

Z

(5) H(f)= flogfdxdv;

and let us investigate the behavior of H under evolution by the Boltz-

mann equation (we only consider classical solutions, possessing all the

integrability one can wish for). Since the transport operator is a ﬁrst-

order diﬀerential operator, one has

ﬂ Z

ﬂdﬂ(6) H(f)=¡ v¢r f(logf +1)=0:xﬂdt transport

On the other hand, from basic symmetry considerations one can prove

that

ﬂ Z Z

ﬂdﬂ(7) H(f)= Q(f;f)(logf +1)=¡ D(f(t;x;¢))dx;ﬂdt Ωcollisions

where the entropy dissipation functional D(f) is nonnegative and

deﬁned by

Z

0 0f f0 0 ⁄(8) D(f)= B(v¡v ;?)(f f ¡ff )log dvdv d :⁄ ⁄ ⁄⁄ ffN N N¡1 ⁄£ £S

Therefore, the H functional acts as a Lyapunov functional for the

Boltzmann equation, and this may suggest a universal behavior for its

solutions. WiththehelpoftheH theorem,letuslookforthestationary

states of (1). Of course, for f = f(x;v) to be a stationary state, it is

necessary that D(f)=0 for all x. Looking back to (8), this implies, if

B >0 a.e. (which is always the case), that

0 0(9) f(v)f(v )=f(v)f(v )⁄⁄

for (almost) all v;v ;? (the x variable is implicit). As was proven by⁄

Boltzmann in the case when f is smooth, equation (9) implies that f

must be a local Maxwellian, i.e. a function of the form

2

jv¡u(x)j

¡

2T(x)e

(10) f(x;v)=‰(x) :

N=2(2…T(x))

The basic idea which underlies Boltzmann’s argument is a conﬂict of

symmetries : by averaging (9) over ?, one ﬁnds that f(v)f(v ) is a⁄

2 2function of v + v and jvj +jv j . This implies that f is a local⁄ ⁄

Maxwellian, because thenrlogf(v)¡rlogf(v ) is always colinear to⁄

v¡v , whence rlogf is an aﬃne function. This kind of geometric⁄

characterization is reminiscent of the well-known characterization of

RR4 C. VILLANI

Gaussian distributions as the only functions g(x) (x 2 R) such that

2 2g(x)g(y) is a function of x +y , and can also be linked to the “com-

peting symmetries” studied by Carlen and Loss [6].

Ofcourse,notalllocalMaxwelliandistributionsarestationarystates.

Iff(x;v)isa localMaxwellianstate, then ofcourse Q(f;f)=0; there-

fore it is a stationary state if and only if

(11) v¢r f =0:x

Plugging (10) into (11), then identifying the powers of v, one arrives

at the system of equations

8

>rT =0>>> ? ¶>< u¢rTTru+ ru=¡2 Id(12) T>>>>r‰ NrT u¢ru>: ¡ + =0:

‰ 2 T T

The ﬁrst of these equations implies that T is a constant. The sec-

ond, together with the boundary condition u¢n = 0 (n stands for the

normal on the boundary) implies, if the domain Ω has no axis of

symmetry, that u· 0 (see for instance Grad [16]). Taking this into

account,thelastequationimpliesthat‰isalsoaconstant. Finally,if‰

and T are constants, and if u=0, then one can identify ‰jΩj with the

total mass of the gas, and (N=2)‰TjΩj with the total kinetic energy of

thegas,andsincethoseareconservedquantities,thisshowsthatthere

is only one possible stationary state, and it can be written as

2

jvj

¡

2Te

f (x;v)=‰1 N=2(2…T)

forsomeconstants‰;T > 0whichareuniquelydeterminedbythetotal

mass and total kinetic energy of the solution.

With the help of Boltzmann’s H theorem and the discussion above,

several theorems related to the trend of equilibrium have been proven.

Essentially, as soon as one is able to build well-behaved solutions, then

one can prove by standard tools the convergence to the unique

stationary state in large time : see in particular Desvillettes [11, 12],

Lions [17], and the references therein. Important variants consider theON THE TREND TO EQUILIBRIUM FOR KINETIC EQUATIONS 5

case where instead of specular reﬂection, one imposes bounce-back re-

ﬂection, or ﬁxed temperature on the walls (in which case the tempera-

tureoftheMaxwellianissolelydeterminedbytheboundarycondition),

see Cercignani [7, 8].

