Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged

particles

Maria-José Cáceres^{1}, José-Antonio Carrillo^{1}, Thierry Goudon^{2}

1Departamento de Matemática Aplicada,

Universidad de Granada, E-18071 Granada, Spain

               E-mail: caceresg@ugr.es, carrillo@ugr.es

2Labo. J.A. Dieudonné, UMR 6621,

Université Nice-Sophia Antipolis, Parc Valrose,

F-06108 Nice cedex 02, France

and INRIA-Sophia, project CAIMAN

                     E-mail: goudon@math.unice.fr

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.

1 Introduction

We wish to investigate the long-time behavior for solutions f of the following kinetic equation

\{ \begin{matrix}∂_{t}f + v · ∇_{x}f − ∇_{x}Φ_{0}(x) · ∇_{v}f = Q(f) & in ℝ^{+} × ℝ^{N} × ℝ^{N}, \\ f_{|t=0} = f_{0} & in ℝ^{N} × ℝ^{N}, \end{matrix}  (1.1)

where the right hand side is given by the following simple linear Boltzmann operator

Q(f) = \frac{1}{τ}(ρM_{θ} − f),  ρ(t, x) = ∫_{ℝ^{N}} f(t, x, v) dv  (1.2)

and M_{θ} is the normalized Maxwellian

M_{θ}(v) = \frac{e^{−v^{2}/2θ}}{(2πθ)^{N/2}} .  (1.3)

This equation arises as a statistical model describing a large system of interacting particles, the unknown f(t, x, v) ≥ 0 being the density of particles occupying position x ∈ ℝ^{N} at time t ≥ 0 and moving with velocity v ∈ ℝ^{N}. The right hand side describes the interactions processes undergone by the particles, while the left hand side describes the transport of the particles when submitted to the spatially dependent force field F (x) = −∇_{x}Φ_{0}(x). Accordingly, in absence of interaction, the particles evolve along the trajectories, obtained as solutions of the ODE’s system

\frac{d}{dt}X(t; x, v) = V (t; x, v), \frac{d}{dt}V (t; x, v) = −∇_{x}Φ_{0}(X(t; x, v))

with initial conditions X(0; x, v) = x, V (0; x, v) = v: if Q = 0, then the solution is given by f(t, x, v) = f_{0}(X(−t; x, v), V (−t; x, v)). Throughout the paper, Φ_{0} is an external potential having a confining effect: we assume that

\{ \begin{matrix}• Φ_{0} ≥ 0, Φ_{0} ∈ C^{∞}(ℝ^{N}), \\ • exp(−Φ_{0}(x)/θ) ∈ L^{1}(ℝ^{N}), \\ • Φ_{0} is a bounded perturbation of a uniformly convex potential on ℝ^{N}: \\ Φ_{0} = Φ_{0}^{uc} + Φ_{0}^{bp} such that \\ there exists λ_{1} > 0 such that(∂_{x_{i}x_{j}}^{2}Φ_{0}^{uc}(x))_{i,j=1,...,N} ≥ λ_{1}I, ∀x ∈ ℝ^{N}, \\ and \\ there exists a and b such that 0 < a ≤ Φ_{0}^{bp}(x) ≤ b < ∞, ∀x ∈ ℝ^{N}. \end{matrix} (1.4)

The regularity hypothesis on the potential will be needed in our main result to derive smoothness estimates on the solutions via smoothness properties of the characteristics. The second hypothesis is the one that allows for non trivial stationary states, thus the one that implies a confinement of the particles. The last one will be needed in order to compare entropy terms and their dissipation.

The collision operator Q is mass-preserving: one has ∫_{ℝ^{N}} Q(f) dv = 0. Accordingly, we can assume without loss of generality that f is a probability density

∫_{ℝ^{N}} ∫_{ℝ^{N}} f(t, x, v) dv dx = ∫_{ℝ^{N}} ∫_{ℝ^{N}} f(0, x, v) dv dx = 1.  (1.5)

The effect of the collision operator is to relax to a local equilibrium f(t, x)=ρ(t, x)M_{θ}(v) ∈ Ker(Q), as a consequence of the dissipation property

− ∫_{ℝ^{N}} Q(f) \frac{f}{M_{θ}} dv = \frac{1}{τ} ∫_{ℝ^{N}} ∫_{ℝ^{N}} M_{θ}(v)M_{θ}(v_{∗})(\frac{f}{M_{θ}} (v) − \frac{f}{M_{θ}} (v_{∗}))^{2} dv_{∗} dv.

In the spatially homogeneous case

∂_{t}f = Q(f) = \frac{1}{τ}(M_{θ} − f).

this immediately determines the asymptotic trend to equilibrium. Indeed, the dissipation relation reads

− ∫_{ℝ^{N}} Q(f) \frac{f}{M_{θ}} dv = \frac{1}{τ} (∫_{ℝ^{N}} \frac{f^{2}}{M_{θ}} dv − 1) = \frac{1}{τ} ∫_{ℝ^{N}} |f − M_{θ}|^{2} \frac{1}{M_{θ}} dv

so that we are led to

\frac{1}{2} \frac{d}{dt} ∫_{ℝ^{N}} \frac{f^{2}}{M_{θ}} dv = \frac{1}{2} \frac{d}{dt} ∫_{ℝ^{N}} |f − M_{θ}|^{2} \frac{1}{M_{θ}} dv = −\frac{1}{τ} ∫_{ℝ^{N}} |f − M_{θ}|^{2} \frac{1}{M_{θ}} dv.

