On the trend to global equilibrium in spatially
inhomogeneous entropy-dissipating systems :
The linear Fokker-Planck equation
Abstract
We study the long-time behavior of kinetic equations in which transport and spatial
confinement (in an exterior potential, or in a box) are associated with a (degenerate)
collision operator, acting only in the velocity variable. We expose a general method,
based on logarithmic Sobolev inequalities and the entropy, to overcome the well-known
problem, due to the degeneracy in the position variable, of the existence of infinitely
many local equilibria. This method requires that the solution is somewhat smooth.
In this paper, we apply it to the linear Fokker-Planck equation, and prove decay to
equilibrium faster than O(t−1/ε ), for all ε > 0.
Contents
1 Introduction and main result
1
2 Heuristics and strategy
9
3 Second derivative of the relative entropy
14
4 Nonlinear interpolations
20
5 Uniform in time hypoellipticity estimates
28
6 Study of a system of differential inequalities
33
7 Conclusion and remarks
38
Appendix: An interpolation lemma
41
1 Introduction and main result
This is the first of two related papers dealing with the long-time behavior
of dissipative kinetic equations of the form
(1.1)
∂f
+ v · ∇x f − F (x) · ∇v f = Q(f ).
∂t
Communications on Pure and Applied Mathematics, Vol. 000, 0001–0045 (199X)
c 199X John Wiley & Sons, Inc.
°
CCC 0010–3640/98/000001-45
2
L. DESVILLETTES AND C. VILLANI
Here the variables t ≥ 0, x ∈ RN and v ∈ RN respectively stand for
time, position and velocity, and the unknown f (t, x, v) ≥ 0 stands for the
density of particles in phase space. Without loss of generality, we shall
N
always assume that f (t, ·, ·) is a probability distribution on RN
x × Rv .
The advection operator v · ∇x describes transport, while Q is a collision
operator, which depends on the type of interaction : the most famous
example may be Boltzmann’s quadratic collision operator [9]. Finally,
N
F : RN
x → R in (1.1) is a macroscopic force, that for simplicity we shall
assume independent of f (no coupling is assumed in this article; coupled
equations will be discussed in [17]).
Following standard physical considerations, we shall always be concerned with collision operators which
- are mass-preserving;
- act only on the velocity dependence : this reflects the physical fact
that collisions are (modelled as) localized in space;
- are dissipative, in the loose sense that collisions tend to make a certain
entropy functional decrease, and this entropy achieves its minimum value
for some subfamily of Gaussian distributions.
Rather than try to give general conditions, we shall focus on a small
number of well-known models.
Due to the dissipation property, the system tends to approach local
equilibrium, that is, at each point x the distribution of velocities tends to
a state which minimizes the entropy functional. At this level of generality,
such a statement is in fact false ! because the advection operator may
prevent collisions from acting efficiently. In the case when F = 0, this
is actually what happens for the Boltzmann equation, and counterexamples [34, 27] show that in general there is no trend to local equilibrium.
Moreover, the whole mass of the gas goes to infinity, and the algebraic decay of various norms of f as t → +∞ was investigated by several authors
(see [13] and the references therein).
On the other hand, we can impose a confinement, that is, for instance,
put F (x) = −∇V (x), where V is, say, strictly convex at infinity. Or,
which is almost a particular case of the latter, we can consider a system
which is enclosed in a box with appropriate boundary conditions. Then,
there is usually (except if some extra symmetry is present in the system) a
unique global steady state, and it is a basic problem to prove (or disprove !)
that solutions to (1.1) converge towards this equilibrium as t → +∞, and
estimate the time it takes to do so.
For the Boltzmann equation for instance, this question has been exten-
SPATIALLY INHOMOGENOUS SYSTEMS
3
sively studied by linearization and tools of spectral theory (see [9] and [13]
for some references). Though these results usually give very satisfactory
answers for perturbation problems, they are now known to adapt rather
badly to the fully nonlinear, inhomogeneous and far-from-equilibrium,
case. There are two main reasons for this :
- they are often naturally set in functional spaces that are much narrower than those adapted to the fully nonlinear study;
- (maybe more importantly) they begin to be efficient when the solution lies in a neighborhood of the equilibrium that is sufficiently small
so that the linearized regime prevails. But they give no explicit clue on
the time necessary for the solution to reach such a neighborhood of the
equilibrium – and sometimes also the size of this neighborhood is not
well-known.
Thus, in our opinion, these methods, that are well suited to a study of
the close-to-equilibrium regime, must be complemented by other methods
which apply in a far-from-equilibrium setting, and control the evolution
of the system until its distance to equilibrium becomes very small. Once
this is done, it is possible to apply efficient linearized tools, as developed
for instance in [2, 39].
Also, it is natural to look for a method which relies as much as possible
on the key physical mechanism, that is the dissipation of entropy. Such
entropy-dissipation based methods have recently received much attention
in the case of spatially homogeneous equations, that is, when f does not
depend on x, and thus only collisions act. Starting at least from the work
of Bakry and Emery fifteen years ago [4], it was understood that explicit
estimates of relaxation to equilibrium could be obtained by the direct
study of the dissipation of entropy, with the help of so-called logarithmic
Sobolev inequalities [32, 19, 4, 20, 33]. The typical example is given by the
spatially homogeneous linear Fokker-Planck equation (whose associated
semigroup is also called the adjoint Ornstein-Uhlenbeck process)
(1.2)
∂f
= ∇v · (∇v f + f v),
∂t
where ∇v · stands for the divergence operator in the v variable. The only
equilibrium is the centered gaussian (or Maxwellian) of temperature 1,
|v|2
(1.3)
e− 2
M (v) =
,
(2π)N/2
and the entropy functional, or free energy, is given by the Kullback relative
4
L. DESVILLETTES AND C. VILLANI
entropy of f with respect to M :
Z
(1.4)
H(f |M ) =
RN
f log
f
dv.
M
The rate of dissipation of H(f |M ) along equation (1.2) is given by the
so-called relative Fisher information of f with respect to M (also known
in the community of particle systems as the “Dirichlet form”),
Z
(1.5)
I(f |M ) =
RN
¯
¯
¯
f ¯¯2
¯
f ¯∇ log ¯ dv.
M
In this context, the basic logarithmic Sobolev inequality of Stam [32] and
Gross [19] reads
(1.6)
1
H(f |M ) ≤ I(f |M ),
2
and thus we get for H(t) ≡ H(f (t, ·)|M ) the differential inequality −Ḣ ≥
2H, whence we obtain exponential decay for H like e−2t . By the CsiszárKullback-Pinsker inequality [11, 24, 30], this readily implies
q
kf (t) − M kL1 ≤
2H(f (0)|M )e−t ,
which is a very satisfying result (optimal as far as the rate in the exponential function is concerned, since the first nonzero eigenvalue of the
Fokker-Planck operator, acting on a suitable L2 -space, is −1).
The same proof, only with different constants, holds when equation (1.2)
is replaced by
¡
¢
∂f
= ∇v · ∇v f + f ∇W (v) ,
∂t
and M (v) is replaced
by e−W (v) , as soon as W is strictly convex at infinity
R −W (v)
and satisfies e
dv = 1. See [1] for an exhaustive study, other
generalizations and a list of references.
Entropy dissipation methods were successfully adapted to various situations with nonlinear collision operators, starting from the pioneering
works of Carlen and Carvalho [7, 8] on Boltzmann’s operator. Since
then, much progress has been done, and recently the papers [16], [35]
proved convergence like O(e−λt ) and O(t−∞ ), respectively for the (spatially homogeneous) Landau and Boltzmann equations, under quite realistic assumptions on the initial data and for so-called over-Maxwellian
cross-sections. Algebraic decay with a fixed, explicit rate is also proven
for a very broad class of cross-sections, see [16, 35, 36].
SPATIALLY INHOMOGENOUS SYSTEMS
5
On the contrary, many previous attempts to tackle the spatially inhomogeneous case with those methods did not succeed. This may seem
surprising, because, after all, the entropy dissipation is just the same in
both cases ! The main results of [16, 35] are quantitative versions of the
“H-theorem”, that is functional inequalities of the form
(1.7)
D(f ) ≥ C(f )H(f |M )α ,
where D stands for the entropy dissipation, and C(f ) depends only on
some size estimates on f (by this we mean moments, Lp norms, lower
bounds, ...), which can often be proven to hold independently of the
problem of trend to equilibrium (see [15] for instance). Just as the logarithmic Sobolev inequality, these functional inequalities may be applied
to solutions of inhomogeneous problems. But since the collision operator
acts only on the velocity variable, this only tells us about the relaxation
to local equilibrium : by this we mean distributions that are in equilibrium
in the velocity variable, but whose macroscopic parameters depend on x
arbitrarily. In fact, for any such distribution, the dissipation of entropy is
equal to 0, and no information can be drawn from the H-theorem. In particular, the strong version of the H-theorem does not prevent the entropy
to form a plateau as time goes on. This difficulty has been known for a
long time (even to Boltzmann ! as pointed out to us by C. Cercignani),
and is discussed with particular attention by Grad [18], Truesdell [37, p.
166–172] and, in the different but related context of hydrodynamic limits
for particle systems, by Olla and Varadhan [29].
Using compactness tools, it is often possible to prove convergence to
global equilibrium, as soon as one can define meaningful solutions and
has sufficient estimates on them (see for instance [12, 5, 3, 26]); but this
method gives no information on the rate of convergence.
In [18], Grad was able to prove convergence to global equilibrium for
the linear Boltzmann equation, using smoothness properties (proved or
assumed, depending on the cases) without looking for a rate. More than
his method, his discussion of the matter – that we encountered only after
the basis for this work was laid down – is very enlightening. We shall
exploit and develop some of his ideas in further research [17].
Our aim here is precisely to show a way to overcome the aforementioned difficulty of the existence of infinitely many local equilibria, using
logarithmic Sobolev inequalities both in velocity and in position space, and
a strategy which is well-suited to nonlinear problems. We shall illustrate
6
L. DESVILLETTES AND C. VILLANI
our approach in this first paper on the linear Fokker-Planck equation,
(1.8)
∂f
+ v · ∇x f − ∇V (x) · ∇v f = ∇v · (∇f + f v),
∂t
where x → V (x) is a smooth potential, strictly convex at infinity. The
steady state is
|v|2
(1.9)
e− 2
f∞ (x, v) = e−V (x) M (v) = e−V (x)
.
(2π)N/2
Though the essential difficulty is already present in this model, it has
at least three advantages on more elaborate equations:
- it leads to (relatively !) simpler computations;
- the local equilibrium associated to a function
f (x, v) depends only
R
on one “macroscopic function”, namely ρ(x) = f (x, v) dv;
- We are able to prove for this equation all the needed smoothness
estimates, using only upper and lower bounds (without restriction of size)
of Gaussian type for the initial distribution function. Establishing these
a priori estimates will in fact be the only place where the linearity of the
equation is really exploited.
More complicated models will be the object of study in the forthcoming work [17] : in particular, the Boltzmann equation, under various
confinement assumptions. There, we shall leave the question of the size
estimates as a (formidable) open problem. As a matter of fact, in the
general case, since even global conservation of energy is still not known
to hold for renormalized solutions of the Boltzmann equation, it is absolutely hopeless to perform any study of this kind. Yet, in some particular
regimes, very smooth solutions have been constructed [28, 6], and maybe
we are not far from our goal in these situations. Anyway, this is, in our
opinion (as it was in Grad’s) a completely distinct problem.
The situation that we study in this paper is the following. We consider
