SHARP ENTROPY DISSIPATION BOUNDS AND
EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR
THE SPATIALLY HOMOGENEOUS BOLTZMANN
EQUATION
G. TOSCANI AND C. VILLANI
Abstract. We derive a new lower bound for the entropy dissipation associated with the spatially homogeneous Boltzmann equation. This bound is expressed in terms of the relative entropy with
respect to the equilibrium, and thus yields a differential inequality
which proves convergence towards equilibrium in relative entropy,
with an explicit rate. Our result gives a considerable refinement
of the analogous estimate by Carlen and Carvalho [9, 10], under
very little additional assumptions. Our proof takes advantage of
the structure of Boltzmann’s collision operator with respect to the
tensor product, and its links with Fokker-Planck and Landau equations. Several variants are discussed.
Contents
1. Introduction
2. Preliminaries : Fokker-Planck and Landau equations
3. Symmetries for Boltzmann and Fokker-Planck equations
4. Integral representation of a lower bound for D
5. Main result
6. Extension to other kernels
7. The Kac model
8. Remarks about Fisher information and entropy dissipation
References
1
2
11
18
23
28
35
37
42
45
2
G. TOSCANI AND C. VILLANI
1. Introduction
This paper deals with the spatially homogeneous Boltzmann equation,

∂f


t ≥ 0, v ∈ RN (N ≥ 2),
 (t, v) = Q(f, f )
∂t
(1)



f (0, ·) = f0 ,
f0 ≥ 0, f0 ∈ L1 (RN ).
The unknown f stands for the probability density of particles in the
velocity space. Q is the so-called Boltzmann collision operator,
Z
Z
³
´
0 0
dv∗
dω B(v − v∗ , ω) f f∗ − f f∗
(2)
Q(f, f ) =
RN
S N −1
where dω is the normalized measure on S N −1 , f 0 = f (v 0 ), and so on,
and

0

v = v − (v − v∗ , ω)ω,
(3)

v 0 = v − (v − v , ω)ω
∗
∗
∗
are the postcollisional velocities of two particles that collide with respective velocities v and v∗ , according to the laws of elastic collision

0
0

v + v∗ = v + v∗
(4)