Combining this relation with the Cauchy-Schwarz estimate, we obtain an exponential convergence to M_{θ} in the L^{1} norm

∫_{ℝ^{N}} |f − M_{θ}| dv ≤ (∫_{ℝ^{N}} |f − M_{θ}|^{2} \frac{1}{M_{θ}} dv)^{1/2} ≤ C_{0} e^{−t/τ}

where C_{0} depends on the initial data.

However, in the space inhomogeneous case, the collision operator pushes the solution to a local equilibrium, while only one distribution ρ(t, x)M_{θ}(v) satisfies (1.1). Indeed, it should fulfill

∂_{t}ρ + v · ∇_{x}ρ + ρ \frac{v}{θ} · ∇_{x}Φ_{0} = 0

which yields

∂_{t}ρ = 0, v · ∇_{x}ρ = −ρ \frac{v}{θ} · ∇_{x}Φ_{0}.

Therefore, the transport operator selects, among all the local equilibrium states, the global steady state

f_{s}(x, v) = ze^{−Φ_{0}(x)/θ}M_{θ}(v),  z = (∫_{ℝ^{N}} e^{−Φ_{0}(x)/θ} dx)^{−1} .  (1.6)

Hence one naturally addresses the following questions:

1) As time becomes large, does the solution f(t, x, v) resembles the steady state f_{s} ?

2) Can one find a rate of convergence for some L^{p} norm ? In particular, does f go to f_{s} exponentially fast ?

The answer to the first question is affirmative, there is convergence at least weakly in L^{1} of f(t, ·) towards f_{s} based on by-now standard arguments [9, 1, 2]. However, strong convergence in L^{1} and moreover, rates of convergence in L^{1} are unknown even for such simple inhomogeneous kinetic equations. The measurement of decay rates for nonlinear diffusion equations has been recently addressed using entropy dissipation methods based on the Bakry-Emery trick to show generalized logarithmic Sobolev inequalities giving explicit rates, see [4]. We refer to [16] for a user friendly review of all of these techniques and for further references of this subject. However, inhomogeneous kinetic problems were not included in any of these works, mainly, because a gradient-flow structure lacks for them. Finally, this situation was overcome and a variation of the entropy dissipation method was applied in [10] to the inhomogeneous kinetic Vlasov-Fokker-Planck operator.

To deal with the rate of convergence for (1.1), we shall use the method introduced by Desvillettes-Villani in [10]. They are interested in similar problems with the Fokker-Planck diffusion operator

Q(f) = div_{v}(θ∇_{v}f + vf) = div_{v}(θM_{θ}∇_{v}(f/M_{θ})),

which shares dissipation properties with the Boltzmann operator (1.2). They obtain convergence to the steady state f_{s} in L^{1} at rate O(t^{−1/ε}) for any ε > 0. Then, in [11], they extend their study to the nonlinear Boltzmann operator. Finally, let us mention that the method of [10] has also been applied for the much simpler neutron-transport equation by Fellner-Schmeiser [13]. A lot of information on the asymptotic trend to equilibrium in kinetic theory can be found in the very nice review of Villani [18].

Beyond the tedious and tricky computations performed in [10], the most convincing argument for the method is how close it is to the physical interpretation of the asymptotic trend to equilibrium (see [14]). The keypoint is to look carefully at the competing effects between the collision operator and the transport operator. Roughly speaking, while the dissipativity of the former pushes the solution to a local equilibrium, the latter drives the solution out of the class of equilibria. Therefore, one introduces non negative quantities H(f|f_{s}) and H̃(f|ρM_{θ}) which measure how far the solution f is from the global or a local equilibrium, respectively. The first step consists in obtaining a differential inequality between these quantities

−\frac{d}{dt}H(f|f_{s}) ≥ κH̃(f|ρM_{θ}),

for some κ > 0. If f reaches an equilibrium at time t_{0}, then (formally) both H̃(f|ρM_{θ}) and its time derivative vanish. Hence, information comes from the second time derivative of H̃(f|ρM_{θ}). Then, the second ingredient of the proof is to remark (up to some computations) that \frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) can be estimated as follows

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ κ′H(f|f_{s}) − R(f).