a smooth potential x 7→ V (x), which is strictly convex at infinity, in the
sense that x 7→ D2 V (x) is uniformly bounded from below by a positive
multiple of the identity matrix as |x| → +∞. This is a simple condition
to ensure that the measure e−V satisfies a logarithmic Sobolev inequality.
In fact, for technical reasons (see the discussion in section 7), we shall
assume that V behaves exactly like a quadratic potential at infinity, that
is
(1.10)
V (x) = ω02
|x|2
+ Φ(x) + V0 ,
2
SPATIALLY INHOMOGENOUS SYSTEMS
7
where ω0 > 0 and Φ goes to 0 at infinity, in the sense that
(1.11)
Φ ∈ H ∞ (RN ) =
\
H k (RN ).
k≥0
No restriction of size is assumed, and V is not necessarily convex. Moreover, without loss of generality V0 is chosen in such a way that
Z
(1.12)
RN
e−V (x) dx = 1.
Our main result is the following (we use the notation (1.9))
Theorem 1.1 Let f0 ≡ f0 (x, v) be a probability density such that for
some a, A > 0,
(1.13)
a f∞ ≤ f0 ≤ A f∞ ,
and let f (t) ≡ f (t, x, v) be the unique (smooth) solution of equation (1.8),
where the potential V satisfies assumptions (1.10) – (1.12). Then, for all
ε > 0 there exists a constant Cε (f0 ), explicitly computable and depending
only on V, f0 , ε, such that
(1.14)
−1/ε
kf (t) − f∞ kL1 (RN
.
N ≤ Cε (f0 ) t
x ×Rv )
Remarks.
1. The assumptions on f0 can probably be relaxed : see section 7 for
a discussion. But even under assumption (1.13), we shall not avoid
technicalities.
2. Our estimate holds in fact not only in L1 , but also in the Schwartz
space of smooth rapidly decreasing functions (that is, for any seminorm defining this space), thanks to standard interpolations.
3. Although we are aware of no explicit result in this direction, we think
it very likely that the use of linear theory yields equivalent or better
(that is, exponential) rates. We are unable to recover an exponential
rate with our method because of the use of interpolations in the
proof (again, see section 7 for some remarks). But, as mentioned
above, one essential feature of our method is that we shall take little
advantage of the linearity of the collision operator, and thus be able
to extend it to many nonlinear situations.
8
L. DESVILLETTES AND C. VILLANI
4. The collision operator can easily be replaced by a more general one,
provided that the advection operator is also changed (if not, the possible steady state is given implicitly by a degenerate (hypo)elliptic
equation, is not a tensor product in x, v, and things become much
more intricate). Namely, our proof will apply, with only minor
changes in the computations, to the more general equation
(1.15)
∂f
+ ∇W (v) · ∇x f − ∇V (x) · ∇v f = ∇v · (∇f + f ∇W (v)),
∂t
when W satisfies exactly the same assumptions as V , thatRis, W (v) =
λ0 |v|2 /2 + Ψ(v) + W0 with λ0 > 0 and Ψ ∈ H ∞ (RN ), e−W = 1.
We shall not carry on the computations in this case.
5. For notational simplicity we also restrict our proof to the case when
ω0 = 1.
The plan of the paper is as follows. In section 2, we give a heuristic
discussion, and we motivate and expose our method. This program is
then applied in sections 3 to 6. Finally, in section 7, we gather all the elements needed to conclude, and then discuss our results as well as possible
improvements, and open questions.
Acknowledgement : Our reflexion on this problem started from
discussions in the Université Paul Sabatier of Toulouse and in the Erwin
Schrödinger Institute of Vienna with Peter Markowich and Jean Dolbeault. Peter Markowich had submitted to us the problem of estimating
rates of decay for the inhomogeneous linear Fokker-Planck equation, and
Jean Dolbeault pointed out to us estimate (2.4). We also are grateful to
Anton Arnold for fruitful discussions.
Finally, both authors acknowledge the support of the European TMR
Project ”Asymptotic Methods in Kinetic Theory”, contract ERB FMBXCT97-0157, and the second author acknowledges the hospitality of the
University of Pavia, where part of this work was written.
SPATIALLY INHOMOGENOUS SYSTEMS
9
2 Heuristics and strategy
We first recall the formula giving the relative entropy, or free energy
functional, which is, according to (1.9),
Z
(2.1) H(f |f∞ ) =
Z
=
f log
RN ×RN
Z
RN ×RN
f log f dv dx+
f
dv dx
f∞
RN ×RN
Z
f V (x) dv dx+
RN ×RN
f
|v|2
N
dv dx+ log(2π).
2
2
For any solution f (t) of (1.8), it is easy to check that the entropy
dissipation, i.e. the negative of the time derivative of the entropy, is also
the relative Fisher information of f with respect to the steady state in
the velocity variable :
¯
¯
¯
Z
(2.2) D(f ) = Iv (f |f∞ ) ≡
RN ×RN
f ¯¯∇v log
Z
=
f ¯¯2
dv dx
f∞ ¯
RN ×RN
¯
¯
¯
f ¯¯2
¯
f ¯∇v log ¯ dv dx.
M
It is always nonnegative, and vanishes if and only f (= f (t, ·)) has the
form ρ(x)M (v), for some function ρ which necessarily satisfies
Z
(2.3)
ρ(x) =
RN
f (x, v) dv.
In words, ρ is the macroscopic density associated with f .
Throughout the paper, for notational simplicity, the notation ρ will
always stand for the left-hand side of (2.3), and thus will always be implicitly coupled to the function f (or f (t, ·, ·)) on consideration. Moreover,
for a given distribution function f (x, v), we shall refer to the distribution
function ρM ≡ ρ(x)M (v) as the local equilibrium associated to f . Thus,
local equilibria are exactly those distributions that make the right-hand
side of (1.8) vanish.
Thanks to the logarithmic Sobolev inequality (using ∇v ρ = 0), we find
Z
Iv (f |f∞ ) =
¯
Z
RN
dx ρ
Z
≥2
RN
dv
Z
RN
dx ρ
RN
¯
f ¯¯2
f ¯¯
∇
log
v
ρ ¯
ρM ¯
Z
f
f
f
dv log
=2
dx dv f log
.
N
N
ρ
ρM
ρM
R ×R
In other words,
(2.4)
D(f ) ≥ 2H(f |ρM ).
10
L. DESVILLETTES AND C. VILLANI
This estimate was pointed out to us by J. Dolbeault, and appears in the
paper by Olla and Varadhan [29]. As explained by these authors, this gives
information on the fact that f will look more andR more like its associated
local equilibrium as time goes to +∞ (think that 0+∞ H(f (t)|ρ(t)M ) dt <
+∞), but no information on the behavior of ρ. The lacking piece of
information is exactly, thanks to the additive property of the relative
entropy,
(2.5)
H(f |f∞ ) − H(f |ρ M ) = Hx (ρ|e−V ),
where
Z
(2.6)
Hx (ρ|e
−V
)=
RN
ρ log
ρ
dx
e−V
is the relative entropy of the density ρ with respect to e−V (in the xvariable).
Among all local equilibria, only one satisfies eq. (1.8). Indeed, a solution of the form ρ(t, x) M (v) must verify
∂t ρ + v · [∇x ρ − ρ∇V ] = 0,
so that separately ∂t ρ = 0 and ∇x ρ = ρ∇V , which finally yields f = f∞ .
We see the trend to equilibrium for (1.8) as the result of a “struggle”
between the collision operator and the antisymmetric operator of transport/confinement. Heuristically, if collisions tend to push the system in
a local equilibrium which is not the “right” one, the antisymmetric part
will drive it out of this local equilibrium, and in fact out of the class of
local equilibria, as is shown by the previous computation. Our task here
will be to quantify this physical intuition, and give a precise mathematical meaning to the formal statement that if the system ever gets trapped
into a local equilibrium which is not global, and thus ceases to dissipate
entropy, then the combined effect of transport and confinement will bring
it at later times to a state which is “far enough from the local equilibrium”
for a lot of entropy to be dissipated.
Similar ideas can already be found in Grad [18]. From the text : “the
H-theorem gives no indication that there actually will be an approach
to absolute equilibrium since it gives no clue to the transition from local
to absolute Maxwellian”, “the question is whether the deviation from a
local Maxwellian, which is fed by molecular streaming in the presence of
spatial inhomogeneity, is sufficiently strong to ultimately wipe out the inhomogeneity”, and also the cryptic remark “a valid proof of the approach
SPATIALLY INHOMOGENOUS SYSTEMS
11
to equilibrium in a spatially varying problem requires just the opposite of
the procedure that is followed in a proof of the H-theorem, viz., to show
that the distribution function does not approach too closely to a local
Maxwellian.” Roughly speaking, his idea was the following. The integral
of the entropy dissipation, as a function of time, is bounded. Thus, for
some large time, the entropy dissipation is small, and f has to look like a
local equilibrium. Moreover, using the linearity of this equation, he could
show that at such a time, the local equilibrium satisfies the same equation
as f , with a small error term. From this he showed that the macroscopic
parameters of the local equilibrium had to be very close to those of the
global equilibrium (remember that the global equilibrium is the only local equilibrium which solves the equation). All the implementation rely
crucially on the linearity of the equation, and on the whole, the argument
is of course not quantitative since there is no way to estimate the time we
have to wait before the entropy dissipation becomes small.
Our goal now is to implement the aforementioned general ideas in
the framework of quantitative entropy dissipation estimates. Thus we
are naturally led to the following problem : when collisions cease to act
efficiently, to what extent can the streaming contribute (indirectly) to the
entropy dissipation ?
To get a more precise idea, let us assume that at some time t0 , f (t0 , x, v) =
ρ(t0 , x)M (v) is a local equilibrium state. Then D(f )|t=t0 = 0. But D(f )
is ¯always nonnegative, therefore its derivative in time also vanishes :
d¯
D(f ) = 0. Thus, in order to prove that the entropy dissipation
dt ¯
t=t0
increases strictly for t > t0 small enough, it would be natural to evaluate the second derivative of D(f ), i.e. the third derivative of the relative
entropy H(f |f∞ ). But this leads to quite heavy computations, with in
addition a subtle interplay between collisions and transport/confinement.
Moreover, it is known from the work of Ledoux [25] that even in the spatially homogeneous case, differentiating three times usually leads to bad
results.
Instead, in order to estimate the entropy dissipation, we shall rather
first apply the logarithmic Sobolev inequality (2.4) and then estimate the
second-order time-derivative of H(f |ρM ) (note that once again
¯
d¯
dt ¯t=t
H(f |ρM ) = 0 since the relative entropy is a nonnegative quantity).
0
Since f (t0 ) = ρ(t0 )M , one has ∂t f |t=t0 = −M v · (∇x ρ + ρ∇V (x)). Let
12
L. DESVILLETTES AND C. VILLANI
us set formally
(2.7)
f (t0 +dt) = ρ(t0 )M −M v·(∇x ρ(t0 )+ρ(t0 )∇V (x)) dt+
∂ 2 f (dt)2
+O(dt)3 .
∂t2 2
We note that, integrating (2.7) in v,
Z
ρ(t0 + dt) = ρ(t0 ) +
RN
(dt)2
∂2f
dv
+ O(dt)3 .
∂t2
2
An easy computation then yields
H(f (t0 + dt)|ρ(t0 + dt)M )
=
1
2
¯
¯
½Z
=
R2N
1
2
µ
ρ(t0 ) M ¯¯v ·
∇x ρ(t0 )
+ ∇V
ρ(t0 )
¯
¯ ∇x ρ(t0 )
½Z
RN
ρ(t0 ) ¯¯
ρ(t0 )
¶ ¯2
¾
¯
¯ dx dv (dt)2 + O((dt)3 )
¯
¯2
¯
+ ∇V (x)¯¯ dx
¾
(dt)2 + O((dt)3 )
1
≡ Ix (ρ(t0 )|e−V ) (dt)2 + O((dt)3 ),
2
where Ix denotes the relative Fisher information in the x space. In other
words,
¯
d2 ¯¯
H(f |ρM ) = Ix (ρ|e−V ).
¯
dt2 ¯t=t
(2.8)
0
This relation is the basis of our program. It is a quantitative measure
of the competition between collisions and transport/confinement, as discussed above. In the case of the more general equation (1.15), with an
arbitrary potential W (v), eq. (2.8) is replaced by
¯
(2.9)
d2 ¯¯
H(f |ρe−W ) ≥ Λ Ix (ρ|e−V ),
¯
dt2 ¯t=t
0
where
Z
Λ=
inf
ω∈S N −1 RN
e−W (v) |∇W (v) · ω|2 dv.
Of course, this information alone doesn’t help a lot, because in practice
the solution f will never, or hardly ever, become exactly equal to a local
equilibrium. But we shall establish a more precise bound, which holds
SPATIALLY INHOMOGENOUS SYSTEMS
13
for densities f which are not local equilibria, but in which an error term
depending on the distance of f to its associated local equilibrium appears :
d2
1
H(f |ρM ) ≥ Ix (ρ|e−V ) − J(f |ρM ),
dt2
2
(2.10)
where J vanishes up to second order (at least formally) with respect to
the variable f − ρ M .
Next, since also H(f |ρM ) behaves like a square norm of f − ρM (Remember the Cziszár–Kullback–Pinsker inequality, H(f |g) ≥ (1/2)kf −
gk2L1 ), we can hope to bound J(f |ρM ) in terms of H(f |ρM ). In full generality this is obviously false, since J will involve gradients of f , while H
does not. This is the point where we need (global) smoothness estimates
of f . Under such estimates, we shall establish nonlinear interpolation
inequalities of the form
J(f |ρM ) ≤ Cε (f )H(f |ρM )1−ε ,
(2.11)
where ε > 0 is arbitrary and Cε depends on the derivatives of higher order
of f .
We end up at this stage with the system of inequalities