By the way, we should mention that at the time of writing, no one

knows how to construct well-behaved solutions in satisfactory general-

ity, andrigorousproofsoftrendtoequilibriumessentiallyconcernonly

the situation in which the gas is spatially homogeneous (no x depen-

dence), or the situation in which it is already very close to equilibrium

at initial time (see for instance [9, paragraph 9.8] or [18]). Of course

this is not a very satisfactory situation...

2. The entropy dissipation program

At the beginning of the nineties, a new direction of research was in-

troducedbyCarlenandCarvalho[5],meetingpreoccupationsofDesvil-

lettes[10]: IsitpossibletoquantifyBoltzmann’s H theoremin

such a way as to obtain eﬀective bounds on the convergence

toequilibrium? Themainmotivationofthisstudywastoreﬁnethe

nonexplicit results of convergence to equilibrium into quantitative esti-

mates, and hopefully show that trend to holds on realistic

scales of time. Another motivation was to keep as close as possible to

physics. Italsoturnedoutthatthisstrategywouldallowtocoversome

situations in which existing non-constructive arguments fail.

To fulﬁll this program in a meaningful way, one should only look for

completely quantitative estimates. In particular, in order to allow for

initial data which are not necessarily very close to equilibrium, one

cannot rest solely on linearization to predict a rate of con-

vergence. Indeed, the principle of linearization allows quantitative

estimates of convergence when the initial datum lies in a close neigh-

borhoodofequilibrium, butcannotpredictthetimewhichisnecessary

for the solution to enter such a neighborhood.

While really a lot remains to be done on the subject, we now have

a rather complete picture of how to fulﬁll this program, after several

contributions by Desvillettes, Toscani and the author [13, 20, 21, 14,

15]. We may add that this is part of a more general trend in PDE’s,

namely the study of convergence to equilibrium by means of functional

inequalities. We refer to [22, chapter 3] for some references about

related problems.6 C. VILLANI

3. Quantifying the effect of collisions

The ﬁrst step in the study is to establish a quantitative version of

Boltzmann’stheoremaboutthecasesofequalityforD(f)=0. Inother

Nwords, if f is a nonnegative function of v2R , we wish to estimate

from below the distance of f to the associated Maxwellian

distribution in terms of D(f). The best that one can hope for in

this business is an entropy-entropy dissipation inequality, i.e. a

functional inequality of the form

f(13) D(f)‚Φ(H(f)¡H(M ));

fwhere M stands for the Maxwellian distribution with the same ﬁrst

moments as f,

2

jv¡uj

¡

2TefM =‰ ;

N=2(2…T)

Z Z Z

2‰= fdv; ‰u = fvdv; N‰T = fjv¡uj dv:

f fIn the sequel, we write H(fjM ) = H(f)¡ H(M ) and call it the

relative entropy.

Ofcourse,thepossiblevalidityofaninequalitysuchas(13)crucially

depends on the cross-section B. By monotonicity, it suﬃces to treat

the case in which B is “small”... For the moment, let us assume that

it is constant, say normalized to 1.

In (13) the function Φ should be explicit with Φ(d) > 0 for d>0.

fIdeally, itwouldbealinearfunction : D(f)‚‚H(fjM ). Itcouldde-

pend on f via some a priori estimates (moments, smoothness...) which

would in principle be established separately.

In the case of diﬀusive equations, inequalities of the form (entropy

dissipation)‚const. (relativeentropy)arecalledlogarithmicSobolev

inequalities,andhavebeentheobjectofalotofstudies(seeinpartic-

ular [1]). In the context of the Boltzmann equation, such an inequality

was conjectured by Cercignani at the beginning of the eighties. But

counterexamples by Bobylev [3], Wennberg [23], Bobylev and Cercig-

nani [4] have shown that such an inequality is false, even if one re-

strictsthemtoclassesoffunctionswithalotofﬁnitemoments, alotof

smoothness, bounded from below by a small Maxwellian distribution...

This problematic is tightly linked to an important variant of Kac’s

problem, which is get uniform estimates on the speed of approach to

equilibrium for some large particle systems. This is explained in the

contribution by E. Janvresse in this issue.

Carlen and Carvalho were the ﬁrst ones to establish an inequality

suchas(13). IntheirresultthefunctionΦonlydependedupon(weak)ON THE TREND TO EQUILIBRIUM FOR KINETIC EQUATIONS 7

smoothness estimates and some moments of order s > 2. They used

this result to obtain the very ﬁrst result of convergence to equilibrium

for the spatially homogeneous Boltzmann equation with explicit rate.