The difficulty is now to relate the remainder R(f) to H̃(f|ρM_{θ}). This step requires smoothness estimates on the solution f and the interpolation method proposed by Desvillettes-Villani induces a loss in the exponent

R(f) ≤ C(f, ε)H̃(f|ρM_{θ})^{1−ε}

for any ε > 0 and C(f, ε) depends on ε and uniform bounds on the derivatives of f. Eventually, we are led to the following system of differential inequalities

\{ \begin{matrix}−\frac{d}{dt}H(f|f_{s}) ≥ κH̃(f|ρM_{θ}), \\ \frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ κ′H(f|f_{s}) − C(f, ε)H̃(f|ρM_{θ})^{1−ε}. \end{matrix}

It can be shown that this system implies the convergence to 0 of H(f|f_{s}) with rate O(t^{1−1/ε}) (see Theorem 2.1 below). We conclude by remarking that H(f|f_{s}) controls the L^{1} norm of f − f_{s}. Let us finally remark that the idea of studying the second derivative in time of the relative entropy is reminiscent of the Bakry-Emery strategy that has been shown to be very successful in the analysis of long-time asymptotics for the nonlinear diffusion equations [4, 5, 6, 7, 12, 16, 17]. According to this strategy our main result states as follows:

Theorem 1.1 Let f_{0} verify

af_{s} ≤ f_{0} ≤ Af_{s},  (1.7)

for some 0 < a < A, f_{0} ∈ C^{∞}(ℝ^{N} × ℝ^{N}), with all its derivatives bounded. Let f be the solution of equation (1.1). Assume that f has all its derivatives, ∂_{x,v}^{α}f, α ∈ ℕ^{2N}, bounded uniformly in time. Then, for all ε > 0 and t > 0 we have

∥f(t) − f_{s}∥_{L^{1}(ℝ^{N} ×ℝ^{N})} ≤ C_{ε}(f_{0})t^{−1/ε}

where C_{ε}(f_{0}) depends on ε, the initial data and the bounds on the derivatives of f.

This result illustrates once again how robust is the method of Desvillettes-Villani. It does not depend on the particular form of the entropy functional, neither on the regularizing properties of the collision operator, which cannot be expected in our case. Of course, a touchy point is to preserve uniform estimates (with respect to time) on the solution. This leads to very tedious arguments in [10] that use the regularizing effect of the collision operator. Here we are able to obtain these estimates for the particular case Φ_{0}(x) = λ|x|^{2} (its proof is sketched in Appendix A).

The paper is organized as follows. The aim of Section 2 is to obtain the crucial system of differential inequalities for the relative entropy H(f, f_{s}) and H̃(f|ρM_{θ}). Assuming the regularity of the solution and the uniform bound in time of the derivatives, this yields Theorem 1.1. To clarify the hypotheses assumed in Theorem 1.1, we include two Appendices: one devoted to the uniform in time estimates of the derivatives for the harmonic potential and the other to show global boundedness of the solution itself.

2 Proof of the asymptotic behavior

Let us define the following relative entropy, which measure how far the solution f is from the global and a local equilibrium, respectively,

\{ \begin{matrix}H(f|f_{s}) = ∫_{ℝ^{N}} ∫_{ℝ^{N}} |f − f_{s}|^{2} \frac{1}{f_{s}} dv dx, \\ H̃(f|ρM_{θ}) = ∫_{ℝ^{N}} ∫_{ℝ^{N}} |f − ρM_{θ}|^{2} \frac{1}{f_{s}} dv dx \end{matrix}  (2.1)

Using only these definitions, we can remark some relations, which will be used in the sequel, • H(f|f_{s}) controls the L^{1} norm

(∫_{ℝ^{N}} ∫_{ℝ^{N}} |f − f_{s}| dv dx)^{2} ≤ H(f|f_{s})

by applying the Cauchy-Schwarz inequality.

• H(f|f_{s}) and H̃(f|ρM_{θ}) can be rewritten as

H(f|f_{s}) = ∫_{ℝ^{N}} ∫_{ℝ^{N}} (\frac{f^{2}}{f_{s}} − f_{s}) dv dx,

and

\begin{aligned}H̃(f|ρM_{θ}) & = & ∫_{ℝ^{N}} ∫_{ℝ^{N}} (\frac{ρ^{2}M_{θ}}{ze^{−Φ_{0}/θ}} + \frac{f^{2}}{f_{s}} − 2\frac{ρf}{ze^{−Φ_{0}/θ}} ) dv dx \\ & = & ∫_{ℝ^{N}} ∫_{ℝ^{N}} \frac{f^{2}}{f_{s}} dv dx − ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx \\ & = & H(f|f_{s}) + 1 − ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx. \end{aligned}  (2.2)

Our aim is to derive the following system of differential inequalities

\{ \begin{matrix}−x′(t) ≥ κ_{1}y(t) \\ y′′(t) + κ_{2}y^{1−ε}(t) ≥ κ_{3}x(t), \end{matrix}  (2.3)

where x(t) = H(f|f_{s}), y(t) = H̃(f|ρM_{θ}), ε ∈ (0, 1) and κ_{1}, κ_{2}, κ_{3} are positive constants. Then, conclusion on the asymptotic behavior of solutions of (1.1) follows by applying Theorem 6.2 in [10].