|v 0 |2 + |v 0 |2 = |v|2 + |v |2 .
∗
∗
(We denote by (a, b) = a · b the scalar product in RN .) On physical
grounds, it is assumed that the nonnegative kernel B(z, ω) (the “cross
section”) depends only upon |z| and (z/|z|, ω). Typical examples are
the (three-dimensional) hard spheres collision kernel
(5)
BHS (z, ω) = |z · ω|,
or more generally the kernels associated to the so-called hard potentials
with cut-off,
(6)
BHP (z, ω) = |z|γ b(α)
where 0 < γ ≤ 1, α ∈ [0, π] is the angle between z and ω, and b ∈
L1 (0, π). For γ < 0, we speak of soft potentials with cut-off; for γ = 0,
we speak of Maxwellian potential with cut-off. More generally, if B
depends only on (z/|z|, ω), we speak of Maxwellian potential. We refer
to [42, 16] for a detailed discussion of other models.
The Boltzmann equation is one of the most popular models in nonequilibrium statistical physics. Soon after its introduction by Maxwell [33],
SHARP ENTROPY DISSIPATION BOUNDS
3
Boltzmann deduced from it the celebrated H-theorem, namely that the
entropy
Z
(7)
H(f ) =
f log f
RN
of any solution f to (1) is nonincreasing with time. From this fact
he gave plausible arguments for these solutions to converge towards a
definite equilibrium state as t goes to infinity (see [7] for instance). Let
us recall them briefly.
Let ϕ(v) be any function of the velocity variable. Multiplying (1) by
ϕ and integrating, we obtain
Z
Z
Z
³
´
(8)
Q(f, f )ϕ =
dv dv∗
dω B f 0 f∗0 − f f∗ ϕ,
R2N
S N −1
where for simplicity we omit the arguments of B. For fixed ω, the
transformation
(9)
Tω : (v, v∗ ) 7−→ (v 0 , v∗0 )
is involutive and has unit Jacobian. Using this change of variables, and
also the transformation
(10)
R : (v, v∗ ) 7−→ (v∗ , v),
we easily obtain
Z
Z
³
´³
´
1
(11) − Q(f, f )ϕ =
dv dv∗ dω B f 0 f∗0 −f f∗ ϕ0 +ϕ0∗ −ϕ−ϕ∗ .
4
Choosing ϕ = 1, vi (1 ≤ i ≤ N ), |v|2 , by (4) we deduce that (11)
vanishes, and hence that the quantities ρ > 0 (mass), u ∈ RN (mean
velocity), T > 0 (temperature) defined by
(12) Z
ρ = f (v) dv,
Z
ρu =
Z
f (v)v dv,
2
ρ(|u| + N T ) =
f (v)|v|2 dv
are preserved with time. Now, choosing ϕ(v) = log f (v) in (11), we get
Z
(13)
− Q(f, f ) log f = D(f )
where D(f ) is the entropy dissipation,
Z
´
³
f 0 f∗0
1
dv dv∗ dω B(v − v∗ , ω) f 0 f∗0 − f f∗ log
.
(14)
D(f ) =
4
f f∗
4
G. TOSCANI AND C. VILLANI
Since (x, y) 7→ (x − y) log(x/y) is a nonnegative function, so is D(f ).
Moreover, at least if B > 0 a.e., D(f ) vanishes if and only if for almost
all v, v∗ , ω,
(15)
f 0 f∗0 = f f∗ .
Boltzmann proved that if f is smooth, the equality (15) implies (with
the notations of (12))
|v−u|2
ρ
− 2T
(16)
f (v) =
e
≡ M f (v).
(2πT )N/2
Such distributions are called Maxwellian.
The assumption of smoothness was later proved to be unessential
(see [17] and the references therein; see also the proof of Perthame [36]
relying on Fourier transform). Lions also gave a direct proof [31]
that (15) implies f ∈ C ∞ (RN ) as soon as f ∈ L1 .
As a conclusion, if f (t) is a solution of (1), then H(f (t)) is strictly
decreasing with time unless f = M f , and this strongly suggests that
(17)
f (t) −−−→ M f
t→∞
in some sense. By the way, the fact that M f is the only minimizer of
H in the class of functions satisfying (12) is a direct consequence of the
identity
Z
³
f
f
f ´
f
H(f ) − H(M ) =
M f 1 − f + f log f
M
M
M
RN
and the positivity of x 7→ 1 − x + x log x.
Once these formal arguments have been cast, it is very difficult to
go further and to prove that (17) actually holds, say in the sense of
the topology induced by the L1 norm. Results have been obtained by
several authors, in particular Arkeryd [3] and Wennberg [49], for hard
(or Maxwellian) potentials with cut-off. More precisely, they prove
(18)
kf (t) − M f k ≤ C(f0 )e−λt
for various weighted Lp norms (p ≥ 1), with a constant C(f0 ) depending on the initial datum. The rate λ is obtained by a compactness
argument, and therefore completely unknown. These results rely on
the study of the linearized Boltzmann equation and the rate is not
given by the entropy dissipation. But, as pointed out by Carlen and
Carvalho [9], such quantities as this rate λ, that are not explicitly computable, may be completely irrelevant from the physical point of view
(as is Poincaré’s recurrence theorem for statistical physics).
Another approach is to try to transform the assertion
D(f ) = 0 ⇐⇒ f = M f
SHARP ENTROPY DISSIPATION BOUNDS
5
into a quantitative result of the form
D(f ) ≥ d(f, M f )
where d is some suitable metric on a space of functions which is stable under the action of the Boltzmann equation (for instance, L12 (RN ),
the space of all L1 functions with finite moments of order 2). Desvillettes [19] was the first to obtain a result in this direction. His lower
bound reads
Z
∀R > 0
D(f ) ≥ KR inf
dv | log f (v) − log M (v)|
M ∈M
|v|≤R
where M is the set of all Maxwellian distributions, and f is assumed
to be bounded below by a fixed Maxwellian distribution. His proof
gives no indication on the way the constant KR (obtained by the use
of the open mapping theorem) depends on R. This result, proven for
a certain class of kernels, was extended by Wennberg [48] to cover the
physically realistic cases. Variants have also been derived for other
kinetic equations.
Another bound below was obtained by Gabetta and Toscani [24]
in the case of Kac’s model, which is a one-dimensional caricature of
Boltzmann’s equation (see section 7). Their bound reads
³
´
D(f ) ≥ θf I(f ) − I(M f )
where θf is a constant depending on f in a complicated way, and
Z
Z
p
|∇f |2
(19)
I(f ) =
= 4 |∇ f |2
f
is the so-called Fisher information of f . Again, the proof does not yield
any indication on the way θf varies with f , and hence the result cannot
be exploited for the trend towards equilibrium.
Some years ago, Carlen and Carvalho [9] were able to derive the first
inequality of the form
¡
¢
(20)
D(f ) ≥ Φ H(f |M f ) ,
where
(21)
H(f |M f ) = H(f ) − H(M f )
is the relative entropy of f with respect to M f , and Φ is a nonnegative
function, strictly increasing from 0 (but very slowly). Φ depends on
a few qualitative properties of f0 (essentially, the existence of a finite
moment of order higher than 2). Though Φ is implicitly defined, its
construction is entirely explicit, and in particular for all ε > 0 one can
6
G. TOSCANI AND C. VILLANI
compute η > 0 such that Φ(η) ≥ ε. The result holds for kernels that
are bounded below, in the sense
B(z, ω) ≥ ν > 0.
In a second paper [10], Carlen and Carvalho adapted their analysis to
the case of the hard spheres potential, under some L∞ -type assumptions on f .
As an immediate consequence of (13) and (21), their result implies
³
´
d
(22)
− H(f |M f ) ≥ Φ H(f |M f )
dt
and this is enough to conclude that
(23)
H(f ) −−−→ H(M f ).
t→∞
But, by the Csiszar-Kullback inequality [18, 29],
p
(24)
kf − M f kL1 ≤ 2H(f |M f ),
and therefore (23) implies at once that f (t) goes towards M f in L1
norm. Moreover, given any number ε > 0 and an initial datum f0 , one
can compute explicitly a time Tε (f0 ) such that kf (t) − M f kL1 ≤ ε for
t ≥ Tε (f0 ).
Several applications of this result have been given : in particular a
rigorous hydrodynamic limit in the large for a model equation in plasma
physics [11] and a proof of trend to equilibrium in weak sense for the
Boltzmann equation for hard spheres, in the case when H(f0 ) = ∞ [1].
Unfortunately, the function Φ given by Carlen and Carvalho is very
intricate (see [10], p. 754), which makes explicit computations rather
difficult, even when moments of high order are finite. Moreover, it is
not a priori clear that Φ possesses positive derivatives (even of high
order) near the origin, and hence the rate of return to equilibrium
predicted by (22) is very slow. It is therefore natural to ask whether
the inequality (22) can be found to hold for a simple function Φ that
grows not too slow (ideally, linearly) near the origin. In fact, in an
older paper, Cercignani [15] conjectured that
(25)
D(f ) ≥ λ(f0 )H(f |M f )
for some λ(f0 ) > 0 depending on the initial datum. Inequality (25)
would imply an exponential decay towards equilibrium. Bobylev [5]
proved that this conjecture cannot hold for Maxwellian molecules if
λ(f0 ) depends only on the mass, momentum and energy of f0 . Indeed,
he exhibited a family of initial data with the same moments up to order 2, for which the trend to equilibrium can be as slow as desired.
SHARP ENTROPY DISSIPATION BOUNDS
7
Wennberg [51] arrived to the same conclusion in the case of hard potentials (6) with 0 < γ ≤ 1, by a direct study of D(f ). Finally, in a
very recent work [6], Bobylev and Cercignani proved the inequality (25)
to be false, for all realistic potentials, even for functions that have an
arbitrary high number of moments close to the equilibrium value, and
are very smooth and bounded below by a given Maxwellian. They conjecture that the only reasonable spaces of functions in which (25) may
hold would be Lp spaces with an inverse Maxwellian weight. A good
theory of existence in such spaces is by very far out of reach at the
moment.
In this paper, we shall derive a new bound of the form (20), with a
function Φ which is at the same time much more simple and increasing
faster near the origin. Namely
(26)
D(f ) ≥ Cε (f ) H(f |M f )1+ε
where Cε depends only on the cross-section, the quantities (12), and
Z
(27)
kf kL1r log L ≡
dv f (v)| log f (v)|(1 + |v|2 )r/2 ,
RN
(28)
kf kL1s ≡
Z
R
f (v)(1 + |v|2 )s/2
N
for some r(ε) > 2, s(ε) > 4 that we shall compute, and K, A such that
(29)
∀v ∈ RN ,
2
f (v) ≥ Ke−A|v| .
We note that the result by Carlen and Carvalho is slightly more
general than ours, since in its most general version it does not require
bounds in L1r log L ∩ L1s but only in L log L ∩ L1s .
Our conclusion holds for all kernels B such that
¶N −2
µ
|z · ω|
(30)
B(z, ω) ≥ ψ(|z|)
|z|
for some smooth function ψ > 0 decaying at most algebraically at
infinity. Consider for instance the case when
ψ(|z|) = (1 + |z|)γ ,
0 < γ ≤ 1.
In this framework, Gustafsson [28] proved uniform (in time) boundedness of the solution to (1) in weighted Lp spaces (p > 1), from