d


− H(f |f∞ ) ≥ 2H(f |ρM )


 dt


2


 d H(f |ρM ) ≥ 1 Ix (ρ|e−V ) − Cε (f )H(f |ρM )1−ε .
2
dt
2
At this point, we can apply the logarithmic Sobolev inequality in the x
variable: for some constant K depending only on V ,
Ix (ρ|e−V ) ≥ K Hx (ρ|e−V ).
(2.12)
Using then the additivity of the entropy, that is equation (2.5), we recover
a closed system of differential inequalities for the two quantities H(f |f∞ ),
H(f |ρM ) (as functions of time) :

d


− H(f |f∞ ) ≥ 2H(f |ρM )


 dt
(2.13)


2


 d H(f |ρM ) ≥ K H(f |f∞ ) − Cε (f ) H(f |ρM )1−ε .
2
dt
2
The last (yet not the easiest) step will be to prove that because it
is a solution of this system, the quantity H(f |f∞ ) decays to 0 as time
14
L. DESVILLETTES AND C. VILLANI
goes to infinity, with the rate t1−1/ε . We then conclude with the CziszárKullback-Pinsker inequality.
Looking back on system (2.13), we understand that the key point was
to work at the same time at the local and the global level, meaning : how
far is f from being at local equilibrium ? how far is ρ from being at global
equilibrium ?
The implementation of this program will turn out to be really tricky.
In the sequel, we shall first establish in section 3 estimate (2.10) for the
second order time-derivative of H(f |ρM ). These computations are not
hard but rather tedious, and have to be conducted with sufficient care
so as not to lose sight of the right order of the error terms. Then in
section 4 we shall establish the interpolation bounds (2.11). The smoothness bounds needed there are established in section 5; the crucial point,
and the major difficulty, is that these bounds have to be uniform in time.
Then the study of system (2.13) is performed in section 6. Finally, in
section 7, we put together the different steps and conclude.
3 Second derivative of the relative entropy
Let us introduce some notations for macroscopic quantities related to a
distribution
function f (x, v). Recalling that the density ρ(= ρf ) is given
R
by ρ(x) = f (x, v) dv, we define the (local) mean velocity u(x), the local
temperature T (x) and the traceless pressure tensor D(x) by the formulas
Z
(3.1)
(3.2)
ρ(x)u(x) =
ρ(x)
RN
f (x, v) v dv,
|u(x)|2 N
+ ρ(x)T (x) =
2
2
Z
RN
f (x, v)
|v|2
dv,
2
Z
(3.3)
ρ(x)u(x) ⊗ u(x) + ρ(x)T (x)Id + D(x) =
RN
f (x, v)v ⊗ v dv,
with Id standing for the N × N identity matrix, and [ξ ⊗ ξ]ij = ξi ξj .
Again, unless some confusion is possible, we shall always write u = u(t, x),
T = T (t, x), D = D(t, x) without recalling the implicit dependence of these
quantities upon f (t, x, v).
In this section, we give an estimate from below on the second-order
time-derivative of the relative entropy H(f |ρM ) of a solution f = f (t, x, v)
of (1.8), with respect to its associated local equibrium ρ(t, x)M (v). We
15
SPATIALLY INHOMOGENOUS SYSTEMS
only deal here with “smooth” solutions. By this we mean (say) functions f which belong to Schwartz’ class, and such that | log f | is a locally
bounded function, increasing at most polynomially at infinity.
Proposition 3.1 Let f = f (t, x, v) be a smooth solution of (1.8), and
ρ, u, T, D be the associated macroscopic fields. Then
µ
¶
d2
H(f |ρM ) ≥ Ix (ρ|e−V ) − Ix (ρ|e−V )1/2 J1 (f |ρM )1/2 + J2 (f |ρM )
dt2
(3.4)
1
1
≥ Ix (ρ|e−V ) − J1 (f |ρM ) − J2 (f |ρM ),
2
2
where
Z
Ix (ρ|e−V ) =
¯
¯ ∇ρ
ρ ¯¯
ρ
¯2
¯
+ ∇V ¯¯ dx,
and the error terms J1 , J2 are defined by
Z
(3.5)
Z
(3.6)
Z
|∇x · (ρu ⊗ u)|2
1
J1 (f |ρM ) = Iv (f |ρM ) +
dx +
ρ|u|2 dx
8
ρ
RN
RN
Z
Z
|∇x · [ρ(T − 1)]|2
|∇ · D|2
+
dx +
dx,
ρ
ρ
RN
RN
J2 (f |ρM ) =
RN
|∇x · (ρu)|2
dx + 2 Iv (f |ρM )1/2 Ix (f |ρM )1/2 .
ρ
Here,
Z
(3.7)
Ix,v (f |ρM ) =
R2N
¯
¯
¯ ∇x,v f
∇x,v (ρM ) ¯¯2
¯
−
f¯
¯ dv dx.
f
ρM
Proof: This is a long calculation that we shall divide in several
steps. By additivity of the relative entropy,
d2
d2
d2
H(f |ρM ) = 2 H(f |f∞ ) − 2 Hx (ρ|e−V ).
2
dt
dt
dt
We shall compute both terms separately.
Step 1 : We handle H(f |f∞ ). Let K, L be the operators defined by
Lf = ∇v · (∇v f + f v),
16
L. DESVILLETTES AND C. VILLANI
Kf = −v · ∇x f + ∇V (x) · ∇v f,
so that ∂t f = Kf + Lf . Then,
d2
H(f |f∞ )
dt2
Z
=
R2N
[log f + 1 − log f∞ ]
∂2f
dv dx +
∂2t
Z
1
f
R2N
µ
∂f
∂t
¶2
dv dx.
Since K and L are linear, ∂ 2 f /∂t2 = K 2 f + L2 f + (KL + LK)f , and
d2
H(f |f∞ )
dt2
(3.8)
·
Z
=
¸·
R2N
Z
+
¸
K 2 f + L2 f + (KL + LK)f dv dx
log f + 1 − log f∞
R2N
¢
1¡
(Kf )2 + (Lf )2 + 2Kf Lf dv dx.
f
According to [33], one has
Z
(3.9)
Z
R2N
[log f + 1 − log f∞ ]L2 f dv dx +
1
(Lf )2 dv dx ≥ 0.
f
R2N
Indeed, this is exactly the expression that would appear as the secondorder time derivative of the entropy in a spatially homogeneous situation
(x is only a parameter, and e−V does not contribute). It is “well-known”
that this expression equals
Z
X
2 Iv (f |ρM ) + 2
1≤i,j≤N
R2N
"
∂2
f
∂vi ∂vj
µ
f
log
ρM
¶#2
dv dx.
Moreover, using the fact that K ∗ = −K, K f∞ = 0 and for any
function g, K log g = K g/g, we get the identity
Z
R2N
¸
·
Z
log f + 1 − log f∞ K 2 f dv dx = −
R2N
(Kf )2
dv dx.
f
Thus,
(3.10)
d2
H(f |f∞ ) ≥
dt2
Z
R2N
·
¸
log f + 1 − log f∞ (KL + LK)f dv dx
Z
+2
R2N
(Kf )(Lf )
dv dx.
f
17
SPATIALLY INHOMOGENOUS SYSTEMS
The first integral in (3.10) can be rewritten as
Z
−2
¡
R2N
¢
K log f + 1 − log f∞ Lf dv dx
Z
+
Z
= −2
R2N
(Kf )(Lf )
dv dx +
f
¡
R2N
Z
¡
R2N
¢
log f + 1 − log f∞ [L, K]f dv dx
¢
log f + 1 − log f∞ [L, K]f dv dx,
with [L, K] = LK − KL, so that in the end,
(3.11)
d2
H(f |f∞ ) ≥
dt2
Z
R2N
(log f + 1 − log f∞ ) [L, K]f dv dx
Z
=
R2N
f [L, K]∗ (log f + 1 − log f∞ ) dv dx.
We now compute [L, K]∗ . We first write K = K1 + K2 , L = L1 + L2 +
L3 , with K1 = −v·∇x , K2 = ∇V (x)·∇v , L1 = ∆v , L2 = v·∇v , L3 = N Id.
We find [L3 , K1 ] = [L3 , K2 ] = [L1 , K2 ] = 0, [L1 , K1 ] = −2 ∇x · ∇v ,
[L2 , K1 ] = K1 , [L2 , K2 ] = −K2 . Thus [L, K]∗ = −2 ∇x · ∇v + v · ∇x +