Theorem 2.1 [10] Let x and y be two nonnegative C^{2} functions defined on ℝ^{+} which satisfy (2.3). Then, there exists a constant C > 0 depending on the initial data x(0), the constants κ_{i} and ε such that

x(t) ≤ \frac{C}{t^{(1−ε)/ε}}

holds for all t > 0.

The first subsection below is devoted to the first part of (2.3); actually we prove that it holds with an equality, therefore we will actually compute exactly the third derivative of H(f|f_{s}). In the second subsection we shall work on the second inequality which is more delicate. It is worth pointing out the following items

• The fact that the external potential Φ_{0} is a bounded perturbation of a uniformly convex potential is needed in order to apply a quadratic-entropy Sobolev inequality (see [3]).

• As mentioned before, we first prove \frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ κH(f|f_{s}) − R(f). In order to estimate the remainder R(f) in terms of H̃(f|ρM_{θ}), we shall need some regularity properties. In particular, in this interpolation step, we shall use the estimate on the initial data 0 < af_{s} ≤ f_{0} ≤ Af_{s}, for some a, A > 0, which will be proved to propagate on the solution, as well as uniform bounds on the derivatives ∂_{x,v}^{α}f, α ∈ ℕ^{2N} (see Lemma 2.5, Proposition 2.3). Clearly, time dependent estimates are available, but, as mentioned in [10], the necessary uniform bounds are far from easy to be obtained.

Finally, in the third subsection we summarize the results obtained, and we end the proof.

2.1 First derivative of the relative entropy H(f|f_{s})

Our first result states that H̃(f, ρM_{θ}) corresponds exactly to the dissipation of the relative entropy H(f|f_{s}).

Lemma 2.2 Let f be the solution of the equation (1.1). Then, the function t ⟼ H(f|f_{s}) is non increasing; precisely, we have

\frac{d}{dt}H(f|f_{s}) = −\frac{2}{τ}H̃(f|ρM_{θ}).

Proof. Recall that f_{s} belongs to the kernel of the differential operator ∇_{x}Φ_{0}∇_{v} − v∇_{x}. Then, by using equation (1.1) and integrating by parts, we realize that the dissipation of H(f|f_{s}) only depends on the collision operator; namely, one gets

\begin{aligned}\frac{d}{dt}H(f|f_{s}) & = & \frac{d}{dt} ∫_{ℝ^{N}} ∫_{ℝ^{N}} \frac{f^{2}}{f_{s}} dv dx = 2 ∫_{ℝ^{N} ×ℝ^{N}} \frac{f}{f_{s}} ∂_{t}f dv dx \\ & = & \frac{2}{τ} ∫_{ℝ^{N}} ∫_{ℝ^{N}} (ρM_{θ} − f) \frac{f}{f_{s}} dv dx. \end{aligned}

The right hand side can be rewritten as

−\frac{2}{τ} (∫_{ℝ^{N}} ∫_{ℝ^{N}} \frac{f^{2}}{f_{s}} dv dx − ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx) .

We conclude by using (2.2).

It is worth pointing out that many other entropies are dissipated by the equation (1.1). Indeed, let h : s ∈ ℝ^{+} ↦ h(s) ∈ ℝ be a convex function which vanishes for s = 1. Let f and g be two probability densities on ℝ^{N} × ℝ^{N}. Then, we can evaluate how f is far from g by using the quantity

H(f|g) = ∫_{ℝ^{N}} ∫_{ℝ^{N}} h(\frac{f}{g} ) g dv dx.

This quantity remains non negative since it rewrites ∫_{ℝ^{N}} ∫_{ℝ^{N}} (h(f/g)−h(1)−h′(1)(f/g − 1)) g dv dx where the integrand is non negative by convexity. For instance, h(s) = |s−1|^{2} defines the entropy used in the paper, h(s) = s ln(s) defines the entropy used in [10]. By using integration by parts and the property (v · ∇_{x} − ∇_{x}Φ_{0} · ∇_{v})f_{s} = 0, we obtain for the evolution of the relative entropy of f with respect to the equilibrium f_{s}

\begin{aligned}\frac{d}{dt}H(f|f_{s}) & = & \frac{1}{τ} ∫_{ℝ^{N}} ∫_{ℝ^{N}}(ρM_{θ} − f) h′(f/f_{s}) dv dx \\ & = & −\frac{1}{τ} ∫_{ℝ^{N}} ∫_{ℝ^{N}}(ρM_{θ} − f) (h′(ρM_{θ}/f_{s}) − h′(f/f_{s})) dv dx ≤ 0. \end{aligned}

Therefore, any relative entropy is dissipated by the equation. This property is also satsified by the Fokker-Planck equation.

However, the choice of the quadratic entropy is shown to be suitable to obtain the system of differential inequalities (2.3). It is not straightforward to see if similar computations work for instance for the entropy based on the function s ln(s). But, noting that ln(x) ≤ x − 1 and using the Cauchy-Schwarz inequality, we remark that

H_{ln}(f|f_{s}) = ∫_{ℝ^{N}} ∫_{ℝ^{N}} f ln(f/f_{s}) dv dx ≤ ∫_{ℝ^{N}} ∫_{ℝ^{N}} f(f/f_{s} − 1) dv dx
≤ (∫_{ℝ^{N}} ∫_{ℝ^{N}} |f/f_{s} − 1|^{2} f_{s} dv dx)^{1/2} (∫_{ℝ^{N}} ∫_{ℝ^{N}} f^{2}/f_{s} dv dx)^{1/2}
≤ C H(f|f_{s})^{1/2}.