Onthisoccasion,theyintroducednewtoolsintheﬁeld: theapparatus

of logarithmic Sobolev inequalities and that of information theory.

Severalyearslater,theCarlen-Carvalhoresultwasnotablyimproved

by Toscani and the author. The improvement concerned both the con-

stants and the shape of the function Φ, which was increasing very

slowly from 0 in the Carlen-Carvalho result. In [20] we have shown the

following estimate. We use the classical notation

Z

2 s=2kfk 1 = f(v)(1+jvj ) dvLs

N

and its natural extension

Z

2 s=2kfk 1 = f(v)log(1+f(v))(1+jvj ) dv:L logLs

N

Theorem. Let B ‚ 1 be a cross-section, and D be the associated

entropy dissipation functional, eq. (8). Let f be a probability density

NonR with unit temperature, and let M be the associated Maxwellian

equilibrium. Let " > 0 be arbitrary, and assume that for some – > 0,

A;K >0,

1 1kfk ;kfk <+1;L L logL

4+2="+– 2+2="+–

2¡Ajvj(14) f(v)‚Ke :

Then, there exists a positive constant C (f), depending only on N, ","

–,kfk 1 ,kfk 1 , A and K, such thatL L logL

4+2="+– 2+2="+–

1+"(15) D(f)‚C (f)H(fjM) :"

As an example (choosing – =1), the following more explicit constant

works :

¡" 1+"(16) D(f)‚KT F H(fjM) ;f "

where K is an absolute constant (not depending on f), T is the “min-f

imal directional temperature” given by

Z

2T =(N¡1) inf f(v)(v¢e) dv;f

N¡1e2S N

and ? ¶

1

2

1F = log +A kfk 1 kfk :" L logLL 3+2="5+2="K

RRR8 C. VILLANI

In short : even if Cercignani’s conjecture is false, and the inequality

D(f) ‚ const:H(fjM) does not holds, one has the slightly weaker

1+"replacement D(f)‚const:H(fjM) for arbitrarily small ">0.

Let us say just a few words about the proof. The argument is some-

what complicated, but it is the only one available at the moment...

It essentially consists in a quantitative version of Boltzmann’s origi-

nal argument, and at the same time rests on the semigroup ideas by

Stam [19], re-introduced in the theory of Logarithmic Sobolev Inequal-

ities by Bakry and Emery [2]. After normalization to reduce to the

case ‰ = 1, u = 0, T = 1, use Jensen’s inequality and the convexity of

(x;y)7!(x¡y)(logx¡logy) to write

? ¶Z Z

1 ff⁄0 0 RD(f)‚D(f)· dvdv ff ¡¡d f f log :⁄ ⁄ ⁄ 0 04 2N ¡d f f⁄

ThenintroducetheFokker-Plancksemigroup(adjointOrnstein-Uhlenbeck)

(S f) deﬁned by the PDEs s‚0

@f

=Δf +r¢(fv):

@s

After a study of the behavior of symmetries under this semigroup, one

proves that

ﬂ ﬂZ 2ﬂ ﬂd 1 r (S F) r (S G)v;v s v;v s⁄ ⁄ﬂ ﬂ¡ D(S f)= (S F +S )G ¡ dvdv ;s s s ⁄ﬂ ﬂds 4 2N S F S Gs s

R

0 0with the notations F = ff , G = ¡f f , and r standing for the⁄ v;v⁄ ⁄

N Ngradient operator in R £R . Here comes the key point : Since G

2 2only depends on v+v andjvj +jv j , one easily checks that the same⁄ ⁄

is true for S G, and that as a consequence,r(S G) lies (pointwise)s s

in the kernel of the linear operator

2N NP :[A;B]2R 7¡!Π(v¡v )[A¡B]2R ;⁄

p

?where Π(z) is the orthogonal projection upon z . SincekPk = 2 as

a linear operator, it follows that

ﬂ • ‚ﬂZ 2ﬂ ﬂd 1 rS f (rS f)s s ⁄ﬂ ﬂ¡ D(S f)‚ (S f)(S f) Π(v¡v ) ¡ dvdv :s s s ⁄ ⁄ ⁄ﬂ ﬂds 8 S f (S f)2N s s ⁄