which uniform bounds in L1s log L are easily extracted. In addition,
it is known that one can choose fixed K and A, depending only on
the mass, energy and entropy of f0 , such that (29) holds for f (t, ·) as
soon as t ≥ t0 > 0 (see [37]). Consequently, our result implies that
for this family of kernels, solutions to the Boltzmann equation decay
8
G. TOSCANI AND C. VILLANI
towards the equilibrium in L1 norm like t−1/ε , for all ε > 0, as soon as
f0 ∈ ∩s>0 Lps . In fact, it would be very likely that the norms of f in
L1s log L become finite for all positive time as soon as the initial entropy
is finite, as it happens for the moments [51].
We emphasize that the bound (26) is optimal, in the sense that,
under our assumptions, the result simply does not hold for ε = 0, as
shown by the previous discussion of Cercignani’s conjecture.
In the case where condition (30) fails to be true, we do not recover
such a strong result as (26). Yet if the set of (z, ω) such that (30) is
violated is of small measure, it is easy to adapt our method and get an
explicit algebraic bound of the form D(f ) ≥ CH(f |M )α . In particular,
this is true for all hard potentials (see section 6).
Up to our knowledge, this is the first result of algebraic decay with
explicit bounds available for the Boltzmann equation. But we wish to
point out that the inequality (26) is more interesting than just a statement concerning solutions of the spatially homogeneous Boltzmann
equation : it is a general functional inequality, that could be applied
in any context, in particular the spatially inhomogeneous Boltzmann
equation, if suitable estimates on the solutions were known.
In addition, our work suggests simple obstructions for (25) not to
hold, linked to the tails of the distribution f , that are generally found
to be the most severe obstacle in rigorous proofs of decay to equilibrium
(Cf. [5] for instance). Also, the comparison of our results to the ones
obtained by Desvillettes and the second author in [20, 21], is a clear
illustration of the physical fact that the tails of distribution may be
an obstacle to the trend to equilibrium in the case of the Boltzmann
equation (which is a kind of jump process), but not in the case of
related diffusion-type equations.
Our approach is based upon several tools that have been known for
a more or less long time in the context of the Boltzmann equation.
The first one is the regularization by the so-called adjoint OrnsteinUhlenbeck semigroup, i.e. the semigroup (St ) generated by the linear
Fokker-Planck equation,
(31)
∂t f = ∇ · (T ∇f + f (v − u)),
u ∈ RN ,
T > 0.
The class of distribution functions satisfying (12) is invariant by this
semigroup. Moreover, solutions of (31) are smooth for all positive
time, and converge towards M f . Therefore, this semigroup gives a
very convenient interpolation between f and M f . It plays a central
role in the proof of Carlen and Carvalho, and also in our analysis.
Our second tool is the use of the tensorial structure of the Boltzmann
equation, and in particular the fact that the entropy dissipation (14) is
SHARP ENTROPY DISSIPATION BOUNDS
9
a convex function of the tensor product f f∗ . In fact, most of our study
will take place in R2N , and it is only in the end that we shall go back
to the one-variable N -dimensional space.
Our third tool, going back at least to Boltzmann, and systematically
used by Desvillettes [19], consists in the introduction of linear operators
that “kill” functions with some symmetries. Typical examples are
¶
h
i µ
|v|2 + |v∗ |2
(32)
(v − v∗ ) ∧ (∇ − ∇∗ ) U v + v∗ ,
= 0,
2
used by Boltzmann to prove that smooth solutions of (15) are Maxwellian
(Cf. [19]), or
µ 2
¶· µ
¶
µ
¶¸
∂
∂2
x+y
x−y
−
(33)
ϕ √
+ϕ √
= 0,
∂y 2 ∂x2
2
2
arising in the context of rescaled convolution (see for instance the work
by Carlen [8] about the cases of equality in the logarithmic Sobolev
inequality).
The last main ingredient of our proof is the so-called Landau collision
operator,
µ
¶
Z
∂
∂f
∂f∗
(34)
QL (f, f ) =
dv∗ aij (v − v∗ ) f∗
−f
,
∂vj
∂vj
∂v∗j
where the symmetric matrix function aij is defined by
¶
µ
zi zj
(35)
aij (z) = δij − 2 Ψ(|z|)
|z|
for some nonnegative function Ψ. Here we have adopted the convention
of Einstein for implicit summation over repeated indices. The collision
operator (34) bears much resemblance with (2), and is in fact obtained
from it by a suitable asymptotic regime (see [43] and the references
therein).
None of these tools is new; but the main feature of our study is the
way they are combined all together.
For the sake of completeness, we shall recall in the next section all
the material concerning equations (31) and (34) that will be needed in
the sequel.
Let us end this introduction with some comments on entropy dissipation methods. These have proved to be very robust and apply
to various contexts, in particular diffusion-type equations ([39], [2]),
where sharp rates of exponential decay towards equilibrium in relative
entropy have been derived for linear and weakly nonlinear models. In
the context of the Boltzmann equation, one of their essential features
10
G. TOSCANI AND C. VILLANI
is monotonicity : namely, if Q1 and Q2 are two Boltzmann operators
of the form (2), with cross sections B1 ≥ B2 , then, in view of (14),
D1 (f ) ≥ D2 (f ).
The advantage of this property was noted by Carlen and Carvalho :
in view of it, all the results that are obtained in an algebraically simplified framework for a peculiar cross section B2 , automatically extend
to all cross sections B1 ≥ B2 .
Carlen, Gabetta and Toscani [12] recently proved exponential convergence to equilibrium with an explicit rate for the Boltzmann equation
with Maxwellian molecules. The rate depends on both the tails and
the smoothness of the solution, and is essentially optimal. But their
method relies on Fourier analysis, and hence seems very difficult to
extend to other kernels. There, the proof makes use of a Lyapunov
functional (distance) introduced in [26], defined in terms of the Fourier
transform, which plays the role of the classical entropy. We note that
the monotonicity properties of this entropy functional have been recently used to give a proof of uniqueness of the solution to (1) for true
Maxwell molecules [41].
The organization of the paper is as follows. Section 2 is devoted
to some preliminary material concerning the linear Fokker-Planck and
the Landau equation. In section 3, we study some symmetry properties enjoyed by the Boltzmann and the Fokker-Planck equation. In
section 4, we establish an integral representation of a lower bound for
D(f ), based on regularization by (St ). In section 5 we state and prove
our main result, namely the bound below (26). In section 6, we show
how to extend our results to various models, including the hard potentials with cut-off. In section 7, we treat Kac’s model as a variant. In
section 8, we do some remarks about the procedure of regularization of
D by the semigroup (St ), and show how it can be linked to the decay of
the Fisher information along the solutions to the Boltzmann equation.
Acknowledgement : The main part of this work was done while
the second author was visiting the Mathematics Departement of the
University of Pavia. It is a pleasure for him to thank the whole Department for their kind hospitality. The first author acknowledges the
partial support of the National Council for Researches of Italy, Gruppo
Nazionale per la Fisica Matematica.
SHARP ENTROPY DISSIPATION BOUNDS
11
2. Preliminaries : Fokker-Planck and Landau equations
From now on, unless otherwise stated, we shall consider only nonnegative ditribution functions f satisfying the normalization
(36)
ρ = 1,
u = 0,
T = 1,
where ρ, u and T are defined by (12). This class of functions is invariant
by all the equations that we shall consider : Boltzmann, Fokker-Planck,
Landau and Kac. Moreover, we shall denote by M the corresponding
Maxwellian distribution,
(37)
M (v) =
Accordingly,
(38)
Z
H(f |M ) =
|v|2
1
− 2
e
.
(2π)N/2
f
f
log M = H(f ) − H(M ),
M
RN M
Z
¯³
v ´ p ¯¯2
¯
(39)
I(f |M ) = 4
f ¯ = I(f ) − I(M ),
¯ ∇+
2
RN
allRthese expressions being well-defined in [0, ∞]. Another form of (39)
is |∇ log(f /M )|2 f , at least when some smoothness is available for f .
Let us recall the basic properties of H(·|M ) and I(·|M ). We refer
to [13, 14] and the references therein for complete proofs of the assertions below, and more material about the Fokker-Planck equation.
Both the relative entropy and the relative Fisher information are
strictly convex, weakly lower semicontinuous (for the L1 topology for
instance) functionals. For the relative entropy, this can be seen directly
by the Legendre-type representation [22, 13]
µZ
¶¾
½Z
ϕ
f ϕ − log
e M
,
(40)
H(f |M ) = sup
where the supremum is taken for instance over all continuous bounded
functions ϕ.
The basic link between H(·|M ) and I(·|M ) is given by the action
of the adjoint Ornstein-Uhlenbeck semigroup, (St )t≥0 , which can be
defined as the semigroup associated to the Fokker-Planck equation.
With the conventions (36), the Fokker-Planck equation simply reads
(41)
∂t f = ∇ · (∇f + f v) ≡ Lf.
The explicit solution of this equation is well-known, and thus
(42)
St f = fe−2t ∗ M1−e−2t ,
12
G. TOSCANI AND C. VILLANI
where we use the notation
(43)
gλ (v) =
1
λN/2
µ
g
v
√
λ
¶
.
Note that M is invariant by the action of this semigroup.
It is well known that t 7−→ St f is continuous for the strong L1
topology. Moreover, assuming that H(f ) < ∞, the function t 7−→
H(St f |M ) belongs to C([0, ∞)) ∩ C 1 (0, ∞), and for all t > 0,
d
H(St f |M ) = −I(St f |M ).
dt
In particular, H(St f |M ) is decreasing with t. What is less known is
the fact that I(St f |M ) is also decreasing. The rate of decay has been
found in [39] :
(44)
(45)
I(St f |M ) ≤ I(f |M )e−2t .
This inequality can also be considered as a direct consequence of the
so-called Blachman-Stam inequality (Cf. [13]), and follows as well by
the important work of Bakry and Emery, based on the so-called Γ2
calculus [4].
We shall use this decay property in the study of the entropy production for Kac equation. Moreover, St f −→ M in relative entropy and