Hence, our result also shows that H_{ln}(f|f_{s}) converges to 0 with any algebraic rate as time goes to infinity.

2.2 Second derivative of the relative entropy H̃(f|ρM_{θ})

This subsection is devoted to the second derivative of H̃(f|ρM_{θ}), that is exactly, in view of Lemma 2.2, the third derivative of the relative entropy H(f|f_{s}). We aim at estimating from below \frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) in terms of H(f|f_{s}) and a non positive remainder involving the following macroscopic quantities associated to f

\{ \begin{matrix}Macroscopic density: ρ(t, x) = ∫_{ℝ^{N}} f(t, x, v) dv, \\ Mean velocity: ρu(t, x) = ∫_{ℝ^{N}} vf(t, x, v) dv, \\ Temperature: ρ(t, x)\frac{|u(t, x)|^{2}}{2} + \frac{N}{2} ρT (t, x) = ∫_{ℝ^{N}} \frac{|v|^{2}}{2} f(t, x, v) dv, \\ Pressure tensor: ρu ⊗ u(t, x) + ρT (t, x)𝕀 + 𝔻(t, x) = ∫_{ℝ^{N}}(v ⊗ v) f(x, v) dv, \end{matrix} (2.4)

where 𝕀 stands for the N × N identity matrix, and, given two vectors a, b ∈ ℝ^{N}, a ⊗ b denotes the N × N matrix with coefficients a_{i}b_{j}. Integrating equation (1.1) with respect to the variable v, after multiplication 1 and v respectively, one is led to the following macroscopic relations

∂_{t}ρ + ∇_{x} · (ρu) = 0  (2.5)

and

∂_{t}(ρu) + Div_{x}(ρu ⊗ u + ρT 𝕀 + 𝔻) + ρ∇_{x}Φ_{0} = −\frac{ρu}{τ}.  (2.6)

(For a matrix valued field A(x), we denote Div_{x}(A) = ∑_{j} ∂_{x_{j}}A_{ij} ∈ ℝ^{N}). Then, we shall prove the following claim.

Proposition 2.3 Let f be a smooth solution of (1.1). Then, there exists λ > 0 such that

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ λθH(f|f_{s}) − R(f)

where the non negative remainder R reads

R(f) = λθH̃(f|ρM_{θ}) + 2 ∫_{ℝ^{N}} \frac{|∇_{x}(ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx  (2.7)
+ \frac{4}{θ} ∫_{ℝ^{N}} \frac{1}{ze^{−Φ_{0}/θ}} [|\frac{3ρu}{τ}|^{2}+ |∇_{x}(ρu ⊗ u)|^{2}+ |Div_{x}(ρ(T − θ)𝕀)|^{2}+ |Div_{x}𝔻|^{2}] dx.

Proof. Proof of Proposition 2.3 relies on the following equality

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) = \frac{4}{τ^{2}} H̃(f|ρM_{θ}) + 2θ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2}ze^{−Φ_{0}/θ} dx − 2∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx
+ 2 ∫_{ℝ^{N}} ∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} ) · (\frac{3ρu}{τ} + Div_{x}(ρu ⊗ u + ρ(T − θ)𝕀 + 𝔻)) dx (2.8)

which will be proved later on. Let us assume temporarily that this relation holds. First, non-negativeness of H̃(f|ρM_{θ}) allows us to write

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ 2θ ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx − 2 ∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx
+ 2 ∫_{ℝ^{N}} ∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} ) · (\frac{3ρu}{τ} + Div_{x}(ρu ⊗ u + ρ(T − θ)𝕀 + 𝔻)) dx

Let us denote by I the last integral in the right hand side. We can estimate it as follows

|I| ≤ θ ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx
+ \frac{4}{θ} ∫_{ℝ^{N}} \frac{e^{\frac{Φ_{0}}{θ}}}{z} (|\frac{3ρu}{τ}|^{2} + |Div_{x}(ρu ⊗ u)|^{2} + |Div_{x}(ρ(T − θ)𝕀)|^{2} + |Div_{x}𝔻|^{2}) dx

where we used the following simple inequalities

\{ \begin{matrix}ab ≤ θ \frac{a^{2}}{2} + \frac{b^{2}}{2θ}, \\ (a + b + c + d)^{2} ≤ 4(a^{2} + b^{2} + c^{2} + d^{2}). \end{matrix}

It follows that

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ θ ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx − 2 ∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx

−\frac{4}{θ} ∫_{ℝ^{N}} \frac{e^{\frac{Φ_{0}}{θ}}}{z} [|\frac{3ρu}{τ}|^{2} + |Div_{x}(ρu ⊗ u)|^{2} + |Div_{x}(ρ(T − θ)𝕀)|^{2} + |Div_{x}𝔻|^{2}] dx

Then, using the fact that Φ_{0} is a bounded perturbation of a uniformly convex potential yields the following Sobolev inequality (see Corollary 2.18 and Theorem 3.2 in [3] for the quadratic entropy)

∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx ≥ λ ∫_{ℝ^{N}} (\frac{ρ^{2}}{ze^{−Φ_{0}/θ}} − ze^{−Φ_{0}/θ}) dx

where λ depends on λ_{1}, a and b (see Theorem 3.2 in [3]). Hence, we conclude by using the relation (2.2) which tells us that the right hand side is nothing but λ(H(f|f_{s})−H̃(f|ρM_{θ})).