∇V (x) · ∇v .
Therefore,
Z
Z
R2N
and
f [L, K]∗ (log f + 1) dv dx = 2
Z
−
R2N
R2N
∇x f · ∇v f
dv dx,
f
Z
f [L, K]∗ log f∞ dv dx = 2
R2N
f v · ∇V (x) dv dx.
This computation ends our first step : putting together the last two
integrals, we find
(3.12)
d2
H(f |f∞ ) ≥ 2
dt2
µ
Z
R2N
f
∇v f
+v
f
¶µ
∇x f
+ ∇V (x)
f
¶
dv dx.
Step 2: We write down the first two macroscopic equations obtained
by integration of (1.8) (with respect to v) against 1 and v. Using notations (3.1)–(3.3), we get
(3.13)
∂ρ
+ ∇x · (ρu) = 0,
∂t
18
L. DESVILLETTES AND C. VILLANI
∂(ρu)
+ ∇x · (ρu ⊗ u + ρT Id + D) + ρ∇V (x) = −ρu.
∂t
(3.14)
Now, we can compute the second derivative of
Z
−V
Hx (ρ|e
)=
Z
ρ log ρ dx +
RN
RN
ρV dx.
Note first that
d2
dt2
(3.15)
Z
Z
RN
ρ log ρ dx = −
|∇x · (ρu)|2
dx −
ρ
Z
Z
d
dt
log ρ ∇x · (ρu) dx
RN
µ
¶
∇x · (ρu ⊗ u) ·
∇x ρ
dx −
ρ
Z
∇x ρ
dx
ρ
RN
RN
RN
Z
Z
Z
∇x ρ
∇x ρ
∇V (x) · ∇x ρ dx −
∇x (ρT ) ·
dx −
(∇x · D) ·
dx.
−
N
N
N
ρ
ρ
R
R
R
=
ρu ·
Then,
d2
dt2
(3.16)
Z
−
ρu · ∇V (x) dx
RN
¶
Z
∇x · (ρu ⊗ u) · ∇V (x) dx −
RN
RN
RN
Z
d
ρV (x) dx =
dt
µ
Z
=−
Z
RN
Z
(∇x · D) · ∇V (x) dx −
RN
∇x (ρT ) · ∇V (x) dx
Z
ρ|∇V (x)|2 dx −
RN
ρu · ∇V (x) dx.
We mention that the first two integrals in the right-hand side of (3.15)
can be combined in several ways. For instance,
Z
RN
|∇x · (ρu)|2
dx −
ρ
Z
=
Z
RN
ρ (∇x · u)2 dx +
Z
=
RN
ρ
X
1≤i,j≤N
µ
Z
RN
¶
∇x · (ρu ⊗ u) ·
£
RN
∇x ρ
dx
ρ
¤
∇x ρ · u(∇x · u) − (u · ∇x )u dx
(∂i uj ∂j ui ) dx ≡
Z
RN
ρ tr(∇u t ∇u) dx.
We now put together (3.15) and (3.16), making systematically the
quantity ∇x ρ/ρ + ∇V (x) appear :
19
SPATIALLY INHOMOGENOUS SYSTEMS
Z
µ
Z
¶·
¸
d2
|∇x · (ρu)|2
∇x ρ
−V
H
(ρ|e
)
=
dx−
∇x ·(ρu⊗u) ·
+ ∇V (x) dx
x
N
N
dt2
ρ
ρ
R
R
·
¸
·
¸
Z
Z
∇x ρ
∇x ρ
−
+ ∇V (x) dx −
+ ∇V (x) dx
ρu ·
(∇x · D) ·
ρ
ρ
RN
RN
µ
¶ µ
¶
Z
∇x (ρT )
∇x ρ
−
ρ
+ ∇V (x) ·
+ ∇V (x) dx.
ρ
ρ
RN
The last term is the important one. It can be rewritten as
Z
−
RN
¯
¯2
Z
¯ ∇x ρ
¯
¯
¯
ρ¯
+ ∇V (x)¯ dx −
ρ
·
RN
¸ ·
¸
∇x ρ
∇x ρ(T − 1) ·
+ ∇V (x) dx.
ρ
Step 3: We now put all the pieces together :
d2
H(f |ρM ) ≥
dt2
¯
¯2
Z
¯ ∇x ρ
¯
¯
¯
ρ¯
+ ∇V (x)¯ dx −
ρ
Z
|∇x · (ρu)|2
dx
ρ
RN
RN
¾ ·
¸
Z ½
∇x ρ
+ ∇V (x) dx
+
∇x · (ρu ⊗ u) + ρu + ∇x · D + ∇x [ρ(T − 1)] ·
ρ
RN
µ
¶ µ
¶
Z
∇v f
∇x f
+2
f
+v ·
+ ∇V (x) dx dv.
f
f
R2N
We conclude the proof by the use of Cauchy-Schwarz inequality. With
obvious symbolic notations,
¯
¯Z
¶
µ
¯
¯
¯ (a + b + c + d) · ∇x ρ + ∇V (x) dx¯
¯
¯
ρ
" Z
#1/2 "Z ¯
#1/2
¯2
¯
¯ ∇x ρ
a2 + b2 + c2 + d2
≤ 4
dx
+ ∇V (x)¯¯ dx
ρ ¯¯
ρ
ρ
"Z
≤2
and
¯Z
¯
¯
¯
R2N
¯Z
¯
≤ ¯¯
µ
R2N
f
µ
f
#1/2
a2 + b2 + c2 + d2
dx
ρ
¶ µ
∇v f
+v ·
f
¶ µ
¶
∇x f
+ ∇V (x)
f
¶
¯
Ix (ρ|e−V )1/2 ,
¯
¯
dx dv ¯¯
¯
∇x ρ
+ ∇V (x) dx dv ¯¯
ρ
¯Z
¯
¶ µ
¶
µ
¯
¯
∇x f
∇x (ρM )
∇v f
¯
+¯
+v ·
−
dx dv ¯¯
f
2N
f
f
ρM
R
∇v f
+v ·
f
20
L. DESVILLETTES AND C. VILLANI
ÃZ
!1/2
¯
¯
¯ !1/2 ÃZ
¯2
¯ ∇x ρ
¯ ∇v f
¯
∇v (ρM ) ¯¯2
¯
¯
¯
≤
f¯
−
ρ¯
+ ∇V (x)¯ dx
f
ρM ¯
ρ
R2N
RN
ÃZ
¯
¯ !1/2 ÃZ
¯
¯ !1/2
¯ ∇v f
¯ ∇x f
∇v (ρM ) ¯¯2
∇x (ρM ) ¯¯2
¯
¯
−
−
+
f¯
f¯
2N
2N
f
ρM ¯
f
ρM ¯
R
R
≤ Iv (f |ρM )1/2 Ix (ρ|e−V )1/2 + Iv (f |ρM )1/2 Ix (f |ρM )1/2 .
Then, the repeated use of Young’s inequality yields proposition 2.
4 Nonlinear interpolations
In this section, we establish the
N
Proposition 4.1 Let f be a smooth function on RN
x ×Rv , with bounded
derivatives of all orders, satisfying (with the notations of section 1.1)
a f∞ ≤ f ≤ A f∞ ,
with a, A positive constants. Let J(f |ρM ) be defined (with the notations (3.1)–(3.3)) by
(4.1)
Z
Z
Z
(ρu)2
1
|∇x · (ρu ⊗ u)|2
|∇x · (ρu)|2
1
J(f |ρM ) =
dx+
dx+
dx
4
ρ
4 RN
ρ
ρ
RN
RN
Z
Z
|∇x [ρ(T − 1)]|2
|∇x · D|2
+
dx +
dx + Iv (f |ρM )
ρ
ρ
RN
RN
1
+ Iv (f |ρ M )1/2 Ix (f |ρ M )1/2 .
2
Then, for all ε > 0, there exists Cε (f ), depending on a, A and kf kW k(ε),∞
for some k(ε) ∈ N, such that
(4.2)
J(f |ρM ) ≤ Cε (f )H(f |ρM )1−ε .
The important point here is that J(f |ρM ) is of order 2 in f − ρM .
Indeed,
Z
Z
ρu =
RN
f v dv =
RN
(f − ρM )v dv;
21
SPATIALLY INHOMOGENOUS SYSTEMS
·Z
¸
Z
ρ(T − 1) =
1
N
RN
f |v|2 dv − ρ|u|2 −
1
=
N
Z
ρM |v|2 dv
RN
R
1 |
(f − ρM )|v| dv −
N
RN
2
RN (f
− ρM )v dv|2
;
ρ
Z
Dij =
RN
Z
(f − ρM )vi vj dv − ρui uj − ρ(T − 1)δij
Ã
|v|2
δij
=
(f −ρM ) vi vj −
N
RN
!
¡R
RN (f
dv−
− ρM )vi dv
¢¡R
RN (f
− ρM )vj dv
¢
ρ
R
+
1 |
N
RN (f
− ρM )v dv|2
δij .
ρ
Thus we easily bound all the terms in (4.1), except the two last ones,
by terms of the form
R
Z
(4.3)
Z
RN
[
RN [f
RN
[∇x
R
RN [f
− ρM ]ϕ(v) dv]2
dx,
ρ
− ρM ]ϕ(v) dv]2
dx,
ρ
Z
RN
¯Z
|∇x ρ|2 ¯¯
ρ5 ¯
R
Z
[
RN [f
RN
Z
[∇x
R
− ρM ]ϕ(v) dv]4
dx,
ρ3
RN [f
RN
− ρM ]ϕ(v) dv]4
dx,
ρ3
¯4
¯
[f − ρM ]ϕ(v) dv ¯¯ dx,
N
R
where ϕ(v) stands for a polynomial in v of degree less than 2.
Since H(f |ρM ) ≥ 12 kf −ρM k2L1 , it would be (almost) routine to do the
2
interpolation, were it not for the fact that 1/ρ is typically of order e|x| /2 !
This makes it very hard to use standard assumptions on moments, or
even rapid decay : note that it is not even a priori clear that the integrals
in (4.3) are finite...
The first term in (4.3) can be estimated in a very satisfactory way,
using only moments. More precisely, let
Z
kf kL1s =
We also denote kM ks =
We begin with the
R
R2N
RN
f (x, v) (1 + |v|2 )s/2 dv dx.
M (v) (1 + |v|2 )s/2 dv.
Proposition 4.2 If |ϕ(v)| ≤ 1 + |v|2 , then
(4.4)
R
µ
¶ ε
Z
1+ε
1
[ RN [f − ρM ]ϕ(v) dv]2
2
1+ε
.
dx ≤ 2H(f |ρM )
kM k2(1+ε)/ε + kf kL1
4(1+ε)/ε
N
ρ
R
22
L. DESVILLETTES AND C. VILLANI
Proof: By the Csiszár-Kullback-Pinsker inequality,
Z
H(f |ρM ) =
Z
RN
ρ
RN
f
f
1
log
dv dx ≥
ρ
ρM
2
Z
RN
1
ρ
¸2
·Z
|f − ρM | dv
RN
dx.
Then, we write
£R
Z
|f − ρM |(1 + |v|2 ) dv
ρ
RN
RN
R
Z
≤
(Z
≤
[
|f − ρM | dv]
R
[
1
1+ε
RN
RN
i
dv
2ε
1+ε
dx
ρ 1+ε
2
)
|f − ρM | dv]
dx
ρ
H(f |ρM )
1+ε
ε
ε
1
1
1+ε
dx
(f + ρM )(1 + |v|2 )
ρ 1+ε
RN
≤2
RN
RN
hR
2
1+ε
¤2
·2
ε
1+ε
1
1+ε


Z
hR
RN
(f + ρM )(1 + |v|2 )