in all Sobolev norms as t → ∞. More precisely [39],
(46)
H(St f |M ) ≤ H(f |M )e−2t .
As a consequence,
(47)
Z
∞
H(f |M ) =
dt I(St f |M ).
0
We shall also use the fact that the Fokker-Planck equation propagates moments. Namely, for any s > 0 and any datum f ∈ L1 , for all
t > 0,
¡
¢
(48)
kSt f kL1s ≤ max(1, 2s−1 ) e−st kf kL1s + (1 − e−2t )s/2 kM kL1s ,
This can be easily seen by remarking that (42) is nothing than the
probability density of the random variable
Xt = e−t X + (1 − e−2t )1/2 W
where X has density f , and W is a normalized Gaussian variable independent of X. Consequently,
¤
£
kSt f kL1s = E (1 + Xt2 )s/2 ,
SHARP ENTROPY DISSIPATION BOUNDS
13
E denoting mathematical expectation. On the other hand,
h
¡
¢2 is/2
1 + e−t X + (1 − e−2t )1/2 W
≤
£ −t
¤s
e (1 + X 2 )1/2 + (1 − e−2t )1/2 (1 + W 2 )1/2
and (48) follows from the inequality (x2 +y 2 )s ≤ max(1, 2s−1 )(x2s +y 2s ).
For the moments of order 2, more can be said : all the quantities of
the form
Z
dv (St f )(v)vi vj
Pij (St f ) =
RN
behave monotonically and converge exponentially fast to their equilibrium value δij as t → ∞.
Finally, the semigroup (St ) has smoothing effects : for all t > 0, St f
is C ∞ and such that
| log St f (v)| ≤ Ct (1 + |v|2 )
(49)
for some constant Ct depending on t. A proof of (49) can be found
in [9].
Let us now prove a less known propagation property.
Proposition 1. Let s > 0, ε > 0, and let f ≥ 0 such that kf kL1s+ε < ∞
and kf kL1s log L < ∞. Then there exists Cs depending only on s, ε and
kf kL1s log L , kf kL1s+ε , such that for all t > 0,
kSt f kL1s log L ≤ Cs .
Proof. Here as in the sequel, we will denote by C, Cs various constants.
We note first that it suffices to obtain a uniform bound on
Z
Ls = dv f (v) log f (v)(1 + |v|2 )s/2 .
Indeed,
Z
Z
2 s/2
(50)
f | log f |(1 + |v| ) ≤ f log f (1 + |v|2 )s/2
Z
+2
f | log f |(1 + |v|2 )s/2 .
f ≤1
But, owing to the inequality x| log x| ≤ y − x log y, that holds for all
0 ≤ x ≤ 1, 0 < y ≤ 1, we see that if f ≤ 1, for any ε > 0
2 )ε/2
f | log f | ≤ e−(1+|v|
and this implies
Z
(51)
f ≤1
+ (1 + |v|2 )ε/2 f,
f | log f |(1 + |v|2 )s/2 ≤ ds,ε + kf kL1s+ε ,
14
G. TOSCANI AND C. VILLANI
where we put
Z
ds,ε =
RN
(1 + |v|2 )s/2 e−(1+|v|
2 )ε/2
dv.
Since the moments of order s + ε are uniformly propagated by (St ), it
is sufficient to consider the first integral in the right hand side of (50),
whence our claim.
For all t > 0, St f is smooth and | log f | is quadratically bounded.
Moreover, the mapping t 7→ Ls (St f ) is continuous (see related arguments in section 5) and continuously differentiable for t > 0. Let us
compute its derivative. In the sequel, we use the abridged notation
f = f (t) for St f . Integrating systematically by parts in all the integrals where only one derivative of f enters, we easily obtain
Z
Z
|∇f |2
2 s/2
∇ · (∇f + f v) log f (1 + |v| ) = −
(1 + |v|2 )s/2
f
Z
£
¤
+ f log f ∆(1 + |v|2 )s/2 − v · ∇(1 + |v|2 )s/2
Z
©
£
¤
ª
+ f ∇ · v(1 + |v|2 )s/2 − ∆(1 + |v|2 )s/2
and
Z
Z
2 s/2
∇ · (∇f + f v) (1 + |v| )
=−
f v · ∇(1 + |v|2 )s/2
Z
+ f ∆(1 + |v|2 )s/2 .
Hence
Z
Z
d
|∇f |2
2 s/2
f log f (1 + |v| ) = −
(1 + |v|2 )s/2
dt
f
·
¸
Z
Ns
s(s − 2)|v|2
s|v|2
2 s/2
+
−
+N kf kL1s .
+ f log f (1+|v| )
(1 + |v|2 )
(1 + |v|2 )2
(1 + |v|2 )
Now, by (51)
·
¸
Z
s(s − 2)|v|2
s|v|2
Ns
2 s/2
+
−
f log f (1 + |v| )
(1 + |v|2 )
(1 + |v|2 )2
(1 + |v|2 )
Z
Z
≤ (N s + s(s − 2))
f log f (1 + |v|2 )s/2 − s
f log f (1 + |v|2 )s/2
f ≥1
f ≤1
Z
h
i
≤ (N s+s(s−2)) f log f (1+|v|2 )s/2 +(N s+s(s−1)) ds,ε + kf kL1s+ε
SHARP ENTROPY DISSIPATION BOUNDS
15
and we obtain
Z
Z
p
d
2 s/2
f log f (1 + |v| ) ≤ −4 |∇ f |2 (1 + |v|2 )s/2 +
dt
Z
h
i
2 s/2
(N s+s(s−2)) f log f (1+|v| ) +(N s+s(s−1)) ds,ε + kf kL1s+ε +N kf kL1s .
Next, since
hp
i sp
p
f (1 + |v|2 )s/4 −
f v(1 + |v|2 )s/4−1 ,
∇ f (1 + |v|2 )s/4 = ∇
2
we write
Z
p
4 |∇ f |2 (1 + |v|2 )s/2 =
¶¯2
Z ¯ µp
Z
¯
¯
2
s/4
2
¯ +s
4 ¯¯∇
f (1 + |v| )
f |v|2 (1 + |v|2 )s/2−2
¯
¶
Z µp
p
v
2 s/4
− 4s ∇
f (1 + |v| )
f (1 + |v|2 )s/4
1 + |v|2
Z
³
´
= I f (1 + |v|2 )s/2 + s2 f |v|2 (1 + |v|2 )s/2−2
¶
µ
Z
v
2 s/2
,
+ 2s f (1 + |v| ) ∇ ·
1 + |v|2
√ √
where we have used the identity 2(∇ g) g = ∇g and integrated by
parts in the last integral.
By Gross’s logarithmic Sobolev inequality [27], written with respect
to the Lebesgue measure, for all functions g ∈ H 1 (RN ), and for any
a > 0,
¶Z
Z
³
´ µ
N
2
2
2
dv |g| log |g| /kgkL2 (RN ) + N + log 2πa
dv |g|2 ≤
2
N
N
R
ZR
≤ 2a
dv |∇g|2
RN
Hence, choosing a = [4N s + 4s(s − 2)]−1 we obtain
·Z
³
´
2 s/2
I f (1 + |v| )
≥ 8(N s + s(s − 2))
f log f (1 + |v|2 )s/2 +
Z
f (1 + |v|2 )s/2 log(1 + |v|2 )s/2 − kf kL1s log kf kL1s +
¶
¸
µ
N
−1
N + log[π(2N s + 2s(s − 2)) ] kf kL1s
2
16
G. TOSCANI AND C. VILLANI
Grouping together all the previous inequalities, we conclude that
Z
Z
d
2 s/2
f log f (1 + |v| ) ≤ −8(N s + s(s − 2)) f log f (1 + |v|2 )s/2 + C,
dt
where C depends on N, s, kf kL1s+ε in a explicitly computable way. By (48),
kf kL1s+ε is bounded uniformly in t, and this implies a uniform bound
for Ls (f (t)).
¤
All these properties would suffice to ensure that (St )t≥0 gives a very
convenient way of smoothing densities in the frame of the Boltzmann
equation. This becomes still clearer in view of the work by Morgenstern [35] upon the case of Maxwellian potentials with cut-off. For
these potentials, under a suitable normalization of B, the Boltzmann
equation simply reads
Z
+
(52)
∂t f = Q (f, f ) − f = dv∗ dσ b(k · σ)f 0 f∗0 − f
where k = (v − v∗ )/|v − v∗ |. First Morgenstern in dimension two,
and subsequently Bobylev in any dimension of the velocity variable [5]
proved that if Q+ is given by (52), then for all δ > 0,
(53)
Q+ (f ∗ Mδ , f ∗ Mδ ) = Q+ (f, f ) ∗ Mδ .
Since Q+ also commutes with the rescaling (43), it follows that
Q+ (St f, St f ) = St Q+ (f, f ),
and as a consequence that the semigroup induced by the Boltzmann
equation with Maxwellian molecules commutes with the adjoint OrnsteinUhlenbeck semigroup. This property was crucial in the analysis of [9]
(see also [47]).
Several other symmetry properties connecting the Boltzmann equation and the Fokker-Planck equation will be studied in the next section
(one can safely assume that we overlooked many others).
The Boltzmann equation with Maxwellian molecules enjoys many
remarkable properties, reminiscent of the Fokker-Planck equation. In
particular, I(f |M ) is decreasing along its solutions. This was proven
in [38] for N = 2, in [9] in the case when b is constant, and in [47] in
the general case. As we shall see in section 8, this decreasing property
can be related to the problem of finding a lower bound for D(f ). This
point had already been noted (by a different argument) by Carlen and
Carvalho.
These peculiar properties of Maxwellian molecules may be somewhat
enlightened (or obscured !) by the study of the so-called asymptotics
of grazing collisions. These are a limiting process under which the
SHARP ENTROPY DISSIPATION BOUNDS
17
Boltzmann equation transforms into a nonlinear diffusion-type equation, called Landau (or sometimes Fokker-Planck !) equation. This
limit was discovered from the formal point of view by Landau [30] in
the study of the plasmas, in the frame of the Coulombian potential.
From the mathematical point of view, a very wide class of potentials
can be considered. We refer to [43] for a detailed study, and further references on the subject. See also [45] for a rigorous variant of Landau’s
original argument.
It turns out that in the case of Maxwellian molecules, the corresponding Landau equation resembles very much the Fokker-Planck equation.
In [44] the following representation was established :
X
(54)
∂t f =
(N − Ti )∂ii f + (N − 1)∇ · (f v) + ∆S f,
i
where an orthonormal basis (ei ) has been chosen such that
Z
(55)
f (v)vi vj dv = δij Ti ,
RN
which is always possible by diagonalization of the quadratic form
Z
q : e 7−→ f (v) (v, e)2 dv
(v ∈ RN ).
The condition (55) is preserved by the equation (54). Ti will be called
the directional temperature of f along the direction ei . Here ∆S denotes the Laplace-Beltrami operator,
´
X³
2
|v| − vi vj ∂ij f (v) − (N − 1) v · ∇f (v).
∆S f (v) =
ij
In particular, in the case of radial distributions, the Landau equation (54) coincides (up to a multiplicative factor N −1) with the FokkerPlanck equation.
The questions of existence, uniqueness, asymptotic behaviour and
some qualitative properties of solutions to equation (54) have been
addressed in [44].
The Landau equation with Maxwellian molecules shares many properties with both the Boltzmann equation and the Fokker-Planck equation (see [46] for instance). We shall be essentially interested in the
associated entropy dissipation,
(56)
Z
p
p
DL (f ) = 2 dv dv∗ |v − v∗ |2 Π(v − v∗ )(∇ − ∇∗ ) f f∗ (∇ − ∇∗ ) f f∗ ,
18
G. TOSCANI AND C. VILLANI
where ∇∗ denotes the gradient with respect to the variable v∗ , and Π(z)
is the orthogonal projection upon the space orthogonal to z,
µ
¶
zi zj
(57)
Πij (z) = δij − 2
|z|
(we use the standard notation Axx = Aij xi xj ).
We refer to [21] for a study of the functional DL . All that we shall
use here is the inequality
(58)
DL (f ) ≥ min (N − Ti ) I(f |M ) = (N − Tf )I(f |M ),
1≤i≤N
where
(59)
Z
Tf = max
e∈S N −1
RN
dv f (v) (v, e)2 .
The inequality (58), established by Desvillettes and the second author,
is clearly reminiscent of the formula (44). See [21] for complete proofs,
and many applications to the trend towards equibrium in the general
case of the Landau equation with hard potentials.
Our study will reveal an unexpected connection between the functionals D and DL , and the semigroup (St ), which will allow us to derive
a bound below for D, starting from the bound (58). To establish this
connection, we have to study more precisely the symmetries of the
equations.
3. Symmetries for Boltzmann and Fokker-Planck
equations
In this section, we give the symmetry properties that will make it
possible to regularize D by the Ornstein-Uhlenbeck semigroup. We
begin with an equivalent representation of (2), obtained by the classical
change of variables (v, v∗ , ω) → (v, v∗ , σ), such that