It remains to justify (2.8). Bearing in mind the definition of H(f|f_{s}) and H̃(f|ρM_{θ}), and (2.2), we can write

\frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) = \frac{d^{2}}{dt^{2}} H(f|f_{s}) − \frac{d^{2}}{dt^{2}} ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx

Then, (2.8) will be a consequence of the following claim.

Lemma 2.4 Let f be a smooth solution of (1.1). Then, we have

\frac{d^{2}}{dt^{2}} H(f|f_{s}) = \frac{4}{τ^{2}} H̃(f|ρM_{θ}) − \frac{4}{τ} ∫_{ℝ^{N}} \frac{ρ∇_{x}(ρu)}{ze^{−Φ_{0}/θ}} dx.  (2.9)

and

\frac{d^{2}}{dt^{2}} ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} = 2 ∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx + \frac{2}{τ} ∫_{ℝ^{N}} \frac{ρ∇_{x} · (ρu)}{ze^{−Φ_{0}/θ}} dx

−2 ∫_{ℝ^{N}} ∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} ) · Div_{x}(ρu ⊗ u + 𝔻 + ρ(T − θ)𝕀) dx

−2θ ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx.

(2.10)

Proof. By combining Lemma 2.2 to (2.2), we obtain

\begin{aligned}\frac{d^{2}}{dt^{2}} H(f|f_{s}) & = & −\frac{2}{τ} \frac{d}{dt}H̃(f|ρM_{θ}) = −\frac{2}{τ} \frac{d}{dt}(H(f|f_{s}) + 1 − ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx) \\ & = & −\frac{2}{τ}( − \frac{2}{τ}H̃(f|ρM_{θ}) − \frac{2}{z} ∫_{ℝ^{N}} ρe^{\frac{Φ_{0}}{θ}} ∂_{t}ρ dx). \end{aligned}

Hence, the mass conservation (2.5) yields (2.9). Next, (2.5) also leads to

\frac{d}{dt} ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx = 2 ∫_{ℝ^{N}} \frac{ρ}{ze^{−Φ_{0}/θ}} ∂_{t}ρ dx = −2 ∫_{ℝ^{N}} \frac{ρ}{ze^{−Φ_{0}/θ}} ∇_{x} · (ρu) dx.

Deriving once again gives

\begin{aligned}\frac{d^{2}}{dt^{2}} ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx & = & 2 ∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx \\ & − & 2 ∫_{ℝ^{N}} \frac{ρ}{ze^{−Φ_{0}/θ}} ∇_{x} · ( − \frac{1}{τ}ρu − ρ∇_{x}Φ_{0} − Div_{x}ℙ) dx \end{aligned} (2.11)

where we used (2.5) for the first term and (2.6) for the second term and ℙ stands for the following pressure tensor

ℙ(t, x) = (𝔻 + ρT 𝕀 + ρu ⊗ u)(t, x) = ∫_{ℝ^{N}}(v ⊗ v) f(t, x, v) dv.

Since ∇_{x}(ρe^{Φ_{0}/θ}) = e^{Φ_{0}/θ}(∇_{x}ρ + \frac{ρ}{θ} ∇_{x}Φ_{0}), last integral in (2.11) by parts becomes

\frac{2}{τ} ∫_{ℝ^{N}} \frac{ρ∇_{x} · (ρu)}{ze^{−Φ_{0}/θ}} dx − 2 ∫_{ℝ^{N}} ρ∇_{x}Φ_{0} · (∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}) \frac{1}{ze^{−Φ_{0}/θ}} dx

−2 ∫_{ℝ^{N}} Div_{x}ℙ · (∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}) \frac{1}{ze^{−Φ_{0}/θ}} dx.

We rewrite the last term as follows

\begin{matrix}−2 ∫_{ℝ^{N}} Div_{x}(𝔻 + ρ(T − θ)𝕀 + ρu ⊗ u) · (∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}) \frac{1}{ze^{−Φ_{0}/θ}} dx \\ −2 ∫_{ℝ^{N}} \frac{θ∇_{x}ρ}{ze^{−Φ_{0}/θ}} · (∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}) dx. \end{matrix}

Thus, (2.11) becomes

\frac{d^{2}}{dt^{2}} ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx = 2 ∫_{ℝ^{N}} \frac{|∇_{x} · (ρu)|^{2}}{ze^{−Φ_{0}/θ}} dx + \frac{2}{τ} ∫_{ℝ^{N}} \frac{ρ∇_{x} · (ρu)}{ze^{−Φ_{0}/θ}} dx

− 2∫_{ℝ^{N}}(ρ∇_{x}Φ_{0} + θ∇_{x}ρ) · (∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}) \frac{1}{ze^{−Φ_{0}/θ}} dx

− 2∫_{ℝ^{N}}Div_{x}(𝔻 + ρ(T − θ)𝕀 + ρu ⊗ u)·(∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0})\frac{1}{ze^{−Φ_{0}/θ}} dx.