 RN
M (1 + |v|2 )
Z
+
dv
hR
1+ε
ε


ρ dx
RN
1+ε
ε
dv
 ε
1+ε


i2
dx


ρ
RN
dx
¸2 Z
dv
f (1 + |v|2 )
RN
 ε
1+ε


i2
ρ
(·Z
RN
1+ε
ε
.
But
hR
Z
RN
RN
f (1 +
1+ε
|v|2 ) ε
ρ
dv
i2
R
(
Z
dx ≤
RN
RN
f dv)
Z
=
R2N
µ
R
RN
f (1 +
|v|2 )2
(1+ε)
ε
(1+ε)
ε
dv
dx
ρ
f (1 + |v|2 )2
¶
dv dx.
The second integral in (4.3) is slightly trickier, because the ratio of
the exponents is lower, which apparently prevents from using the same
tricks. At this point we shall use the upper and lower bounds for f , to
bound these fourth-order error terms by second-order ones.
We prove the
23
SPATIALLY INHOMOGENOUS SYSTEMS
Proposition 4.3 Assume that a f∞ ≤ f ≤ A f∞ and |ϕ(v)| ≤ 1 + |v|2 .
Then
Z
R
− ρM ]ϕ(v) dv]4
dx
ρ3
RN
µ ¶2
µ
¶ ε
1+ε
1
A
2
2
1+ε
≤ 2H(f |ρM )
+ kf kL1
(N + 1) kM kL1
.
4(1+ε)/ε
2(1+ε)/ε
a
(4.5)
[
RN [f
Proof: It suffices to note that ρ(x) ≥ a e−V (x) , hence f ≤ (A/a)ρM ,
so that
R
[
RN [f
− ρM ]ϕ(v) dv]2
≤
ρ2
µ
=
µ
A
a
A
a
¶2
¶2 µZ
RN
M (v)(1 + |v|2 ) dv
¶2
(N + 1)2 ,
and apply Proposition 4.2.
The other terms are more delicate because they involve derivatives of
f . If these derivatives were not rapidly decaying, again we would be in
trouble. Thus we shall use an interpolation lemma, namely proposition
A.2 of the Appendix.
With this lemma at hand, we shall reduce the estimate of the third
term in (4.3) to that of the first.
Proposition 4.4 Assume that a f∞ ≤ f ≤ A f∞ , and f has bounded
derivatives of all orders. Then, for all ε > 0 there exists Cε (f ) (depending
on V, A, a, and kf kW k(ε),∞ (R2N ) for some k(ε) ∈ N), such that
(4.6)
(Z
)1−ε
R
R
Z
( RN [f − ρM ]ϕ(v) dv)2
[∇x RN [f − ρM ]ϕ(v) dv]2
dx ≤ Cε (f )
dx
.
ρ
ρ
RN
RN
Proof: Note first that thanks to our hypotheses on V , we know that
for all η > 0,
¯
2 ¯¯
¯
¯V (x) − V0 − |x| ¯ ≤ η
¯
2 ¯
when |x| is large enough.
Therefore, there exists b, B > 0 satisfying
(4.7)
∀x ∈ RN ,
b e−
|x|2
2
≤ e−V (x) ≤ B e−
|x|2
2
.
24
L. DESVILLETTES AND C. VILLANI
Then,
∀x, v ∈ RN ,
f (x, v) ≤ A B e−
|x|2 +|v|2
2
,
N
and we can apply proposition A.2 of the appendix in R2N = RN
x × Rv to
get, for all δ > 0, k, l ∈ N,
∀x, v ∈ RN ,
|∇kx ∇lv f (x, v)| ≤ Ck,l,δ (f ) e−(1−δ)
Using this estimate for l = 0, we see that g ≡
for all δ > 0, k ∈ N,
∀x ∈ RN ,
(4.8)
R
|x|2 +|v|2
2
.
RN (f −ρM )ϕ(v) dv
|x|2
2
|∇k g(x)| ≤ Ck,δ (g) e−(1−δ)
satisfies
.
We now note that
eV (x)
1
eV (x)
≤
≤
.
A
ρ(x)
a
∀x ∈ RN ,
(4.9)
Therefore, in order to prove proposition 4.4, we just have to prove that
(for any ε > 0 small enough)
µZ
Z
RN
e
V (x)
2
|∇g| dx ≤ Cε (g)
RN
e
¶1−ε
V (x) 2
g dx
.
The following simple method was suggested to us by M. Ledoux. Let
Lg ≡ ∆g + ∇V (x) · ∇g.
It is immediately checked that
Z
RN
Z
(Lg)h eV (x) dx =
Z
RN
g(Lh) eV (x) dx = −
RN
(∇g · ∇h) eV (x) dx.
Hence
Z
Z
RN
|∇g|2 eV (x) dx = −
RN
g(Lg) eV (x) dx
µZ
≤
Z
¶1/2 µZ
RN
g 2 eV (x) dx
¶1/2
RN
(Lg)2 eV (x) dx
,
Z
RN
(Lg)2 eV (x) dx =
RN
g(L2 g) eV (x) dx
µZ
≤
2
RN
V (x)
g e
¶1/2 µZ
dx
RN
2
2
(L g) e
V (x)
¶1/2
dx
,
25
SPATIALLY INHOMOGENOUS SYSTEMS
and so on. Iterating the process, we find
µZ
Z
2
RN
|∇g| e
V (x)
2
dx ≤
RN
g e
V (x)
¶1−
dx
1
2k
µZ
RN
(L
2k−1
¶
2
g) e
V (x)
dx
1
2k
.
Then, we notice that
Z
2k−1
RN
(L
Z
2
g) e
V (x)
dx ≤ Ck
k−1
sup (Dj g)2 (x) (1 + |x|2 )2
RN j≤2k
Z
≤ Ck,δ (g)
since thanks to (4.7), eV (x) ≤ b−1 e
|x|2
2
2
RN
eV (x) dx
e−(1−δ)|x| e(1+δ)
|x|2
2
dx,
.
Then, we handle the following term in (4.3).
Proposition 4.5 Assume that a f∞ ≤ f ≤ A f∞ , and f has bounded
derivatives of all orders. Then, for all ε > 0 there exists Cε (f ) (depending
on V, A, a, and kf kW k(ε),∞ (R2N ) for some k(ε) ∈ N), such that
(4.10)
)1−ε
(Z
R
R
Z
[∇x RN [f − ρM ]ϕ(v) dv]4
[∇x RN [f − ρM ]ϕ(v) dv]2
dx
.
dx ≤ Cε (f )
ρ3
ρ
RN
RN
Proof: Using estimates (4.7) and (4.8), we see that for all δ ∈ (0, 1)
there is a constant Cδ (f ) such that
¯
Z
¯
¯∇x
¯
¯
¯
RN
[f − ρM ]ϕ(v) dv ¯¯ ≤ Cδ (f ) ρ1−δ .
We obtain therefore
¯
Z
¯
¯∇x
¯
R
¯ 2+3δ
¯ 1−δ
2+3δ
≤ Cδ (f ) 1−δ ρ2+3δ .
[f − ρM ]ϕ(v) dv ¯¯
N
Thus, we get
Z
[∇x
R
RN [f
RN
≤ Cδ (f )
2+3δ
1−δ
2+3δ
− ρM ]ϕ(v) dv]4
dx ≤ Cδ (f ) 1−δ
3
ρ
Z
[∇x
RN
R
RN [f
Z
R
[∇x
RN [f
RN
·
− ρM ]ϕ(v) dv]2(1−3δ)
∇x
ρ1−3δ
¸ δ(1−6δ)
Z
RN
2−7δ
− ρM ]ϕ(v) dv] 1−δ
dx
ρ1−3δ
[f − ρM ]ϕ(v) dv
1−δ
dx
26
L. DESVILLETTES AND C. VILLANI
≤ Cδ (f )
2+3δ
1−δ
[∇x
R
RN [f
RN
µZ
×
µZ
·
RN
¶1−3δ
− ρM ]ϕ(v) dv]2
dx
ρ
¶3δ
¸2 1/6−δ
Z
∇x
RN
[f − ρM ]ϕ(v) dv
1−δ
dx
.
Taking ε = 3δ (and δ < 1/6), we get the required estimate.
We finally take into account the last term of (4.3).
Proposition 4.6 Assume that a f∞ ≤ f ≤ A f∞ , and f has bounded
derivatives of all orders. Then, for all ε > 0 there exists Cε (f ) (depending
on V, A, a, and kf kW k(ε),∞ (R2N ) for some k(ε) ∈ N), such that
(4.11)
(Z
)1−ε
R
R
Z
|∇x ρ|2 [ RN [f − ρM ]ϕ(v) dv]4
[ RN [f − ρM ]ϕ(v) dv]2
dx ≤ Cε (f )
.
dx
ρ5
ρ
RN
RN
Proof: Using estimates (4.7) and (4.8), we see that for all δ ∈ (0, 1)
there is a constant Cδ (f ) such that
|∇x ρ| ≤ Cδ (f ) ρ1−δ .
Moreover,
¯Z
¯
¯
¯
¯
¯
[f − ρM ]ϕ(v) dv ¯¯ ≤ C ρ.
N
R
Thus, we get
R
Z
|∇x ρ|2 [
− ρM ]ϕ(v) dv]4
dx ≤ Cδ (f )2
ρ5
RN [f
RN
R
Z
≤ C 2+5δ Cδ (f )2
R
Z
≤
Cδ0 (f )
RN
µZ
≤ Cδ0 (f )
[
RN
R
[
RN [f
RN [f
[
RN
RN [f
R
Z
[
RN [f
RN
− ρM ]ϕ(v) dv]4
dx
ρ3+2δ
− ρM ]ϕ(v) dv]2−5δ
dx
ρ1−3δ
− ρM ]ϕ(v) dv]2−6δ
ρ1−3δ
·Z
¸δ
RN
[f − ρM ]ϕ(v) dv
¶1−3δ µ Z
− ρM ]ϕ(v) dv]2
dx
ρ
RN
dx
·Z
RN
¸1/3
[f − ρM ]ϕ(v) dv
Taking ε = 3δ (and δ < 1/6), we get the required estimate.
It only now remains to estimate the term Ix,v (f |ρM ) in (4.1). There
too, the decay of f and ρM will cause trouble, and we use assumptions
of upper and lower bounds.
¶3δ
dx
.
27
SPATIALLY INHOMOGENOUS SYSTEMS
Proposition 4.7 Assume that a f∞ ≤ f ≤ A f∞ , and f has bounded
derivatives of all orders. Then, for all ε > 0 there exists Cε (f ), depending
only on ε, V, a, A, kf kW k(ε),∞ for some k(ε) ∈ N, such that
Z
(4.12)
R2N
¯
¯
µZ
¯
f ¯¯2
¯
f ¯∇x,v log
dx dv ≤ Cε (f )
ρM ¯
R2N
f
f log
dx dv
ρM
¶1−ε
.
Proof: By proposition A.2 of the appendix applied to f , we get for
2
2
all δ > 0, |∇ log f | = |∇f |/f ≤ Cδ (f )eδ(|x| +|v| )/2 . In fact, since the higer
1 p1 ..(∇r f )pr
order derivatives of log f are bounded by terms of the form (∇ ff )p1 +..+p
,
r
we get for all p ∈ N,
2 +|v|2 )/2
|∇p log f | ≤ Cδ,p (f ) eδ(|x|
.
The same estimate holds for |∇p log(ρM )|. Applying the same strategy
as in the proof of Proposition 4.4, we find
Z
R2N
¯
¯
Z
¯
f ¯¯2
¯
dx
dv
≤
A
f ¯∇x,v log
ρM ¯
e
R2N
(Z
≤ Cε,δ (f )
R2N
¯
¯
¯
f ¯¯2
¯∇x,v log ρM ¯ dx dv
−(V (x)+|v|2 /2) ¯
)1−ε
¯
¯
¯
f ¯¯2
¯ log ρM ¯ dx dv
½Z
−(V (x)+|v|2 /2) ¯
e
R2N
e
−(V (x)+|v|2 /2) δ(|x|2 +|v|2 )
e
¾ε
dx dv
.
Then,
Z
R2N
≤
C
a
e
¯
¯
¯
¯
Z
¯
¯
1
f ¯¯2
f ¯¯2
¯