v 0 = v + v∗ + |v − v∗ | σ



2
2
(60)



v 0 = v + v∗ − |v − v∗ | σ.
∗
2
2
In these variables,
Z
Z
´
³
e − v∗ , σ) f 0 f∗0 − f f∗ ,
(61)
Q(f, f ) =
dv∗
dσ B(v
RN
S N −1
e − v∗ , σ) = (2|k · ω|)N −2 B(v − v∗ , ω), and the notation
where B(v
v − v∗
k=
|v − v∗ |
SHARP ENTROPY DISSIPATION BOUNDS
19
will be systematically used in the
R sequel.
R We recall that dω and dσ are
normalized in such a way that dω = dσ = 1.
We also set
(62)
X = (v, v∗ ),
X 0 = (v 0 , v∗0 ) = Tω X = Uσ X
where Uσ is associated to the transformation (60). Note that for fixed
σ, Uσ : R2N → R2N is not bijective. For any function G(X), we write
t
Tω G = G ◦ Tω ,
t
Uσ G = G ◦ Uσ .
We now state a lemma which is due to Boltzmann himself.
Lemma 1. Let f ∈ L1 (RN ). Then the average
Z
dσf 0 f∗0
(63)
G(v, v∗ ) =
S N −1
depends only on m = v + v∗ and e = (|v|2 + |v∗ |2 )/2. More generally,
this result holds for any average of the form
Z
(64)
G(X) =
dσ t Uσ F (X),
S N −1
where F ∈ L1 (R2N ).
The proof is immediate, since, in view of (60), the average (63)
depends only on the sphere with center q = (v + v∗ )/2 and radius
r = |v − v∗ |/2. The set of (N + 1) scalar variables (q, r) is clearly
equivalent to (m, e). We note that G ∈ L1 (R2N ) since for any ϕ ∈ L∞ ,
¯Z
¯ ¯Z
¯
¯
¯ ¯
¯
t
¯
¯
¯
¯
¯ 2N Gϕ¯ = ¯ 2N N −1 dσ dX Uσ F (X)ϕ(X)¯
R
R ×S
¯Z
¯
Z
¯
¯
t
¯
=¯
dX F (X) dσ Uσ ϕ(X)¯¯ ≤ kF kL1 kϕkL∞ .
R2N
Now, the heart of the matter lies in the following property. We
denote by T the operation of tensor product, and by A the average
operation (64). Moreover, we use the same symbol St for the action of
the adjoint Ornstein-Uhlenbeck semigroup in RN and in R2N . When
no confusion is possible, we also use the symbol M for the Maxwellian
in 2N variables : M (X) = M (v)M (v∗ ).
Proposition 2. The diagram
R
A
T
f −−−→ F = f f∗ −−−→ G = dσf 0 f∗0



S
S
S
(65)
y t
y t
y t
T
St f −−−→
is commutative.
St F
A
−−−→
St G
20
G. TOSCANI AND C. VILLANI
Remark. Let DN be the set of functions in L1 (RN ) satisfying conditions (36). Then T maps DN into D2N , and A maps D2N into itself.
Proof. We first prove that the action of St commutes with the tensorization T . Since St is the composition of a convolution by a Maxwellian
distribution Mλ(t) and a rescaling of the velocity space, it is sufficient
to check the property for these two operations. Since for all µ > 0,
µX = (µv, µv∗ ), obviously (f f∗ )λ = (fλ )(fλ )∗ , which proves the second
part of the proposition, and shows at the same time that we only need
to consider the convolution by M instead of Mλ . On the other hand,
(66)
M (X) = M (v)M (v∗ ),
and this directly implies that for all function g,
(M ∗ g)(M ∗ g)∗ = M ∗ (gg∗ ).
We now prove that in R2N , St commutes with A. Since X 7−→
Uσ (X) is homogeneous of degree one, the rescaling (43) commutes with
A. Therefore, we just have to check that A also commutes with the
convolution by M .
Let us set
q=
w + w∗
,
2
r = |w − w∗ |,
`=
w − w∗
.
|w − w∗ |
Thus,
r
w = q + `,
2
r
w∗ = q − `,
2
r
w0 = q + σ,
2
r
w∗0 = q − σ.
2
Then, for any function F (v, v∗ ),
³Z
(67) M ∗
t
dσ Uσ F
Z
´
=J
³
r
r ´
dq rN −1 dr d` dσ F q + σ, q − σ
2
2
³
r
r ´
M v − q − `, v∗ − q + ` ,
2
2
where J denotes some Jacobian (remember that dσ is the normalized
measure on S N −1 ). On the other hand,
Z
(68)
Z
t
dσ Uσ (M ∗ F ) = J
dq r
r ´
r
dr d` dσ F q + `, q − `
2
2
³
r ´
r
M v 0 − q − `, v∗0 − q + ` .
2
2
N −1
³
SHARP ENTROPY DISSIPATION BOUNDS
21
Exchanging ` and σ in (67), we see that we only have to prove that
for all q, r, `, v, v∗ ,
Z
r
r
dσ M (v − q − σ, v∗ − q + σ)
(69)
2
2
µ
¶
Z
r v + v∗ |v − v∗ |
r
v + v∗ |v − v∗ |
= dσ M
+
σ − q − `,
−
σ−q+ ` .
2
2
2
2
2
2
Changing v into v − q and v∗ into v∗ − q, we reduce to the case when
q = 0. Using now the property
µ
¶ µ
¶
z + z∗
z − z∗
√
√
(70)
M (z, z∗ ) = M
M
,
2
2
√
we let M ((v + v∗ )/ 2) appear as a multiplicative factor of both sides,
and we just have to prove that
¶ Z
¶
µ
µ
Z
v − v∗ − rσ
|v − v∗ |σ − r`
√
√
dσ M
= dσ M
.
2
2
Up to a constant, the left hand side is
Z
|v−v∗ |2
r2
r
−
−
4
e 4
dσ e− 2 (v−v∗ )·σ ,
e
while the right hand side is
−
e
|v−v∗ |2
4
2
Z
− r4
e
Since, by rotational invariance,
S N −1 , the conclusion follows.
r
dσ e− 2 |v−v∗ |`·σ .
R
dσe−r`·σ does not depend on ` ∈
¤
Corollary 2.1. Let G(X) depend only on m = v + v∗ and e = (|v|2 +
|v∗ |2 )/2. Then, for all t > 0, St G depends only on m and e.
In fact, a direct proof of this corollary is immediate : by density
and linearity, it is sufficient to consider only the case when G(X) =
G1 ((v +v∗ )/2)G2 (|v −v∗ |/2). In view of (70), St = (St G1 )(St G2 ). Since
G2 is radial by assumption, and since M is radial, so is St G2 , which
completes the proof.
In the case N = 2, more can be said. In the representation (61),
one can take as new variable the angle θ between σ and k. Thus the
transformation X 7−→ X 0 can be seen as a rotation in R2 , or, more
precisely
¶
µ
¶
µ
v + v∗
v + v∗
, v − v∗ 7−→
, Rθ (v − v∗ )
(71)
2
2
22
G. TOSCANI AND C. VILLANI
where Rθ denotes the standard rotation by angle θ in oriented R2 . By
extension, we shall denote by Rθ the application given by (71).
Proposition 3. Assume N = 2. Then for each θ ∈ R/(2πZ), the
diagram
tR
T
(72)
f −−−→ F = f f∗ −−−θ→ t Rθ F = f 0 f∗0 = F 0



S
S
S
y t
y t
y t
T
St f −−−→
tR
−−−θ→
St F
(St F )0
is commutative. In short, St (f 0 f∗0 ) = (St f )0 (St f )0∗ .
Proof. It suffices to note that
Z
t
M ∗ ( Rθ F )(X) =
=
=
=
ZR
dY t Rθ F (Y )M (X − Y )
2N
2N
ZR
ZR
2N
R2N
dY F (Rθ Y )M (X − Y )
dY F (Y )M (X − Rθ−1 Y )
dY F (Y )M (Rθ X − Y )
= (M ∗ F )(Rθ X),
where we have used the fact that a rotation has unit Jacobian, and
that M is invariant under t Rθ .
¤
The particular character of the dimension 2 was already noticed in
related problems [38, 47]. Unfortunately, it is difficult to see how this
property could generalize to higher dimensions. For example, in the
case N = 3, even if one chooses a system of spherical coordinates
(r, θ, φ) with axis k, there is no canonical way to choose the coordinate
φ, and it is not clear whether this can be done in such a manner that
φ be well defined independently of k. The analog of proposition 3 is
however valid if one replaces Rθ by
Z 2π
Jθ : F 7−→
dφ t Uσ F,
0
where in the right hand side the coordinates of σ in the local spherical
system of axis v − v∗ are (θ, φ).
We conclude this section by noting that a similar lemma holds with
the transformations Tω .
SHARP ENTROPY DISSIPATION BOUNDS
23
Proposition 4. For each ω ∈ S N −1 , the diagram
tT
T
(73)
f −−−→ F = f f∗ −−−ω→ t Tω F = f 0 f∗0 = F 0