However, we remark that, the third term of the right hand side recasts as

−2θ ∫_{ℝ^{N}} |∇_{x}ρ + \frac{ρ}{θ}∇_{x}Φ_{0}|^{2} \frac{1}{ze^{−Φ_{0}/θ}} dx = −2θ ∫_{ℝ^{N}} |∇_{x}(\frac{ρ}{ze^{−Φ_{0}/θ}} )|^{2} ze^{−Φ_{0}/θ} dx,

which ends the proof.

As we said at the beginning of the Section, now we aim at estimating R(f) in terms of H̃(f|ρM_{θ}). We must recall that it is at this point where we need to assume that all derivatives of the solution are bounded uniformly in time and the initial data verifies 0 < af_{s}(x, v) ≤ f_{0}(x, v) ≤ Af_{s}(x, v), for some 0 < a < A. Under these assumptions we obtain

Lemma 2.5 Let f be a smooth function on ℝ_{t} × ℝ_{x}^{N} × ℝ_{v}^{N}, satisfying

\{ \begin{matrix}0 < af_{s}(x, v) ≤ f(t, x, v) ≤ Af_{s}(x, v) \\ sup_{x,v} |∂_{x,v}^{α}f(t, x, v)| ≤ C_{p}  for any α ∈ ℕ^{2N}, |α| = p. \end{matrix}  (2.12)

Then, for all ε ∈ (0, 1), there exists a constant C(ε, f) depending on a, A and C_{0}, ..., C_{p(ε)} for some p(ε) ∈ ℕ such that

R(f) ≤ C(ε, f) H̃(f|ρM_{θ})^{1−ε}

holds.

Proof. We must estimate all the terms in R(f) by expressions of the form

C(ε, f) H̃(f|ρM_{θ})^{1−ε},

where C(ε, f) stands for different constants in each term. Let us start by focusing on the last two terms in the definition (2.7) of R(f). These terms can be estimated by expressions of the form

\{ \begin{matrix}∫_{ℝ^{N}}[∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{2} \frac{e^{Φ_{0}/θ}}{z} dx, ∫_{ℝ^{N}}[∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{4} \frac{e^{Φ_{0}/θ}}{z ρ^{2}} dx, \\ ∫_{ℝ^{N}} \frac{|∇_{x}ρ|^{2}}{ρ^{4}ze^{−Φ_{0}/θ}} [∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{4} dx, \\ ∫_{ℝ^{N}}[∇_{x}∫_{ℝ^{N}}[f − ρM_{θ}]φ(v)dv]^{2}\frac{e^{Φ_{0}/θ}}{z} dx, ∫_{ℝ^{N}}[∇_{x}∫_{ℝ^{N}}[f − ρM_{θ}]φ(v)dv]^{4}\frac{e^{Φ_{0}/θ}}{z ρ^{2}} dx, \end{matrix} (2.13)

where φ(v) stands for a polynomial in v of degree less than 2. We are left with the task of estimating these quantities. This can be done by following the interpolation procedure described in [10], Section 4. However, we can also reduce our estimates directly to the results described in a series of Propositions in Section 4 of [10] (except constants). These results provides bounds on the quantities

\{ \begin{matrix}∫_{ℝ^{N}}[∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{2} \frac{1}{ρ} dx, ∫_{ℝ^{N}}[∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{4} \frac{1}{ρ^{3}} dx, \\ ∫_{ℝ^{N}} \frac{|∇_{x}ρ|^{2}}{ρ^{5}} [∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{4} dx, \\ ∫_{ℝ^{N}}[∇_{x}∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{2} \frac{1}{ρ} dx, ∫_{ℝ^{N}}[∇_{x}∫_{ℝ^{N}}[f − ρM_{θ}]φ(v) dv]^{4} \frac{1}{ρ^{3}} dx. \end{matrix} (2.14)

Since we assume 0 < af_{s} ≤ f ≤ Af_{s}, we get aze^{−Φ_{0}/θ} ≤ ρ ≤ Aze^{−Φ_{0}/θ}, and thus

(Aze^{−Φ_{0}/θ})^{−1} ≤ ρ^{−1} ≤ (aze^{−Φ_{0}/θ})^{−1},

so that terms in (2.13) are dominated by terms (2.14), and thus, nonlinear interpolations in Section 4 of [10] leads to bounds of the type C(ε, f) H̃(f|ρM_{θ})^{1−ε}.