¯ log ρM ¯ dx dv ≤ a 2N ρM ¯ log ρM ¯ dx dv
R
Ã
¯
¯2 !
¶
µ
¯
¯
f
f
f
f
¯
log
−
+1
1 + ¯¯ log
dx dv
ρM
ρM
ρM
ρM
ρM ¯
−(V (x)+|v|2 /2) ¯
Z
R2N
C
≤ (1 + | log(A/a)|2 )
a
µ
Z
R2N
ρM
C
= (1 + | log(A/a)|2 )
a
¶
f
f
f
log
−
+1
ρM
ρM
ρM
Z
R2N
f log
dx dv
f
dx dv.
ρM
Here we have used the elementary inequality | log u|2 ≤ C(u log u − u +
1)(1 + | log u|2 ).
Proposition 4.1 is now a direct consequence of propositions 4.2 to 4.7.
28
L. DESVILLETTES AND C. VILLANI
5 Uniform in time hypoellipticity estimates
In this section, we prove the following result.
N
1
N
Proposition 5.1 Let f ∈ C(R+
t , L (Rx × Rv )) be a solution of equation (1.8), with V (x) satisfying assumptions (1.10), (1.12). Then, for any
t0 > 0, f lies in L∞ ([t0 , +∞); Cb∞ (RN × RN )), i.e. has all its derivatives
in x and v bounded, uniformly for t ≥ t0 > 0.
Remark. The same proof holds for the more general equation (1.15).
Before proving Proposition 5.1, we first establish a convenient representation formula. Remember that for simplicity, we take ω0 = 1 in (1.10).
Let us rewrite equation (1.8) as
(5.1)
∂t f + v · ∇x f − x · ∇v f − ∇v · (∇v f + v f ) = ∇Φ(x) · ∇v f,
and denote by
(5.2)
fˆ(t, ξ, η) =
Z
RN ×RN
e−i (x·ξ+v·η) f (t, x, v) dv dx
the Fourier transform of f . Eq. (5.1) becomes
(5.3)
[f .
∂t fˆ + η · ∇ξ fˆ + (η − ξ) · ∇η fˆ + |η|2 fˆ = i η · ∇Φ
We introduce the caracteristic differential system associated to the firstorder differential part of the left hand side of (5.3) :
(5.4)
ξ˙ = η,
(5.5)
η̇ = η − ξ,
the solution of which is given by the flow
(5.6)
Ã√ !
à √ !!
Ã√ !
3
3
1
3
3
cos
t − sin
t
ξ + sin
t η,
2
2
2
2
2
Ã√
Ã√ !
à √ !! #
Ã√ !
1
3
3
3
3
t ξ+
cos
t + sin
t
η
− sin
2
2
2
2
2
2 t
Tt (ξ, η) = √ e 2
3
"Ã √
≡ [Tt1 (ξ, η), Tt2 (ξ, η)].
29
SPATIALLY INHOMOGENOUS SYSTEMS
The solution of equation (5.3) can be written in the semi–explicit form
(5.7) fˆ(t, ξ, η) = fˆ0 (T−t (ξ, η)) e−
+i
Z t
0
Rt
0
2 (ξ,η)|2 dσ
|Tσ−t
2
[f (s, Ts−t (ξ, η)) e−
Ts−t
(ξ, η) ∇Φ
Rt
s
2 (ξ,η)|2 dσ
|Tσ−t
ds.
After the change of variables σ → t − σ, s → t − s, we end up with the
so-called Duhamel representation of fˆ :
fˆ(t, ξ, η) = fˆ0 (T−t (ξ, η)) e−
(5.8)
+i
Z t
0
Rt
0
2 (ξ,η)|2 dσ
|T−σ
2
[f (t − s, T−s (ξ, η)) e−
T−s
(ξ, η) ∇Φ
Rs
0
2 (ξ,η)|2 dσ
|T−σ
ds.
We now state and prove two crucial lemmas.
Lemma 5.2 There exists K > 0, such that for any s ≥ 0, ξ, η ∈ RN , one
has
(5.9)
Z s
0
µ
2
|T−σ
(ξ, η)|2 dσ ≥ K
¶
inf(s, 1)3 |ξ|2 + inf(s, 1) |η|2 .
Proof: It is obviously enough to prove the lemma for s ∈ [0, s0 ] for
some s0 < 1. But for s ∈ [0, s0 ], we have
Z s
0
4
2
|T−σ
(ξ, η)|2 dσ ≥ e−1
3
Ã√
!
√
√
√
Z s ¯¯
3
3
3
1
3
¯
σ) ξ +
cos(
σ) − sin(
σ)
¯sin(
2
2
2
2
2
0 ¯
√
√
µ
√
sin( 3 s)
sin( 3 s)
2 −1
2
√
√
(s −
) |ξ| + (1 − cos( 3 s) +
− s) ξ · η
≥ e
3
3
3
√
¶
√
1 sin( 3 s)
1
1
2
√
+(
+ s + cos( 3 s) − ) |η|
2
2
2
3
µ
(5.10)
=
¶
2 −1
e
α1 (s) (s3 |ξ|2 ) + 2 α2 (s) (s2 ξ · η) + α3 (s) (s |η|2 ) ,
3
where
(5.11) α1 (s) =
s−
√
sin(√ 3 s)
3
,
s3
α2 (s) =
√
1 − cos( 3 s) +
2 s2
√
sin(√ 3 s)
3
−s
,
¯2
¯
¯
η ¯ dσ
¯
30
L. DESVILLETTES AND C. VILLANI
(5.12)
α3 (s) =
√
1 sin(√ 3 s)
2
3
√
+ s + 12 cos( 3 s) −
s
1
2
.
Then, α1 (0) = 1, α2 (0) = 3/4, α3 (0) = 3/2.
The eigenvalues of the matrix
Ã
!
α1 (s) α2 (s)
α2 (s) α3 (s)
M(s) =
are strictly positive for s = 0, and by continuity, are bounded below by
K > 0 for s ∈ [0, s0 ] if s0 is small enough.
For such parameters s, we get
Z s
(5.13)
0
2 −1
e K (s3 |ξ|2 + s |η|2 ),
3
2
|T−σ
(ξ, η)|2 dσ ≥
and the lemma is proven.
Lemma 5.3 Let s0 ∈ [0, 1] and
(5.14)
Ls0 (ξ, η) =
Z s0
0
3
(s |ξ| + |η|) e−K (s
|ξ|2 +s |η|2 )
ds.
Then there exists C > 0 (depending only on K) such that
(5.15)
|Ls0 (ξ, η)| ≤
C
.
1 + |ξ|1/3 + |η|
Proof: Thanks to the change of variables u = s |ξ|2/3 and v = s |η|2 ,
we get
Z +∞
0
s |ξ| e−K (s
Z +∞
0
(5.17)
|ξ|2 +s |η|2 )
≤ |ξ|−1/3
(5.16)
and
3
|η| e−K (s
3
ds ≤
Z +∞
0
|ξ|2 +s |η|2 )
≤ |η|−1
0
0
s |ξ| e−K s
3
|ξ|2
ds
3
u e−K u du,
ds ≤
Z +∞
Z +∞
Z +∞
0
2
|η| e−K s |η| ds
e−K v dv.
31
SPATIALLY INHOMOGENOUS SYSTEMS
On the other hand, if we denote
(5.18)
C1 =
3
u3/2 e−K u ,
sup
C2 =
u∈[0,+∞)
v 1/2 e−K v ,
sup
v∈[0,+∞)
we find
Z +∞
0
≤ C1 |η|
Z +∞
0
(5.20)
s |ξ| e
−1
(5.19)
and
−K (s3 |ξ|2 +s |η|2 )
|η| e−K (s
3
Z +∞
0
|ξ|2 +s |η|2 )
≤ C2 |ξ|−1/3
ds ≤ C1
0
0
2
s−1/2 e−K s |η| ds
v −1/2 e−K v dv,
ds ≤ C2
Z +∞
Z +∞
Z +∞
0
3
s−1/2 e−K s
|ξ|2
ds
3
u−1/2 e−K u du.
Grouping estimates (5.16), (5.17), (5.19) and (5.20), we conclude the proof
of lemma 5.3.
Proof of Proposition 5.1: By mass conservation,
(5.21)
sup sup |fˆ(t, ξ, η)| ≤ kf0 kL1 (RN ×RN ) .
t≥0 ξ,η∈RN
We shall show that if
sup |fˆ(t, ξ, η)| ≤
t≥0
Ck
(1 + |ξ|2 + |η|2 )k
(k ∈ R+ ), then for any t0 > 0,
(5.22)
sup |fˆ(t, ξ, η)| ≤
t≥t0
Ck0
1
(1 + |ξ|2 + |η|2 )k+ 6
.
The conclusion will follow by induction.
We first note that in view of (5.21) and lemma 5.2, estimate (5.22)
holds with fˆ replaced by
A(t, ξ, η) = fˆ0 (T−t (ξ, η)) e−
Rt
0
2 (ξ,η)|2 dσ
|T−σ
.
32
L. DESVILLETTES AND C. VILLANI
Thus, according to the Duhamel representation, we only need to estimate
(5.23)
B(t, ξ, η) =
Z t
0
2
[f (t − s, T−s (ξ, η)) e−
T−s
(ξ, η) ∇Φ
Rs
0
2 (ξ,η)|2 dσ
|T−σ
ds.
With Ck denoting various constants depending on one another, we
have
¯Z
¯
¯
¯
d
ˆ
¯
[
|∇Φf (t, ξ, η)| = ¯ ∇Φ(ξ∗ )f (t, ξ − ξ∗ , η) dξ∗ ¯¯
Z
≤
|ξ∗ |≤ 21 |ξ|
d ∗ )|
dξ∗ |∇Φ(ξ
Ck
(1 + |ξ|2 + |η|2 )k
Z
+
|ξ∗ |≥ 12 |ξ|
Ck
d 1
k∇Φk
L
(1 + |ξ|2 + |η|2 )k
≤
Ck
+
(1 + |ξ|2 )k (1 + |η|2 )k
d ∗ )|
dξ∗ |∇Φ(ξ
Ck
(1 + |η|2 )k
Z
RN
d ∗ )| dξ∗ .
(1 + |ξ∗ |2 )k |∇Φ(ξ
Since
Z
RN
d ∗ )|(1 + |ξ∗ |2 )k dξ∗
|∇Φ(ξ
·Z
≤
RN
d ∗ )|2 (1 + |ξ∗ |2 )2k+N +1 dξ∗
|∇Φ(ξ
¸1/2 ·Z
RN
dξ∗
(1 + |ξ∗ |2 )N +1
¸1/2
≤ Ck kΦkH 2k+N +2 ,
we find
[ (t, ξ, η)| ≤
sup |∇Φf
t≥0
Ck
.
(1 + |ξ|2 + |η|2 )k
Let s0 ≤ inf(1, t0 ) be an intermediate time that will be chosen later
on. For t ≥ t0 , we write
|B(t, ξ, η)| ≤
≤
Z t
s0
Z t
0
2
[f (t
|T−s
(ξ, η)| |∇Φ
− s, T−s (ξ, η))| e
3
−
Rs
0
2 (ξ,η)|2 dσ
|T−σ
ds
3 +s |η|2 )
0
2 e−s/2 (s|ξ| + |η|) ds Ck e−K(s0 |ξ|
+
Z s0
0
(s|ξ| + |η|) ¡
Ck
1 + |T−s
¢k
(ξ, η)|2
3 |ξ|2 +s|η|2 )
e−K(s
ds.
33
SPATIALLY INHOMOGENOUS SYSTEMS
By continuity of the flow t 7→ Tt (ξ, η), and its linearity with respect
to (ξ, η), we can choose s0 ∈ (0, inf(t0 , 1)) in such a way that for all
s ∈ [0, s0 ],
1
|T−s (ξ, η)|2 ≥ (|ξ|2 + |η|2 ).
2
Then, for t ≥ t0 ,
3
3
2
|B(t, ξ, η)| ≤ Ck (|ξ| + |η|) e−K(s0 |ξ| +s0 |η| )
Z s0
Ck
3
3
2
(s|ξ| + |η|)e−K(s |ξ| +s|η| ) ds.
+
2
2
k
(1 + |η| + |ξ| ) 0
The last integral is bounded by
conclude by Lemma 5.3.
R +∞
0
3 |ξ|3 +s|η|2 )
(s|ξ| + |η|)e−K(s
, and we
6 Study of a system of differential inequalities
In this section, we consider the system with two unknowns x(t), y(t) ∈
C 2 (R+ , R+ )