S
S
S
y t
y t
y t
T
St f −−−→
St F
tT
(St F )0
−−−ω→
is commutative.
The proof is the same as for proposition 3.
Remarks.
(1) The properties of invariance under t Rθ or t Tω characterize the
Maxwellian distribution. Hence no other convolution regularization than Maxwellian could yield the same conclusion.
(2) As pointed out to us by Desvillettes, these propositions give
an immediate proof that Maxwellian distributions are the only
solutions in L12 of the equation (15). Indeed, let f be such
a solution. Then, in view of proposition 4, so is St f for any
t > 0. Since St f is smooth, classical proofs relying for instance
on the “killing operator” (32) prove that St f is the Maxwellian
distribution M f . By the continuity of St at time 0, this is also
true for f .
4. Integral representation of a lower bound for D
In this section, we fix a cross section
¶N −2
µ
|z · ω|
,
(74)
B(z, ω) = ψ(|z|) 2
|z|
e σ) = ψ(|z|),
i.e. B(z,
with the variables (60). The entropy dissipation for the kernel B reads
Z
Z
³
´
1
f 0 f∗0
(75)
D(f ) =
dv dv∗ ψ(|v − v∗ |) dσ f 0 f∗0 − f f∗ log
.
4
f f∗
The assumptions on the nonnegative function ψ shall be precised later
on.
By Rthe joint convexity of the function (x, y) 7→ (x − y) log(x/y), and
since dσ = 1, we get
Z
F
1
dX ψ(|v − v∗ |)(F − G) log ≡ D(f ),
(76)
D(f ) ≥
4
G
where we use the notations (62), and
(77)
F (v, v∗ ) = f f∗ ,
G=
Z
dσ t Uσ F = AF.
24
G. TOSCANI AND C. VILLANI
Our aim here is to establish an integral representation for D(f ). To
this end, we shall regularize D by (St ) and compute (d/dt)D(St f ). At
first sight this seems a formidable job to do, since f appears no less
than eight times in (75). But applying proposition 2, we see that it is
equivalent to apply St to the functions F and G appearing in the right
hand side of (76).
For all positive time t > 0, St F and St G are smooth and their
logarithm is bounded by a quadratic expression. This is enough to
justify all the manipulations below. The following lemma will enable
us to compute very simply the time-derivative of D(St f ). It yields
actually the commutator between derivation along St and the function
(x, y) 7−→ (x − y) log(x/y), which is homogeneous of degree 1.
Proposition 5. Let F and G be smooth functions with logarithms
quadratically bounded. Then
(78)
¯
¯2
¯
¯
¯ ∇F
d ¯¯
St F
∇G
¯ +
(St F − St G) log
= −(F + G) ¯¯
−
¯
dt t=0
St G
F
G ¯
¯
¸
·
d ¯¯
F
.
St (F − G) log
dt ¯t=0
G
Proof. In the sequel, ∇ stands for ∇X . Let L denote the Fokker-Planck
operator in R2N , and let us compute
F
(79) L(F − G) log + (F − G)
G
µ
LF
LG
−
F
G
¶
·
¸
F
− L (F − G) log
.
G
We shall show that this expression is equal to the first term in the right
hand side of (78). Indeed, expanding (79), we find
³
´
F
∇ · ∇(F − G) + (F − G)X log +
G
µ
¶
∇ · (∇F + F X) ∇ · (∇G + GX)
(F − G)
−
F
G
µ
¶
·
¸
∇F
∇G
F
F
+X −
− X + (F − G)X log
−∇· ∇(F − G) log + (F − G)
G
F
G
G
SHARP ENTROPY DISSIPATION BOUNDS
25
·
¸
·
¸
F
F
=
∇ · ∇(F − G) log − ∇ · ∇(F − G) log
G
G
·
¸
·
¸
F
F
+
∇ · (F − G)X log − ∇ · (F − G)X log
G
G
¶
¶
µ
·µ
¸
F −G
F −G
+
∇ · (∇F + F X) − ∇ ·
(∇F + F X)
F
F
¶
·µ
¶
¸
µ
G−F
G−F
∇ · (∇G + GX) − ∇ ·
(∇G + GX)
+
G
G
µ
¶
∇F
∇G
= − ∇(F − G) ·
−
F
G
µ
¶
∇F
∇G
− (F − G)X ·
−
F
G
µ
¶
∇G G∇F
− (∇F + F X) · −
+
F
F2
µ
¶
F ∇G
∇F
+
− (∇G + GX) · −
.
G
G2
Expanding the last expression, we see that all the terms containing
X cancel out. As for the other ones, we obtain in the end
|∇F |2 |∇G|2
∇F · ∇G
∇G · ∇F
|∇F |2
|∇G|2
−
−
+2
+2
−G
−
F
F
G
F
G
F2
G2
¯
¯2
¯ ∇F
∇G ¯¯
= −(F + G) ¯¯
−
.
F
G ¯
¤
This relation may seem somehat miraculous. In section (8), we shall
try to connect it with other known properties in kinetic theory.
With proposition 5 at hand, it is immediate to compute the time
derivative of D(St f ). Let us write
L∗ : g 7−→ ∆X g − X · ∇X g,
for the adjoint of L. We note that
µ
g ∈ D0 (R2N )
ψ 0 (|v − v∗ |)
∆X ψ(|v−v∗ |) = 2∆v ψ(|v−v∗ |) = 2 ψ (|v − v∗ |) + (N − 1)
|v − v∗ |
¶
00
X·∇X ψ(|v−v∗ |) = v·∇v ψ(|v−v∗ |)−v∗ ·∇v ψ(|v−v∗ |) = |v−v∗ |ψ 0 (|v−v∗ |).
Hence we can safely use the notation
¶
µ
(N − 1)
∗
00
− |z| ψ 0 (|z|),
(80)
(L ψ)(|z|) = 2ψ (|z|) + 2
|z|
,
26
G. TOSCANI AND C. VILLANI
and by (76) and (78) we get for all t > 0,
(81)
¯
¯2
Z
¯ ∇St F
∇St G ¯¯
d
1
¯
D(St f ) = −
dX ψ(|v − v∗ |)(St F + St G) ¯
−
dt
4
St F
St G ¯
Z
1
St F
+
dX (L∗ ψ)(|v − v∗ |)(St F − St G) log
,
4
St G
provided that
St F
∈ L1 (R2N ),
St G
(82)
ψ(|v − v∗ |)(St F − St G) log
(83)
(L∗ ψ)(|v − v∗ |)(St F − St G) log
St F
∈ L1 (R2N ).
St G
We remark that, thanks to condition (49) and the result of Proposition 1, conditions (82) and (83) are propagated by St , under weak
assumptions on the growth of ψ and L∗ ψ. In more detail, we assume for simplicity that ψ is bounded below by a fixed number ν > 0,
F ψ ∈ L1 (R2N ), and
(84)
L∗ ψ(|v − v∗ |) ≤ Cψ(|v − v∗ |) ≤ C1 (1 + |X|2 )α/2 ,
for some α > 0. Note that the condition (84) is always satisfied if
ψ(|z|) behaves like a power of |z| at infinity, since then, the dominant
term as |z| → ∞ of L∗ ψ(|z|) is −|z|ψ 0 (|z|). Then, if
(85)
kf kL12+α log L < ∞,
both (82) and (83) hold uniformly in time for all t ≥ t0 > 0 (because the
Ornstein-Uhlenbeck adjoint semigroup generates pointwise Maxwellian
lower bounds).
Finally, we prove that continuity at time 0 of the mapping t 7→
D(St f ) follows under the same conditions (84) and (85).
First, by the convexity of D and the strong continuity of (St ) at
time 0,
D(f ) ≤ limt→0 D(St f ).
Therefore it is sufficient to check that D(f ) ≥ limt→0 D(St f ).
Let us denote by Mε the centered Maxwellian with temperature ε,
and define
(Gψ) ∗ Mε
(F ψ) ∗ Mε
,
Gε =
Fε =
ψ
ψ
Since ψ is bounded below, it is clear that (Fε , Gε ) −→ (F, G) strongly
in L1 × L1 as ε → 0, and (Fε ψ, Gε ψ) −→ (F ψ, Gψ) as well. We note
SHARP ENTROPY DISSIPATION BOUNDS
27
that
D(f ) = D(F ψ, Gψ),
where
1
D(F, G) =
4
Z
dX (F − G) log
F
.
G
Let us use for a while the notation D(f ) = D(F ). In view of the choice
of Fε , Gε ,
D(F ) ≤ limε→0 D(Fε ψ, Gε ψ) = D((F ψ) ∗ Mε , (Gψ) ∗ Mε ) ≡ D(Fε ).
But D is a translation-invariant functional, and (F ψ)∗Mε is an average
of translates of F ψ. By convexity of D,
³
´
D (F ψ) ∗ Mε , (Gψ) ∗ Mε ≤ D(F ψ, Gψ),
and hence
(86)
D(Fε ) → D(F )
as ε → 0.
By the preceding proof, since Fε , Gε are smooth (hence t 7−→ D(St f )
is continuous at t = 0), we see that for all t > 0,
Z
Z
1 t
Sτ F ε
D(Fε ) ≥ D(St Fε ) −
dτ (L∗ ψ)(Sτ Fε − Sτ Gε ) log
.
4 0
Sτ Gε
Hence, since by assumption L∗ ψ ≤ Cψ (and since Sτ Gε is an average
of Sτ Fε ),
Z
C t
D(Fε ) ≥ D(St Fε ) −
dτ D(Sτ Fε ).
4 0
By Gronwall’s lemma,
D(St Fε ) ≤ D(Fε )eCt/4 .
Letting ε go to 0, by the convexity of D and (86), we get
D(St F ) ≤ D(F )eCt/4 .
This is sufficient to conclude that limt→0 D(St F ) ≤ D(F ).
On the other hand, as t goes to infinity, D(St F ) goes to 0 (because
St F −→ M ), at least when
(87)
F log F ψ(1 + |X|2 ) ∈ L1 (R2N ).
But this follows by condition (85). Thus, we conclude with the following
28
G. TOSCANI AND C. VILLANI
Theorem 6. Representation formula for D. Assume that ψ is
bounded below, and conditions (84), (85) are satisfied at t = 0. Then
(88)
1
D(f ) =
4
Z
¯
¯2
¯ ∇St F
¯
∇S
G
t
¯
dt dX ψ(|v − v∗ |)(St F + St G) ¯¯
−
St F
St G ¯
0
Z
Z
1 ∞
St F
−
dt
dX (L∗ ψ)(|v − v∗ |)(St F − St G) log
.
4 0
St G
∞
Z
Remark. We are confident that this formula holds under more general
assumptions, but this will be more than enough for our purposes.
5. Main result
We are now ready to prove our main result.
Theorem 7. Let f satisfy the normalization (36). Let ψ(|z|) ≥ (1 +
2
|z|)γ for some real number γ ≤ 0. Assume that f (v) ≥ Ke−A|v| ,
and that kf kL12+s+ε log L < ∞, kf kL14+s+ε < ∞ for some s > 0, ε > 0.
Then there exists a constant Cs (f ) depending only on s, ε, γ, K, A,
kf kL12+s+ε log L and kf kL14+s+ε , such that
(89)
D(f ) ≥ Cs (f ) H(f |M )1+
2−γ
s
,
where D(f ) is the lower bound for the entropy dissipation given by (76).
Remark. To treat potentials that are “essentially” bounded below, in
the sense of section 6, like hard potentials, it will be sufficient to apply
this theorem with γ = 0. However, we choose a function ψ which may
be decaying to show that the theorem also holds for soft potentials.
Proof. Let ψ1 be a smooth convex function with ψ1 (|z|) = 1 for |z| ≤ 1,
ψ1 (|z|) = |z|2 for |z| ≥ 2, |z|2 /2 ≤ ψ1 (|z|) ≤ 1 + |z|2 . Since L∗ (|z|2 ) =
4N − 2|z|2 , we can impose that
|L∗ ψ1 (|z|)| ≤ C(1 + |z|2 ).
Let us set ψR (|z|) = ψ1 (|z|/R). Hence,
¢
C ¡
|L∗ ψR (|z|)| ≤ 2 1 + |z|2 1|z|≥R .
R
Let DR (f ) be the functional D associated to ψR .
Since ψR (|z|) ≥ |z|2 /(2R2 ), and since
|v − v∗ |2 ≥ R2 =⇒ |v|2 + |v∗ |2 ≥
R2
,
2
SHARP ENTROPY DISSIPATION BOUNDS
29
we obtain by Theorem 6
(90)
¯
¯2
Z ∞ Z
¯ ∇X S t F
1
∇X St G ¯¯
2
¯
DR (f ) ≥
dt
dX |v−v∗ | (St F +St G) ¯
−
8R2 0
St F
St G ¯
Z ∞ Z
¡
¢
C
St F
− 2
.
dt
dX 1 + |X|2 1|X|≥R/√2 (St F − St G) log
R 0
St G
Next, since ψR (|z|) = 1 for |z| ≤ R, we can write
(1 + |z|)γ ≥ (1 + R)γ ψR (|z|) − (1 + R)γ
Accordingly,
(91)
·
γ
D(f ) ≥ (1 + R)
1
DR (f ) −
4R2
|z|2
1|z|≥R .
R2
Z
2
dX |X| 1|X|≥R/
√
¸
F
.
2 (F − G) log
G
Summing up, by (90) and (91) we have decomposed D(f ) into three
parts, two of which involve only large values of X. Now we shall estimate the principal part of DR (f ). The heart of the whole argument
lies in the following
Proposition 8. Let F = f f∗ and let G be a function depending only
on m = v + v∗ and e = (|v|2 + |v∗ |2 )/2. Assume that f and G are
smooth, with logarithms quadratically bounded, and that f satisfies the
normalization (36). Then
¯
¯2
Z
¯ ∇X F
¯
∇
G
X
2
¯ ≥ 1 (N − Tf ) I(f |M ),
(92)
dX |v − v∗ | (F + G) ¯¯
−
F
G ¯
2
where
Z
Tf = sup
dv f (v)(v, e)2 .
e∈S N −1
Proof. Let us write ∇X = [∇, ∇∗ ]. Then,
·
µ
¶¸
∇f ∇f
∇X F
=
,
.
(93)
F
f
f ∗
On the other hand,
(94)
·
¸
∇X G
∇m G
∂ e G ∇m G
∂e G
=
+v
,
+ v∗
(m, e).
G
G
G
G
G
Let us consider the “killing operator” P (v, v∗ ) defined by
(95)
P : [A, B] 7−→ Π(v − v∗ ) (A − B),
30
G. TOSCANI AND C. VILLANI
where Π is given by (57). In view of (94), clearly,
∇X G
= 0.
(96)
P
G
For all (v, v∗ ), kP k ≤ 2, where k·k denotes the norm in the sense of matrices. Here we see precisely the advantage of the Ornstein-Uhlenbeck
regularization in our approach : it enables us to use the operator (95),
which is pointwise bounded, instead of (32), which is definitely not.
Let us set
∇X F
∇X G
K(X) =
−
.
F
G
In view of (93) and (96),
¶
µ
¶¶
µ
µ
∇X F
∇f
∇f
P K(X) = P
= Π(v − v∗ )
−
.
F
f
f ∗
Since |P K|2 ≤ 4|K|2 , we find
¯
¯2
Z
¯ ∇X F
∇X G ¯¯
2
¯
dX |v − v∗ | (F + G) ¯
−
F
G ¯
Z
Z
1
≥ dX |v − v∗ |2 F |K|2 ≥
dX |v − v∗ |2 F |P K|2
4
¯
µ
µ
¶ ¶¯2
Z
¯
¯
∇f
∇f
1
2
¯
¯ .
dv dv∗ |v − v∗ | f f∗ ¯Π(v − v∗ )
=
−
4
f
f ∗ ¯
Apart of the factor 1/2, this is the entropy dissipation for the Landau
equation with Maxwellian molecules. It suffices to apply the inequality (58) to conclude.
¤
Now, by the properties recalled in section 2, for all t > 0,
T(St f ) ≤ Tf ,
with equality only when all the directional temperatures of f are equal
to 1. Hence, setting
λ = N − Tf ,
