Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus


14 pages
ON THE TREND TO EQUILIBRIUM FOR KINETIC 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 (1) ∂f∂t + v · ?xf = Q(f, f) (2) Q(f, f) = ∫ R N dv? ∫ SN?1 d? B(v ? v?, ?)(f ?f ?? ? ff?). 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, connected, bounded domain ? of RN , then v varies in the tangent space to ? at x, which can be identified to RN . At each time t, one can think of f(t, ·, ·) as the probability distribution of the particles in phase space ??RN . The main distinctive feature of kinetic theory lies in the keeping track of the dependence of the density upon the velocity variable. There are two integro-differential operators appearing in (1).

  • into account

  • constructive arguments fail

  • boltzmann's argument

  • time reversal

  • specular reflection

  • transport operator

  • interaction between

  • pre-collisional velocities

Voir plus Voir moins

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
(1) +v¢r f =Q(f;f)x
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
There are two integro-differential operators appearing in (1). One
is the transport operator, v¢r . Its presence reflects the fact thatx
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.
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.
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
v+v jv¡v j⁄ ⁄> 0v = + ?>< 2 2
>> 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.
(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
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
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
(10) f(x;v)=‰(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
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].
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
>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
is only one possible stationary state, and it can be written as
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-
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-
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,
2TefM =‰ ;
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.
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
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.
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
2 s=2kfk 1 = f(v)(1+jvj ) dvLs
and its natural extension
2 s=2kfk 1 = f(v)log(1+f(v))(1+jvj ) dv:L logLs
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
2T =(N¡1) inf f(v)(v¢e) dv;f
N¡1e2S N
and ? ¶
1F = log +A kfk 1 kfk :" L logLL 3+2="5+2="K
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⁄
(S f) defined by the PDEs s‚0
=Δf +r¢(fv):
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
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 ;⁄
?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
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
This shows that, up to error terms which are important only at large
? ¶
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) :
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
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,
H(fjf )= flog ;1
and at the same time the relative entropy with respect to the local
equilibrium state, Z
ffH(fjM )= flog :
In general, one can expect this system to be of second-order. Entropy-
entropy dissipation inequalities will typically result in an inequality
f fi(18) ¡ H(fjf )‚K H(fjM ) ;1 1
withsomeexponentfi‚1. Thenonehastofighttoproveaninequality
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.