0


−x (t) ≥ A1 y(t),
(6.1)


y 00 (t) + A y 1−ε (t) ≥ A x(t)
2
3
for some ε ∈ (0, 1) (not necessarily small), and A1 , A2 , A3 positive constants. Our aim is to obtain decay estimates for x(t).
A natural approach to this problem would be to compare solutions
of the system of inequalities to solutions of the corresponding system of
equations. But here such an attempt would be doomed, because solutions
of the linear system (say) −x0 = y, y 00 + y = x, have a tendency to
oscillate strongly. The point is that in doing so, we would forget the
crucial assumption that x and y are nonnegative.
The proof that we present is essentially based on the following intuition : while y remains strictly positive, x has to decay. It may happen
that y becomes very small, but then, if x itself is not too small, y 00 will be
strictly positive (by the second inequality of (6.1)). And there will hold
the following alternative : either the value of y will soon grow again, or
it will stay low for some time, but then it will grow very fast (because it
is a strictly convex function of time), and in both cases x has to decay.
Quantify this vague idea turns out to be somewhat technical, but in the
34
L. DESVILLETTES AND C. VILLANI
end we shall recover the same rate of decay (though of course with worse
constants) as if there was no term y 00 in the second inequality of (6.1).
We begin with the
Lemma 6.1 Let y be a nonnegative C 2 function on some time interval
[T1 , T2 ], satisfying the differential inequality
x0
(6.2)
y 00 (t) + A2 y 1−ε (t) ≥ A3
2
for some constants ε ∈ (0, 1), A2 , A3 > 0, x0 ∈ (0, 1). Then one can find
constants C1 , C2 > 0, depending only on A2 , A3 , ε, such that either
ε
T2 − T1 ≤ 8 C1 x02(1−ε) ,
(6.3)
or
(6.4)
Z T2
T1
1
y(t) dt ≥ C2 (T2 − T1 ) x01−ε .
Proof: We split our time interval [T1 , T2 ] into intervals on which y
is “small” or “large”. Consider the intervals I such that on I, A2 y 1−ε <
A3 x0 /4, and there exists some s ∈ I such that A2 y 1−ε < A3 x0 /8 : these
will be the intervals on which y is “small”. Since y 0 is bounded, there is
a finite number of such intervals I = (τ, θ). As far as the end-points τ, θ
are concerned, there are three possible cases :
- either A2 y 1−ε (τ ) = A2 y 1−ε (θ) = A3 x0 /4. The corresponding intervals are denoted by I1 = (τ1 , θ1 ), . . . , In−1 = (τn−1 , θn−1 ).
- either A2 y 1−ε (τ ) < A3 x0 /4 and τ = T1 . The corresponding interval,
if it exists, is denoted by I0 = (τ0 , θ0 ).
- either A2 y 1−ε (θ) < A3 x0 /4 and θ = T2 . The corresponding interval,
if it exists, is denoted by In = (τn , θn ). Of course it may happen that
I0 = In .
On each interval Ii = (τi , θi ), one has y 00 ≥ A3 x0 /4. Therefore y is
strictly convex and admits a unique minimum at a time t̄i ∈ (τi , θi ), such
that (by definition of Ii ) A2 y(t̄i )1−ε ≤ A3 x0 /8. Then, integrating twice,
we get
x0
x0
(t − t̄i ),
y(t) ≥ A3
(t − t̄i )2 .
(6.5) ∀t ∈ (τi , θi ),
y 0 (t) ≥ A3
4
8
Using this estimate for t = τi , and then t = θi , we obtain the following
bound on the size of the interval Ii :
ε
(6.6)
θi − τi ≤ C1 x02(1−ε) ,
SPATIALLY INHOMOGENOUS SYSTEMS
35
1
where C1 = 2 (8/A3 )1/2 (A3 /(4A2 )) 2(1−ε) .
We now get rid of the case when θn−1 − τ1 ≤ (1/2) (T2 − T1 ), which
means that a large fraction of the time interval falls before τ1 or after
θn−1 . Suppose first that the measure of I0 ∪ In is bigger than half the
measure of (T1 , τ1 ) ∪ (θn−1 , T2 ) (by convention, the measure of an interval
which does not exist is 0....). Then, by (6.6),
ε
T2 − T1 ≤ 8 C1 x02(1−ε) ,
(6.7)
and (6.3) holds. Suppose now that the measure of I0 ∪ In is smaller
than half the measure of (T1 , τ1 ) ∪ (θn−1 , T2 ). Then, on a set of measure
≥ (T2 − T1 )/4, holds A2 y 1−ε ≥ A3 x0 /8, whence
Z T2
(6.8)
T1
µ
A3 x0
1
y(t) dt ≥ (T2 − T1 )
4
A2 8
1
¶ 1−ε
.
1
≥ C3 (T2 − T1 ) x01−ε ,
with C3 = (1/4)(A3 /(8A2 ))1/(1−ε) .
We now turn to the “general” case, when θn−1 − τ1 ≥ (1/2) (T2 − T1 ),
and we shall only be concerned with what happens on (τ1 , θn−1 ).
Note first that since (by convexity) y 0 (θi ) > 0 and y 0 (τi+1 ) ≤ 0, there
exists a first time t0i ∈ (θi , τi+1 ) such that y 0 (t0i ) = 0. (Here we assume
that i < n − 1; the remaining cases can be treated as above.) Then,
thanks to Jensen’s inequality,
(6.9)
Z t0
i
θi
y(t) dt ≥
Z t0 µ
i
θi
y 00 (t)
1 00
−
A2 y (t)≤0
1
¶ 1−ε
dt
−1/(1−ε)
≥ (t0i − θi )−ε/(1−ε) A2
where we have used the fact that y 0 (θi ) = −
But we also know that
(6.10)
Z t0
i
θi
µ
y(t) dt ≥
(t0i
R t0i 00
R t0i
00
θi y (t) dt ≤ θi −y (t)1y 00 (t)≤0 dt.
A3 x0
− θi )
A2 4
1
¶ 1−ε
.
Mixing estimates (6.9) and (6.10), we end up with
(6.11)
Z t0
i
θi
µ
y(t) dt ≥
A−1
2
A3 x0
4A2
y 0 (θi )1/(1−ε) ,
¶ε/(1−ε)
y 0 (θi ).
36
L. DESVILLETTES AND C. VILLANI
We now try to obtain a lower bound for y 0 (θi ). Thanks to estimate
(6.5), we know that
x0
(6.12)
y 0 (θi ) ≥ A3
(θi − t̄i ).
4
On the other hand, by definition of τi , A2 y(t̄i )1−ε ≤ A3
since y is convex on (τi , θi ),
x0
8 .
Moreover,
y(θi ) − y(t̄i ) ≤ y 0 (θi ) (θi − t̄i ),
(6.13)
so that
µ
(6.14)
0
−1/(1−ε)
y (θi ) ≥ (1 − 2
A3 x0
)
A2 4
1
¶ 1−ε
(θi − t̄i )−1 .
Mixing (6.12) and (6.14), we get the lower bound
(6.15)
µ
¶
A3 1/(2(1−ε)) (1−ε/2)/(1−ε)
x0
.
y 0 (θi ) ≥ (1 − 2−1/(1−ε) )1/2 (A3 /4)1/2
4 A2
From (6.11) and (6.15), we obtain
(6.16)
Z τi+1
τi
y(t) dt ≥
Z t0
i
θi
(1+ε/2)/(1−ε)
y(t) dt ≥ C4 x0
,
A3 (1+ε/2)/(1−ε)
where C4 = A−1
(1 − 2−1/(1−ε) )1/2 (A3 /4)1/2 .
2 ( 4A2 )
In order to conclude, we now separate two cases :
• If τi+1 − θi ≥ θi − τi (that is, the regime when y is large prevails),
then
(6.17)
Z τi+1
τi
µ
y(t) dt ≥
A3 x0
A2 8
1
¶ 1−ε
(τi+1 − θi )
1
≥
2
µ
A3 x0
A2 8
1
¶ 1−ε
(τi+1 − τi ).
• If τi+1 − θi ≤ θi − τi (that is, the regime when y is small prevails),
from estimates (6.16) and (6.6) follows
(6.18)
Z τi+1
τi
(1+ε/2)/(1−ε)
y(t) dt ≥ C4 x0
θi − τi
ε/(2(1−ε))
C1 x0
C4 1/(1−ε)
≥
x
(τi+1 − τi ).
2 C1 0
SPATIALLY INHOMOGENOUS SYSTEMS
37
Denoting
µ µ
(6.19)
C5 = inf
1
2
A3
8 A2
1
¶ 1−ε
¶
C4
,
,
2 C1
we get in both cases (remember that x0 ∈ (0, 1)),
(6.20)
Z τi+1
τi
1
y(t) dt ≥ C5 x01−ε (τi+1 − τi ).
Adding estimate (6.20) for i = 1, .., n − 1, we get (6.4) with C2 =
inf(C3 , C5 /2), and the proof of the lemma is ended.
We now prove the main theorem of this section.
Theorem 6.2 Let x, y be two nonnegative C 2 functions defined on R+
and satisfying
(6.21)
−x0 (t) ≥ A1 y(t),
(6.22)
y 00 (t) + A2 y 1−ε (t) ≥ A3 x(t).
for some constants ε ∈ (0, 1) and A1 , A2 , A3 > 0.
Then there exists C6 > 0 depending only on x(0), A1 , A2 , A3 and ε,
such that for all t > 0,
(6.23)
x(t) ≤
C6
.
(1−ε)/ε
t
Remark. If ε = 0, the conclusion can be replaced by : x(t) ≤ x(0) e−λt
for some λ > 0.
Proof: Note first that x decreases on R+ . Assume, without loss of
generality, that x0 ∈ (0, 1) (this amounts to changing A1 and A3 ). We
denote by tx0 , Tx0 the unique times such that x(tx0 ) = x0 , x(tx0 + Tx0 ) =
x0 /2 (these times may be infinite at this level). By eq. (6.21),
(6.24)
x0
≥ A1
2
Z tx +Tx
0
0
tx0
y(t) dt.
38
L. DESVILLETTES AND C. VILLANI
Then, thanks to eq. (6.22), the hypotheses of lemma 6.1 are satisfied
with T1 = tx0 and T2 = tx0 + Tx0 . Using this lemma, we know that either
ε/(2(1−ε))
Tx0 ≤ 8 C1 x0
or
Z tx +Tx
0
0
tx0
,
1
y(t) dt ≥ C2 x01−ε Tx0 .
Introducing
µ
¶
1
C7 = max 8 C1 ,
,
2C2 A1
(6.25)
we get in all cases (remember that x0 ≤ 1),
ε
− 1−ε
(6.26)
Tx0 ≤ C7 x0
.
We now consider the sequence of times Ti such that x(Ti ) = 2−i . Thanks
to the previous analysis, we see that Ti+1 −Ti ≤ C7 2iε/(1−ε) (in particular,
Ti < +∞). Then, Tn − T0 ≤ C8 2nε/(1−ε) , for C8 = C7 /(2ε/(1−ε) − 1).
Finally, for all n ∈ N,
x(C8 2nε/(1−ε) ) ≤ 2−n .
(6.27)
Using the fact that x is decreasing, it is then easy to conclude that for
any time t ≥ 0,
(6.28)
x(t) ≤
(1−ε)/ε
where C6 = 2 C8
C6
,
t(1−ε)/ε
, which ends the proof of Theorem 6.2.
7 Conclusion and remarks
Proof of Theorem 1.1: Let f be a solution of equation (1.8), with
initial datum f0 satisfying assumption (1.13). The maximum principle
implies that for all t ≥ 0,
(7.1)
a f∞ ≤ f (t, ·) ≤ A f∞ .
Thanks to proposition 5.1, we know that for any t0 > 0, the derivatives
of any order in x and v of f are uniformly bounded for t ≥ t0 . Using
proposition A.2 of the appendix, we see that they also are rapidly decaying
SPATIALLY INHOMOGENOUS SYSTEMS
39
when |x| → +∞ or |v| → +∞. Together with estimate (7.1), this implies
that our solution to eq. (1.8) is smooth in the sense of proposition 3.1.