we can apply the preceding proposition and recover
¯
¯2
Z ∞ Z
¯ ∇X St F
¯
∇
S
G
1
X
t
2
¯
¯
dt
dX
|v
−
v
|
(S
F
+
S
G)
−
∗
t
t
¯
R2 0
St F
St G ¯
Z ∞
λ
λ
≥
I(S
f
|M
)
=
H(f |M ),
t
2R2 0
2R2
where we have used the relation (47).
It is shown in [21] that λ ≥ λ0 (f ), where λ0 depends only on H(f )
and the normalization (36). Indeed, if (ei ) is any orthonormal basis,
SHARP ENTROPY DISSIPATION BOUNDS
31
P
then
i Ti = N . Therefore, to control λ from below it suffices to
control all the directional temperatures Ti from below. The finiteness of
H(f ) and of some moments of f suffices to prevent f from concentrating
on a hyperplane (v, e) = 0.
√
Proposition 9. Estimate of the tails. Let R ≥ 2, and
Z
F
(97)
e(R) = dX |X|2 1|X|≥R/√2 (F − G) log .
G
Z ∞ Z
St F
(98)
E(R) =
dt dX |X|2 1|X|≥R/√2 (St F − St G) log
,
St G
0
Then for all s > 0, ε > 0,
Cs+ε
,
Rs
where Cs+ε depends only on the normalization (36), s, ε, kf kL12+s+ε log L ,
2
kf kL14+s+ε , K and A such that f ≥ Ke−A|v| .
e(R) + E(R) ≤
Remark. We have used the inequality (1+|X|2 )1|X|>R/√2 ≤ 2|X|2 1|X|>R/√2
to get E.
Proof. We begin with e(R). Throughout the proof, C will denote various constants depending only on the aforementioned quantities.
2
2
Since f ≥ Ke−A|v| , by tensorization F ≥ Ke−A|X| , and also G ≥
2
Ke−A|X| since Maxwellian distributions satisfy the equation (15). Since
(x − y) log(x/y) ≤ x log x if x ≥ y ≥ 1, and | log F |, | log G| ≤ C(1 +
|X|2 ) if F, G ≤ 1, we can write
Z
e(R) ≤ dX |X|2 1|X|≥R/√2 F log F
Z
+ dX |X|2 1|X|≥R/√2 G log G
Z
+C dX |X|2 (F + G)(1 + |X|2 )1|X|≥R/√2 .
R
Since G = dσ F 0 and |X 0 |2 = |X|2 , we write, using the convexity
of x 7−→ x log x and the change of variables dσ dX = dσ dX 0 ,
Z
dX |X|2 1|X|≥R/√2 G log G
µZ
Z
≤
dσ
¶
dX |X|
2
1|X|≥R/√2 F 0
log F
0
Z
=
dX |X|2 1|X|≥R/√2 F log F.
32
G. TOSCANI AND C. VILLANI
In the same manner,
Z
Z
2
2
√
dX |X| 1|X|≥R/ 2 (1 + |X| )G = dX |X|2 1|X|≥R/√2 (1 + |X|2 )F.
Hence,
Z
dX |X|2 1|X|≥R/√2 F log F
e(R) ≤ C
Z
dX |X|4 1|X|≥R/√2 F.
+C
Writing
1|X|≥R/√2 ≤ 1|v|≥R/2 + 1|v∗ |≥R/2 ,
|X|2 = |v|2 + |v∗ |2 ,
|X|4 ≤ C(|v|4 + |v∗ |4 ),
log F = log f + log f∗ ,
we obtain
¶
¶ µZ
2
dv |v| 1|v|≥R/2 f | log f |
dv∗ f∗ (1 + |v∗ | )
e(R) ≤ C
¶
µZ
¶ µZ
2
2
dv∗ f∗ (1 + |v∗ | )1|v∗ |≥R/2
dv |v| f | log f |
+C
¶
µZ
¶ µZ
4
dv∗ f∗
dv |v| 1|v|≥R/2 f
+C
¶
µZ
¶ µZ
4
dv∗ f∗ 1|v∗ |≥R/2
dv |v| f
+C
µZ
2
´
C ³
1
1
1
1
1
1
≤ r kf kL2+r log L + kf kL2+r kf kL2 log L + kf kL4+r + kf kLr kf kL4 .
R
Let us now turn to E(R). First,
2
St (e−A|v| ) = Me−2t /(2A) ∗ M1−e−2t = M1+[(2A)−1 −1]e−2t ,
and since St is a linear positive transformation,
2
St f ≥ Ce−A0 |v| ,
where
A0 = sup
½
1
1 + [(2A)−1 − 1]e−2t
¾
, 0≤t<∞
= max(1, 2A).
As a consequence, we also have a fixed Maxwellian lower bound for
St F and St G.
SHARP ENTROPY DISSIPATION BOUNDS
33
We estimate E(R), taking into account the fact that St F −→ M ,
St G −→ M . By the elementary inequality
(f − g) log
f
≤ |f − M || log f | + |g − M || log g|
g
+ C|f − M |(1 + |X|2 ) + C|g − M |(1 + |X|2 )
(which is easy to obtain distinguishing between the cases f ≥ g ≥ M ,
M ≥ f ≥ g, 2 ≥ f ≥ M ≥ g, f ≥ 2 ≥ M ≥ g and so on), we have
Z ∞ Z
dt dX |X|2 1|X|≥R/√2 |St F − M || log St F |
(99) E(R) ≤
Z0 ∞ Z
(100)
+
dt dX |X|2 1|X|≥R/√2 |St G − M || log St G|
0
Z ∞
Z
(101)
+ C
dt dX |X|2 1|X|≥R/√2 |St F − M |(1 + |X|2 )
Z 0∞ Z
(102)
+ C
dt dX |X|2 1|X|≥R/√2 |St G − M |(1 + |X|2 ).
0
By convexity and changes of variables in the same spirit as before,
we reduce to the problem of estimating only (99) and (101). We begin
with (101). By the results recalled in section 2, for all t > 0,
p
kSt F − M kL1 ≤ 2H(F |M )e−t
(of course H(F ) = 2H(f ) is finite), and for all r > 0,
kSt F − M kL1r ≤ Cr (kf kL1r ) + kM kL1r .
Hence, for any ε > 0, K > 0, separating between small and large
|X|,
Z
C
dX (1 + |X|)r |St F − M | ≤ CK r kSt F − M kL1 + ε kSt F − M kL1r+ε
K
Cr+ε
≤ CK r e−t +
.
Kε
where Cr+ε depends only on kF kL1r+ε (i.e. on kf kL1r+ε and the normal1/(r+ε) t/(r+ε)
ization (36)). Choosing K = Cr+ε
e
, we get
ε
kSt F − M kL1r ≤ Ce− r+ε t ,
and therefore
Z ∞
Z ∞ Z
C
4
√
dt kSt F − M kL14+s
dt
dX |X| 1|X|≥R/ 2 |St F − M | ≤ s
R 0
0
µ
¶
C 4+s+ε
≤ s
.
R
ε
34
G. TOSCANI AND C. VILLANI
Finally, we handle the integral (99). Applying the same strategy as
above and using proposition 1, it is sufficient to prove that
Z
dX |St F − M || log St F ||X|2 −−−→ 0
t→∞
with an exponential rate.
To that purpose, we use the elementary inequality
(103)
¯
¯
µ
¶
¯
¯
x
x
|x−y|| log x| ≤ |x−y| ¯¯log ¯¯ 1x≤y +|x−y|| log y|+C x log + y − x .
y
y
To prove (103), it suffices to note that
¯
¯
¯
¯
x
|x − y|| log x| ≤ |x − y|| log y| + |x − y| ¯¯log ¯¯ ,
y
and to bound the second term. By homogeneity, we just have to check
that if z = (x/y) ≥ 1, then
(z − 1) log z ≤ C(z log z + 1 − z).
This last inequality is easily obtained (note that both functions have
vanishing derivatives of first order for z = 1).
By Hlder’s inequality, then (103) applied with x = St F and y = M ,
Z
|St F − M || log St F ||X|2
¶ε/(1+ε) ³
µZ
≤C
|St F − M || log St F |
µZ
≤ Cε
kf kL12+2ε log L kf kL14+2ε
´1/(1+ε)
1
¶ 1+ε
dX |St F − M |(1 + |X| ) + H(St F |M )
.
2
The right hand side converges exponentially fast to 0, and the desired
conclusion follows by the same arguments as before.
¤
End of the proof of Theorem 7 :
By propositions 8 and 9,
¶
µ
E(R) e(R)
λ
γ
H(f |M ) −
− 2
D(f ) ≥ D(f ) ≥ C(1 + R)
R2
R2
R
·
¸
Cs+ε
≥ C(1 + R)γ−2 H(f |M ) − s .
R
√
1/s
Choosing R = max(2−1/s Cs+ε H(f |M )−1/s , 2), we get the desired result.
¤
SHARP ENTROPY DISSIPATION BOUNDS
35
6. Extension to other kernels
Theorem 7 covers essentially all kernels B(z, ω) that are locally
bounded below by |z · ω|/|z| (in dimension 3; (|z · ω|/|z|)N −2 in the
general case). The cross-section |z · ω|, for instance, corresponding
to the hard-spheres potential, does not satisfy this assumption for |z|
near 0. In order to obtain a result for such potentials, we just have to
follow the strategy applied by Carlen and Carvalho [10], namely cut
out (small) portions of the velocity space where B is small. In this
example, we write
|z · ω| ≥ ²
|z · ω|
|z · ω|
− ²1|z|≤²
,
|z|
|z|
and we estimate from above the entropy dissipation associated to ψ(|z|) =
1|z|≤² , i.e.
Z
Z
³
´
f 0 f∗0
χ(²) = dv dv∗ 1|v−v∗ |≤² dσ f 0 f∗0 − f f∗ log
.
f f∗
As ² goes to 0, χ(²) −→ 0, and we have to estimate explicitly the
corresponding rate of convergence. Here, it is not clear whether the
2
assumptions f ∈ L1s log L ∩ L1r and f (v) ≥ Ke−A|v| suffice to provide
such an estimate. But as soon as, say, L2 bounds are available for f ,
this can be done easily. Indeed, let
D² (R) = {(v, v∗ ) ∈ R2N ; |v − v∗ | ≤ ², |v|2 + |v∗ |2 ≤ R}.
Then, for all R > 0, s > 0,
(104)
C
χ(²) ≤ s
R
Z
2
³
´
f 0 f∗0
0 0
1
f f∗ − f f∗ log
f f∗
Z
³
´
f 0 f∗0
0 0
+
f f∗ − f f∗ log
.
f f∗
D² (R)×S N −1
2 (s+1)/2
(|v| + |v∗ | )
|v|2 +|v∗ |2 ≥R2
Using the Maxwellian lower bound, we write systematically
³
´
f 0 f∗0
≤ f 0 f∗0 log(f 0 f∗0 )+f f∗ log f f∗ +C(f f∗ +f 0 f∗0 )(|v|2 +|v∗ |2 ),
f 0 f∗0 −f f∗ log
f f∗
and changing primed to unprimed variables we reduce to estimating
Z
(105)
f f∗ log(f f∗ )(|v|2 + |v∗ |2 )s/2
R2N
and
Z
(106)
f f∗ log(f f∗ ).
D² (R)
36
G. TOSCANI AND C. VILLANI
Writing log f f∗ = log f + (log f )∗ , we easily see that (105) is bounded
by a constant depending on the norm of f in L1s log L. On the other
hand, if f ∈ L2 , then f f∗ ∈ L2 (R2N ), and this is enough to provide an
explicit estimate of (106) in terms of the Lebesgue measure of D² (R).
In the end, it suffices to choose a convenient R, depending on ².
We do not enter into the details of this computation, since several
estimates of this type can be found in [10] for the Boltzmann equation,
and in [21] for the Landau equation with hard potentials. The same
technique also allows to cover all kernels B for which
(107)
¯(
)¯
µ
¶N −2
¯
¯
|z · ω|
¯
¯
, |z| ≤ R ¯ = O(Rβ ²δ )
¯ (z, ω)/B(z, ω) ≤ ²ψ(|z|)
¯
¯
|z|
for some positive numbers β < N , δ > 0, as R goes to infinity and ²
goes to 0 – that is, precisely those kernels such that the set of points
for which they are not bounded below is of very small measure.
As a conclusion, for all kernels B satisfying (107), our method enables
to obtain an algebraic estimate of the form
(108)
D(f ) ≥ Cα (f ) H(f |M )α ,
with α and Cα (f ) explicitly computable, and depending on β, δ in (107),
γ in (7), and various norms of f , say in weighted L2 .
Let us briefly comment on the conditions of theorem 7. As proven
recently by Pulvirenti and Wennberg [37], a Maxwellian lower bound
for the solution to (1) in the case of hard potentials is available at any
positive time, provided the initial datum has bounded energy and entropy. On the other hand, the Ornstein-Uhlenbeck semigroup produces
such bounds. But, as show the numerical applications in [10], these are
terribly small, and it is not clear whether they would be useful.
Essentially, for hard potentials, solutions automatically have a good
decay at infinity, and with bounds that are uniform in time. In particular, weighted Lp -bounds on the solution propagate uniformly in time
if sufficiently many moments exist at time t = 0. A fairly complete
study of these uniformly boundedness properties in Lpr was done by
Gustafsson in [28]. This result was improved by Wennberg in [50].
In any case, provided the conditions of Gustafsson theorems are satisfied, both weighted L2 - norms and the Maxwellian lower bound are
available, uniformly in time. This allows to transform the inequality (108) into a theorem of decay to equilibrium with explicit rate.
On the contrary, for soft potentials it is an open problem whether the
bounds can be found to hold uniformly in time.
SHARP ENTROPY DISSIPATION BOUNDS
37
Let us also do some comments on the estimate below for λ = N − Tf
in Theorem 7. Even though λ is estimated below in terms only of
H(f ) and the normalization (36), this gives very poor estimates, as
show the numerical applications in [21]. Indeed, the entropy is very
bad at controlling the concentration on sets of small measure. Here
again, working in an L2 framework enables far better estimates.
An interesting feedback effect was studied in [21] : under suitable
assumptions, as time goes by, solutions of the Boltzmann equation
converge towards the Maxwellian distribution, say in L12 , and hence all
the directional temperatures of f (t) converge towards the equilibrium
value 1. Therefore, the constant λ essentially becomes better with time,
and is equal to N −1 asymptotically. This effect is a direct consequence
of the nonlinearity of the Boltzmann equation.
By the way, note that in the particular case of radial solutions, one
has always Tf = 1, hence λ = N − 1.
7. The Kac model
In this section, we show how the previous analysis can be extended
to the Kac model. We recall that Kac’s collision operator reads, for a
distribution f (v), v ∈ R,
Z 2π
³
´
(109)
QK (f, f ) =
dθ dv∗ f 0 f∗0 − f f∗
0
where we note by dθ the normalized measure on (0, 2π), and