2Up to a factor jv¡v j in the integrand, which only matters at large⁄

velocities, the right-hand side coincides with the Landau entropy

dissipation functional for Maxwellian molecules (the Landau

equationistheusualreplacementoftheBoltzmannequationinplasma

physics). It was shown in [13] that the Landau entropy dissipation

functional is bounded below by a constant times ¡(d=ds)H(S fjM).s

RRRON THE TREND TO EQUILIBRIUM FOR KINETIC EQUATIONS 9

This shows that, up to error terms which are important only at large

velocities,

? ¶

d d

(17) ¡ D(S f)‚const: ¡ H(S fjM) :s s

ds ds

After integrating this with respect to s from 0 to +1, one recovers

the desired inequality. It is in the treatment of the error terms that

the exponent 1 is changed into an exponent 1+". See [20] for details,

or [22] for a more complete review.

Once this theorem is established for the constant cross-section, vari-

ants can be established to cover more general situations.

† “Soft potentials” : Assume now that

¡ﬂB(v¡v ;?)‚(1+jv¡v j) ; ﬂ > 0:⁄ ⁄

Then, for all ">0, eq. (16) still holds with

? ¶

1 2F = log +A kfk kfk 1 :1" L logLL 3+(2+ﬂ)="5+(2+ﬂ)="K

† “Hard potentials” : Assume now that

?B(v¡v ;?)‚jv¡v j ; ? > 0:⁄ ⁄

pAssume moreover that f 2 L , for some p > 1, and • large•

enough. Then, there exists ﬁ > 1, C > 0, depending on N, ?,

pp, •,kfk , and on A;K in (14), such thatL•

ﬁD(f)‚CH(fjM) :

Suchinequalitiesallowonetoelegantlysolvetheproblemoftrendto

equilibrium for the spatially homogeneous Boltzmann equation, under

various assumptions on the cross-section. In particular, in the case

?where the cross-section decays at inﬁnity, say B = (1+jvj) for 0 >

¡•? >¡2, this method can be applied to prove convergence like O(t )

for any •>0, while standard compactness arguments fail to deal with

this situation.

But the main interest of these inequalities is that they can be ap-

plied to the much more interesting study of the spatially inhomoge-

neous Boltzmann equation. We shall explain this in the next section.

To conclude the present paragraph, we should mention (i) that these

problemsoflowerboundsarequitedelicatebecausevery,veryfewgen-

eral methods have been identiﬁed for their study; (ii) that the problem10 C. VILLANI

addressed above is related to the Kac spectral gap problem and its

variants, as described by E. Janvresse in the present volume.

4. When fluid dynamics comes into play, or : the role of

the x variable

The role of the position variable in the trend to equilibrium for the

Boltzmann equation is subtle. On one hand, the collision mechanism

whichisatthebasisofthedecreaseoftheH functionaldoesnotacton

the x variable. On the other hand, the transport operator plays a key

role in preventing the system fromfalling into a local equilibrium,

i.e. a local Maxwellian which would not necessarily be the unique

stationary state. It is the combined eﬀect of transport and collisions

which makes the system converge to the stationary state.

To quantify this, the author, together with L. Desvillettes, has sug-

gestedthefollowingapproach. Oneshouldtrytoestablishasystemof

diﬀerential inequalities involving the relative entropy with respect

to the stationary state,

Z

f

H(fjf )= flog ;1

f1

and at the same time the relative entropy with respect to the local

equilibrium state, Z

ffH(fjM )= flog :

fM

In general, one can expect this system to be of second-order. Entropy-

entropy dissipation inequalities will typically result in an inequality

like

d

f ﬁ(18) ¡ H(fjf )‚K H(fjM ) ;1 1

dt

withsomeexponentﬁ‚1. Thenonehastoﬁghttoproveaninequality

like

2d f f ﬂ(19) H(fjM )‚K H(fjf )¡CH(fjM ) ;2 12dt

with some exponent ﬂ • 1, and the constants K , K , C may depend1 2

on estimates about f.

Theconjunctionof (18)and(19), togetherwiththenonnegativityof

fH(fjM ), H(fjf ) implies, by a hand-made theorem, that H(fjf )1 1

converges to 0 with an explicit rate. If ﬁ and ﬂ are very close to 1,

¡•then the rate is O(t ) for some very large exponent •.

This strategy was implemented in [14] for a simpler model, the spa-

tially inhomogeneous Fokker-Planck equation with conﬁning potential.