Thus, this proposition applies to our solution. For the same reason, the
functional inequality given by proposition 4.1 can be applied to f (t, ·)
with a constant Cε (f ) which does not depend on t (when t ≥ t0 ). Thus,
the quantities x(t) = H(f (t)|f∞ ) and y(t) = H(f (t)|ρ(t)M ) satisfy the
system of differential inequalities (6.1) with A1 = 2, A2 = Cε (f ) and A3 =
K/2, where K is the constant in the logarithmic Sobolev inequality (2.12).
Thanks to theorem 6.2, we see that for all ε > 0 there exists Cε (f0 ) such
that
x(t) = H(f (t)|f∞ ) ≤ Cε (f )t−1/ε .
Then, the Cziszár-Kullback-Pinsker inequality then implies the conclusion
of our theorem.
We now wish to briefly discuss this result, and point out several open
problems.
1) Smoothness bounds are only needed at the level of the interpolations. The hypoellipticity of linear operators of the form ∂t + v · ∇x − ∆v
is a standard topic [38, 22], which has been systematically studied by
Hörmander [21] for instance. In particular, his celebrated theorem of hypoellipticity applies here to show that solutions become immediately C ∞
(and would also apply for much more general linear operators). But we
are aware of no study of the uniformity in time of these bounds. We were
unable to modify the proofs of Hörmander, or the simpler variant given
later by Köhn [23, 10] to obtain such bounds. This is why we use Fourier
transform methods in section 5, and this is the only reason why we assume that the confining potential behaves qudratically at infinity. In all
the rest of the paper, it would be sufficient that V be uniformly convex
at infinity, since all we really need is to apply a logarithmic Sobolev inequality for the measure e−V (x) dx. We think it very likely, though, that
uniform hypoellipticity also holds for such a (smooth) potential.
2) A natural question is whether the upper bound assumption f0 ≤
A f∞ can be relaxed. Let us only mention that at present, we are aware
of no other simple condition Rof control of high-order moments, i.e. for instance the quantities supt≥0 f (t, x, v)(1 + |x|2 + |v|2 )p dx dv for p > 2. It
is clear that our understanding of the tail behavior for solutions of equation (1.8) is extremely poor. This is quite in contrast to the homogeneous
40
L. DESVILLETTES AND C. VILLANI
theory, in which uniform boundedness of all moments (which are initially
finite) is known.
3) The lower bound a f∞ ≤ f0 is a natural assumption to prevent a
possible vanishing of the local density ρ. Troubles may come from large
positions, where the density is very small. This problem disappears when
the system is enclosed in a box, with periodic conditions for instance.
But even in this case, large velocities are still delicate to handle in the
proof of Proposition 4.7, because there remains the problem of estimating
∇x (log f ) at large velocities (the term with ∇v (log f ) is nonessential because it is in fact exactly the entropy dissipation). Again, it is likely that
a lower bound, maybe not so strong as the one of (1.13), automatically
holds for positive times, though we are aware of no result in this direction.
4) In the case of a quadratic confinement q
potential ω02 |x|2 /2, the optimal rate is exponential, and given by Re(1 − 1 − 4ω02 )/2, in dimension 1
(see [31]). Our method is unable to recover an exponential rate, because
of the use of interpolation.
5) In the computations of section 3, we threw away a piece of information which is the second derivative of the entropy due to the effect of the
collision operator only. In the homogeneous case, this quantity dominates
the entropy dissipation itself, which is extremely useful. But here, this
term will not involve gradients with respect to x, so that apparently we
cannot hope that it will help getting rid of the interpolations.
6) As mentioned previously, one advantage of the method of proof
is that we do not really care of the linearity of the equation (except
for establishing smoothness bounds, of course). The only crucial point,
as regards the collision operator, is that there is a lower bound for the
entropy dissipation in terms of the relative entropy with respect to the
local equilibrium. Corresponding variants of the method will be sketched
in [17].
7) A final comment concerns the complexity of the implementation.
Even though the general principles of our method are rather simple, we
have seen that it leads to quite complicated computations – and we only
considered a simple model with only one parameter of local equilibrium !
However, as we have tried to show, many of the computations can be conducted in a rather systematic way, and some of our intermediate propositions are general enough that they can be used in many similar problems.
SPATIALLY INHOMOGENOUS SYSTEMS
41
On the whole, our general mathematical scenario follows the physical intuition, and this should not be concealed by the heavy calculations.
Appendix: An interpolation lemma
In this section, we give more or less standard estimates of interpolation.
Lemma A.1 : Let f be a C 2 function from RN to R, and φ, ψ two
nonincreasing functions from R+ to R∗+ , such that
∀x ∈ RN ,
(A.1)
∀x ∈ RN ,
(A.2)
P
|f (x)| ≤ φ(kxk2 ),
k∇∇f (x)k∞ ≤ ψ(kxk2 ),
2 1/2 ,
1≤i≤N Xi )
where kXk2 = (
kXk∞ = sup1≤i≤N |Xi |, and k∇∇f (x)k∞ =
sup1≤i,j≤N |∂ij f (x)| (and so on). Then,
(A.3)
q
∀x ∈ RN ,
k∇f (x)k∞ ≤ 2 N φ(kxk2 )ψ(kxk2 ).
Proof: Let
It (x) = k∇f (x + t r(x)u(x)) − ∇f (x)k∞ ,
RN ,
where u(x) ∈
ku(x)k2 = 1, and r(x) ≥ 0 will be chosen later on. For
all t ∈ [0, 1], x ∈ RN ,
¯
¯
¯X Z 1
¯
¯
¯
It (x) = sup ¯
∂ij f (x + s t r(x)u(x)) ds ui (x)¯ t r(x).
¯
¯
0
1≤j≤N
i
Let us set u(x) = e(x) sgn[e(x) · x], where ||e(x)||2 = 1 shall be chosen
later on. We use the convention that (say) sgn (0) = 1. Thus we always
have kx + s t r(x)u(x)k2 ≥ kxk2 , and by (A.2), since ψ is nonincreasing,
√
It (x) ≤ N t ψ(kxk2 ) r(x).
On the other hand, since also φ is nonincreasing,
2 φ(kxk2 ) ≥|f (x + r(x)u(x)) − f (x)|
¯Z 1
¯
¯
¯
¯
=¯
∇f (x + t r(x)u(x)) dt · u(x)¯¯ r(x)
0
Z 1
≥ |∇f (x) · e(x)|r(x) −
≥ |∇f (x) · e(x)|r(x) −
≥ |∇f (x) · e(x)|r(x) −
0
k∇f (x + t r(x)u(x)) − ∇f (x)k2 dt r(x)
√ Z
N
0
1
It (x) dt r(x)
N
ψ(kxk2 ) r2 (x).
2
42
L. DESVILLETTES AND C. VILLANI
We now choose e(x) = ∇f (x)/k∇f (x)k2 if ∇f (x) 6= 0, and we get
k∇f (x)k2 ≤
2 φ(kxk2 ) N
+ ψ(kxk2 )r(x).
r(x)
2
We conclude by setting r(x) =
p
4/N
p
φ(kxk2 )/ψ(kxk2 ).
We deduce from this lemma the
Proposition A.2 : Let f ∈ C ∞ (RN , R), N ≥ 1, be a function
satisfying the following estimates :
1. there exist K0 , D > 0 such that
∀x ∈ RN ,
(A.4)
2
|f (x)| ≤ K0 e−D |x| ,
2. for all p ∈ N∗ , there exists Kp > 0 such that
∀x ∈ RN ,
(A.5)
k∇p f (x)k∞ ≤ Kp .
Then, for all k ∈ N, ε > 0, there exists Lε,k > 0 (depending on the Ki ,
0 ≤ i ≤ N (ε, k), and D) such that
(A.6)
∀x ∈ Rn ,
2
k∇k f (x)k∞ ≤ Lε,k e−(D−ε) |x| .
Proof: Since this proof is quite similar to the proof of theorem B.2
and corollary B.3 of [14], we only sketch it briefly. Suppose that for all
k ∈ [0, p],
(A.7)
2
∀x ∈ RN ,
k∇k f (x)k∞ ≤ Kk,q e−Dk,q |x| .
Then, thanks to lemma A.1,
(A.8)
∀x ∈ RN ,
2
k∇k f (x)k∞ ≤ Kk,q+1 e−Dk,q+1 |x| ,
where K0,q+1 = K0,q , Kp,q+1 = Kp,q and for all k ∈ [1, p − 1], Kk,q+1 =
p
2 N Kk−1,q Kk+1,q .
In the same way, D0,q+1 = D0,q , Dp,q+1 = Dp,q and for all k ∈ [1, p−1],
Dk,q+1 = 21 (Dk−1,q + Dk+1,q ).
Let us now set D0,0 = D, K0,0 = K0 , Dk,0 = 0, Kk,0 = Kk for
k ∈ [1, p]. With this choice of initial data, we easily check that D1,q
converges towards D (1 − 1/p) when q tends to infinity. Since p and q
can be taken as large as desired, the proposition is proven for k = 1. The
conclusion follows by a trivial induction.
SPATIALLY INHOMOGENOUS SYSTEMS
43
Acknowledgment. Our reflexion on this problem started from discussions in the Université Paul Sabatier of Toulouse and in the Erwin
Schrödinger Institute of Vienna with Peter Markowich and Jean Dolbeault. Peter Markowich had submitted to us the problem of estimating
rates of decay for the inhomogeneous linear Fokker-Planck equation, and
Jean Dolbeault pointed out to us estimate (2.4). We also are grateful to
Anton Arnold for fruitful discussions.
Finally, both authors acknowledge the support of the European TMR
Project ”Asymptotic Methods in Kinetic Theory”, contract ERB FMBXCT97-0157, and the second author acknowledges the hospitality of the
University of Pavia, where part of this work was written.
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