0

v = v cos θ − v∗ sin θ
(110)

v 0 = v sin θ + v cos θ.
∗
∗
Hence the postcollisional velocities are simply obtained by a rotation
of angle θ in the space (v, v∗ ).
There is only one collisional invariant in the KacR model, namely
e = (|v|2 +|v∗ |2 )/2. It is clear that for any function f , dθf 0 f∗0 depends
only on e.
Proposition 3 extends trivially to the Kac model, and all the subsequent analysis can be done. But the peculiarity of the dimension 1 is
that there is no corresponding Landau equation, since the orthogonal
projection Π defined by (57) is meaningless. Therefore, we have to
change the proof of proposition 8 because we cannot rely on the use of
the linear operator (95).
38
G. TOSCANI AND C. VILLANI
This can be linked to the following elementary observation. For any
vector function g on RN (think of g as ∇ log f ), the property
·
¸
·
¸
∀(v, v∗ ), g(v) − g(v∗ ) = λv,v∗ (v − v∗ ) =⇒ ∀v, g(v) = λv + µ
(λ ∈ R, µ ∈ RN ), holds in any dimension N ≥ 2, but is obviously false
if N = 1.
The following proposition is thus a replacement for Proposition 3.
Proposition 10. Assume N = 1. Let f be a smooth function with
logarithm quadratically bounded, with unit mass and temperature, and
let G be a function of e. Then
¯
¯2
Z
¯ ∇F
∇G ¯¯
2
¯
(111)
dX |X| F ¯
≥ I(f |M ).
−
F
G ¯
Proof. This time,
· 0
¸
∇G
G
G0
= v (e), v∗ (e) .
G
G
G
Hence we can apply the “killing operator”
P (v, v∗ ) : [A, B] 7−→ v∗ A − vB.
For all (v, v∗ ), the square norm of P is bounded by 2(|v|2 + |v∗ |2 ) =
2|X|2 . Defining
∇F
∇G
K(v, v∗ ) =
−
,
F
G
we see that
f 0 (v)
f 0 (v∗ )
P K(v, v∗ ) = v∗
−v
.
f (v)
f (v∗ )
Hence,
¯
¯2
µ
¶2
Z
Z
¯
f 0 (v)
∇G ¯¯
1
f 0 (v∗ )
2 ¯ ∇F
dv dv∗ v∗
dX F |X| ¯
−
≥
−v
f f∗
F
G ¯
2
f (v)
f (v∗ )
µZ
¶ µZ
¶ µZ
¶ µZ
¶
f 0 (v)2
2
0
0
=
dv∗ f (v∗ )|v∗ |
dv
−
dv vf (v)
dv∗ v∗ f (v∗ ) .
f (v)
Integrating by parts the second term, we obtain simply
I(f ) − 1 = I(f |M ).
¤
SHARP ENTROPY DISSIPATION BOUNDS
39
At this point we can apply the ideas of the previous section, concluding in particular with the algebraic decay to equilibrium in relative
entropy of the solution to Kac equation. On the other hand, due to the
particular symmetries of this one-dimensional model, Theorem 7 can
be improved in a number of ways.
Theorem 11. Assume N = 1. Let f ∈ L12 (R) satisfy the normalization (36). If in addition, for some s > 0, kf kL12+s < ∞, and
Z
f 0 (v)4
(112)
I2 (f ) = dv
< ∞,
f (v)3
then for all ε > 0 there exists a constant Cs,ε (f ) depending only on s,
ε, kf kL12+s and I2 (f ), such that
D(f ) ≥ Cs,ε (f ) H(f |M )1+(2+ε)/s .
(113)
Remark. It is known that if I2 (f0 ) is finite, then I2 (ft ) remains bounded,
uniformly in time, if ft is the solution to the Kac equation with initial
datum f0 : more precisely [25, 40],
©
ª
I2 (St f ) ≤ max I2 (f0 ), CI(f0 )2 ,
where C is numerical.
Proof. Let us repeat the argument of Proposition 10. We consider
a function f which is smooth and whose logarithm is quadratically
bounded. Let G be a function of e. Then, for any R > 0,
(114)
Z
dX F
¯2
¯
¯2
¯
Z
¯
¯ ∇F
∇G ¯¯
∇G ¯¯
1
2 ¯ ∇F
¯
dX F |X| ¯
−
≥
¯ F − G ¯ ≥ 2R2
F
G ¯
|v|≤R, |v∗ |≤R
¶2
µ
Z
1
f 0 (v∗ )
f 0 (v)
dv dv∗ v∗
−v
f f∗ =
4R2 |v|≤R,|v∗ |≤R
f (v)
f (v∗ )
µZ
¶ µZ
¶
1
f 0 (v)2
2
dv∗ f (v∗ )|v∗ |
dv
=
2R2
f (v)
|v|≤R
|v∗ |≤R
µZ
¶ µZ
¶
1
0
0
dv vf (v)
dv∗ v∗ f (v∗ ) .
−
2R2
|v|≤R
|v∗ |≤R
≡
Writing
Z
1
Ξ(f ).
2R2
f 0 (v)2
dv
= I(f ) −
f (v)
|v|≤R
Z
dv
|v|>R
f 0 (v)2
,
f (v)
40
G. TOSCANI AND C. VILLANI
Z
Z
0
dv vf 0 (v),
dv vf (v) = −1 −
|v|≤R
|v|>R
we see that
Z
Z
f 0 (v)2
(115) Ξ(f ) = I(f |M ) +
dv |v| f (v)
dv
f (v)
|v|>R
|v|>R
Z
Z
0
2
f (v)
− I(f )
dv |v|2 f (v)
−
dv
f
(v)
|v|>R
|v|>R
µZ
¶2
Z
0
0
−2
dv vf (v) −
dv vf (v) .
2
|v|>R
By Cauchy-Schwarz inequality,
µZ
¶2 Z
0
(116)
dv vf (v) ≤
|v|>R
|v|>R
Z
2
dv |v| f (v)
|v|>R
dv
|v|>R
f 0 (v)2
.
f (v)
Thus,
Ξ(f ) ≥ I(f |M ) − GR (f ),
where we define
(117)
Z
GR (f ) = I(f )
Z
Z
f 0 (v)2
dv |v| f (v) +
dv
−2
dv vf 0 (v)
f (v)
|v|>R
|v|>R
|v|>R
·
¸
Z
Z
f 0 (v)2
2
2
0
=I(f |M )
dv |v| f (v) +
|v| f (v) + 2vf (v) +
f (v)
|v|>R
|v|>R
¶2
µ 0
Z
Z
f (v)
+ v f (v).
=I(f |M )
dv |v|2 f (v) +
f (v)
|v|>R
|v|>R
2
Now, we estimate GR (St f ) for t > 0. First, clearly,
Z
Cs
I(St f |M )
dv |v|2 (St f )(v) ≤ I(St f |M ) s ,
R
|v|>R
where Cs depends only on kf kL12+s .
Next, for all ε > 0,
¶2
µ
· Z
Z
ε
(St f )0 (v)
(St f )0 (v)2
1+ε
+ v (St f )(v) ≤ I(St f |M )
dv
2
dv
(St f )(v)
St f (v)
|v|>R
|v|>R
1
¸
Z
1+ε
+2
dv |v|2 (St f )(v)
|v|>R
≤ I(f |M )
ε
1+ε
2εt
− 1+ε
e
· Z
2
Cs
(St f )0 (v)2
+2 s
dv
St f (v)
R
|v|>R
1
¸ 1+ε
,
SHARP ENTROPY DISSIPATION BOUNDS
41
where we have used (45). By Cauchy-Schwarz inequality,
µZ
¶1/2
Z
(St f )0 (v)2
1/2
dv
≤ I2 (St f )
dv St f (v)
.
St f (v)
|v|>R
|v|>R
Next, we use the fact that I2 is bounded uniformly in time for solutions of the one-dimensional Fokker-Planck equation [40] :
©
ª
I2 (St f ) ≤ max e4 I2 (f ), (1 − e−2 )−2 I2 (M ) .
The proof of this inequality relies on the method developed by Lions
and the first author [32] for proving refined estimates of the central
limit theorem.
Combining this with the boundedness of kf kL12+s , we easily obtain
ε
2εt
Cs
Cs,ε
I(St f |M ) 1+ε + s/(1+ε) e− 1+ε .
s
R
R
Since (by Theorem 6 with ψ ≡ 1)
Z ∞
1
D(f ) ≥
dt [I(St f |M ) − GR (St f )],
2R2 0
(118)
GR (St f ) ≤
by (118) we see that if R ≥ Rs (depending only on s and on kf kL12+s ,
with Rs ≥ 1),
·
¸
1 1
Cs,ε
(119)
D(f ) ≥
H(f |M ) − s/(1+ε) ,
2R2 2
R
where Cs,ε depends only on I2 (f ), kf kL12+s , s and ε. Note indeed that if,
for some a > 0, we denote by t0 the first time t such that I(St f |M ) ≤ a,
then (this is a rough estimate !)
Z +∞
Z t0
Z +∞
ε
ε
ε
I(St f |M ) 1+ε dt =
I(St f |M ) 1+ε dt +
I(St f |M ) 1+ε dt
0
0
t0
Z
+∞
1
− 1+ε
≤a
I(St f |M ) dt + a
0
Z
+∞
ε
1+ε
ε
e−2 1+ε t dt,
0
and we just have to choose Cs a−1/(1+ε) = 1/2 to make sure that the
estimate (119) holds.
−s/(1+ε)
Choosing in (119) R−s/(1+ε) = min{(4Cs,ε )−1 , H(f |M ), Rs
, we
get the result (with 2ε in place of ε).
¤
Theorem 11 gives a lower bound on the entropy production for Kac
equation with a constant depending on the functional I2 . The boundedness of this functional substitutes the bounds on L1s log L, and also
the moment condition is better. On the other hand, this functional has
42
G. TOSCANI AND C. VILLANI
been shown to be uniformly bounded in time (cf. [23]) along the solution to Kac equation. Hence, if the initial datum f0 for Kac equation
is such that I2 (f0 ) < ∞, the decay to equilibrium follows.
McKean [34] has studied the rate of convergence to equilibrium in the
Kac model, conjecturing the existence of a lower bound of type (113),
without specifying the nature of the positive term on the right side.
McKean’s paper contains a number of general ideas, including the validity of formula (53), as well as the introduction of Fisher information
(that he called Linnik functional), and its connection with the trend to
equilibrium. For his study of the decay, McKean used the regularization of the solution f (t), taking the convolution with a Maxwellian of
small energy : f δ = f ∗Mδ . Then I2 (f δ ) is bounded, and the conditions
of the previous theorem are automatically satisfied if some moment of
order higher than 2 is finite.
8. Remarks about Fisher information and entropy
dissipation
We begin with a trivial assertion. Let (B t )t≥0 be a semigroup commuting with the adjoint Ornstein-Uhlenbeck semigroup (St )t≥0 , and let
D be the associated entropy dissipation functional. Then
(120)
¯
¯
d ¯¯
d ¯¯
D(St f ) = ¯ I(B t f ).
¯
dt t=0
dt t=0
Indeed, in view of the commuting property, both terms are equal to
¯
¯
d ¯¯
d ¯¯
H(B s St f ).
¯
¯
dt t=0 ds s=0
In other words, the evolution of the entropy dissipation along the
adjoint Ornstein-Uhlenbeck semigroup is given by the evolution of the
Fisher information along the semigroup (B t ).
As a first application, we can recover in a straightforward way the
first term in the right-hand side of (88), without the use of formula (78).
More precisely, we shall show that for any two smooth functions F and
G,
(121)
¯
¯ Z
¯2
Z
¯ ∇F
St F
∇G ¯¯
d ¯¯
¯
dX (St F − St G) log
.
= − dX (F + G) ¯
−
dt ¯t=0
St G
F
G ¯
SHARP ENTROPY DISSIPATION BOUNDS
43
To this purpose, we introduce the linear system


∂t F = (G − F )
(122)

∂ G = (F − G).
t
It is clear that, if we set H = H(F )+H(G), then the time-dissipation
of H associated to the system (122) is the functional
Z
F
D(F, G) = (F − G) log .
G
But the semigroup associated to the system (122) obviously commutes with the system


∂t F = LF
(123)

∂ G = LG,
t
where L stands for the linear Fokker-Planck operator, since LF −LG =
L(F − G).
Hence, to prove (121), it is sufficient to compute the time-derivative
of I = I(F ) + I(G) along solutions of the system (122). This computation is immediate and yields the desired result.
Let us now choose for (B t ) the semigroup associated to the Boltzmann equation with Maxwellian molecules, and think the other way.
In view of Bobylev’s lemma [5], (B t ) commutes with (St ). Hence, in all
the cases when the entropy dissipation D is directly seen to be decreasing under evolution by the adjoint Ornstein-Uhlenbeck semigroup, we
recover by the remark above an alternative proof that I is decreasing
along solutions of the Boltzmann equation (note that it suffices to deal
with smooth functions, because of the regularizing properties of the
Fokker-Planck equation).
That D is decreasing along (St ) can be seen directly at least in three
different cases, with the help of the results of section 3.
The case N = 2.
For Maxwellian molecules in two dimensions, we can write, following
the notations of proposition 3,
Z π
D(f ) =
ζ(θ) D(F, t Rθ F )
0
for some nonnegative function ζ. For each θ, D(F, t Rθ F ) is decreasing
along the adjoint Ornstein-Uhlenbeck semigroup, and hence D also.
As a consequence, we have a new proof of the result by Toscani [38]
44
G. TOSCANI AND C. VILLANI
that I is decreasing along solutions of the Boltzmann equation with
Maxwellian molecules in two dimensions.
The case B(z, ω) constant.
If B is a constant, then we can write, following the notations of
proposition 4,
Z
dω D(F, t Tω F ),
D(f ) =
S N −1
and conclude as before. This gives a new proof of the result by Carlen
and Carvalho [9] that I is decreasing along solutions of the Boltzmann
equation with constant kernel.
In fact, in both of the previous cases, one can also adapt and simplify the argument given by McKean for Kac’s model : to prove that
D(f ∗ Mδ ) ≤ D(f ) (which implies the decreasing property of I by differentiation in δ), write j(x, y) = (x − y) log(x/y) and note that, by
Jensen’s inequality, for given θ (or ω, with Tω in place of Rθ ),
j(F ∗ Mδ , (t Rθ F ) ∗ Mδ ) ≤ j(F, t Rθ F ) ∗ Mδ .
Integration with respect to dv dv∗ , and use of the translational invariance of D yield then
D((F ∗ Mδ , (t Rθ F ) ∗ Mδ ) ≤ D(F, t Rθ F ),
and the conclusion follows.
e σ) constant.
The case B(z,
e is
For simplicity we treat the case N = 3. If B = |z · ω|/|z|, then B
constant, and
Z
Z
F
D(f ) =
dω dX |k · ω|(F − Gω ) log
,
Gω
S N −1
with Gω = t Tω F . Applying proposition 5, we see that it suffices to
prove that for each ω,
Z
F
≤ 0.
(124)
dX L∗ (|k · ω|) (F − Gω ) log
Gω
Let us compute L∗ (|k · ω|). First, using the general formula
(125)
∇v [b(k · ω)] =
1
b0 (k · ω)Πk⊥ ω,
|v − v∗ |
we find that
∇X (|k · ω|) =
sgn(k · ω)
[Πk⊥ ω, −Πk⊥ ω] .
|v − v∗ |
SHARP ENTROPY DISSIPATION BOUNDS
45
This term is well-defined only for k · ω 6= 0, but since we can restrict
to functions F and G that are smooth, this does not matter here. As
a consequence, X · ∇X (|k · ω|) is a multiple of
v · Πk⊥ ω − v∗ · Πk⊥ = (v − v∗ ) · Πk⊥ ω = 0.
A similar computation shows that
v − v∗
∆X (|k · ω|) = − 2
sgn(k · ω) · Πk⊥ ω
(126)
|v − v∗ |3
µ
¶
1
4
(127)
δ(k·ω)=0 Πk⊥ ω · Πk⊥ ω
+
|v − v∗ | |v − v∗ |
4
(128)
−
|k · ω|.
|v − v∗ |2
where to compute the last term we have used formula (125) and the
relation sgn(u)u = |u|. The expression (126) is 0 because v − v∗ and
Πk⊥ ω are orthogonal. The contribution of (127) to the integral (124) is
also 0 because when (k · ω) = 0, then F = Gω , and F , Gω are smooth.
Finally, the expression (128) is nonpositive. Gathering all of this, we
obtain that the inequality (124) actually holds.
We do not know if by this method one can recover the general theorem that Fisher’s information is decreasing along solutions of the Boltzmann equation with Maxwellian molecules in any dimension [47]. But
we found rather striking this connection with the problem of finding a
lower bound for the entropy dissipation.
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Department of Mathematics, University of Pavia, via Abbiategrasso 209,
27100 Pavia, ITALY. e-mail [email protected]
École Normale Suprieure, DMI, 45 rue d’Ulm, 75230 Paris Cedex 05,
FRANCE. e-mail [email protected]