Actually, in [10], terms in (2.14) are evaluated by C(ε, f)J(f|ρM_{θ})^{1−ε}, where

J(f|ρM_{θ})=∫ ∫ f log \frac{f}{ρM_{θ}} dx dv.

However the crucial property of J used within the proof is that

J(f|ρM_{θ}) ≥ b ∫_{ℝ^{N}} \frac{1}{ρ}[ ∫_{ℝ^{N}} |f − ρM_{θ}| dv]^{2} dx,

holds. We can check that such a property is also fulfilled by H̃(f|ρM_{θ}), since

H̃(f|ρM_{θ}) ≥ ∫_{ℝ^{N}} \frac{1}{ze^{−Φ_{0}/θ}} [ ∫_{ℝ^{N}} |f − ρM_{θ}| dv]^{2} dx,

and using 0 < af_{s} ≤ f ≤ Af_{s}

H̃(f|ρM_{θ}) ≥ a ∫_{ℝ^{N}} \frac{1}{ρ}[ ∫_{ℝ^{N}} |f − ρM_{θ}| dv]^{2} dx.

Finally, it remains to estimate the first term in the definition (2.7) of R(f) as

λθH̃(f|ρM_{θ}) ≤ CH̃(f|ρM_{θ})^{1−ε}

with C constant. It can be seen as an easy consequence of the tail control: af_{s} ≤ f ≤ Af_{s}. We can write

λθH̃(f|ρM_{θ}) = λθH̃(f|ρM_{θ})^{ε} H̃(f|ρM_{θ})^{1−ε}.

Remark that

H̃(f|ρM_{θ}) = ∫_{ℝ^{N} ×ℝ^{N}} \frac{f^{2}}{f_{s}} dv dx − ∫_{ℝ^{N}} \frac{ρ^{2}}{ze^{−Φ_{0}/θ}} dx

can be estimated as

H̃(f|ρM_{θ}) ≤ A^{2} ∫_{ℝ^{N} ×ℝ^{N}} f_{s} dv dx − a^{2} ∫_{ℝ^{N}} ze^{−Φ_{0}/θ} dx = A^{2} − a^{2}

(the right hand side is > 0). Then, we conclude that

λθH̃(f|ρM_{θ}) ≤ CH̃(f|ρM_{θ})^{1−ε},

with C = λθ(A^{2} − a^{2})^{ε}.

Remark 2.6 In order to apply previous lemma, we should prove hypotheses (2.12) on the solutions of (1.1). The proof of the first hypothesis is postponed to Appendix B. In fact, it can be shown the following result: Let f_{0} : ℝ^{N} × ℝ^{N} → ℝ be a probability density such that for some a, A > 0,

af_{s}(x, v) ≤ f_{0}(x, v) ≤ Af_{s}(x, v).

Let f be the unique solution of equation (1.1) with initial data f_{0}. Then f satisfies

af_{s}(x, v) ≤ f(t, x, v) ≤ Af_{s}(x, v).

It means that is suffices to assume that the initial data is estimated from above and below by f_{s} to preserve this property during time. For uniform in time estimates on all the derivatives of f the reader can see Appendix A for the harmonic potential.

Remark 2.7 We should point out that nonlinear interpolation results of Section 4 in [10], and thus, Lemma 2.5 rely on the following smoothness result due to Desvillettes [8], (see also [10, Proposition A.2]). Let φ ∈ C^{∞}(ℝ^{D}). Assuming that

0 ≤ φ(t, x) ≤ A e^{−Δ|x|^{2}}

for A, Δ > 0 and that

sup_{x∈ℝ^{D}} |∂^{α}φ(x)| ≤ K_{p},

holds for any α ∈ ℕ^{D}, |α| = p ∈ ℕ, yields for any k ∈ ℕ, ε > 0,

|∂^{α}φ| ≤ C(k, ε) e^{−(Δ−ε)|x|^{2}}

where the constant C(k, ε) depends on Δ and a finite number of K_{i}’s.

2.3 End of the proof of asymptotic behavior

In the previous subsections we have obtained the necessary tools to prove Theorem 1.1. It remains to put together the pieces. Lemma 2.2, Proposition 2.3 and Lemma 2.5 lead to the system of differential inequalities

\{ \begin{matrix}−\frac{d}{dt}H(f|f_{s}) = \frac{2}{τ}H̃(f|ρM_{θ}) \\ \frac{d^{2}}{dt^{2}} H̃(f|ρM_{θ}) ≥ λθH(f|f_{s}) − C_{ε}(f)H̃(f|ρM_{θ})^{1−ε}. \end{matrix}

Then, by using Theorem 2.1, we conclude that, for any ε > 0, there exists a constant C(ε, f) > 0 such that

H(f|f_{s}) ≤ C(ε, f_{0}) t^{−1/ε}

We end the proof by using

∥f(t) − f_{s}∥_{L^{1}}^{2} ≤ H(f|f_{s}).

It is worth ending the paper by mentioning the very recent work of Hérau-Nier [15]. They prove the asymptotic trend to equilibrium for the Fokker-Planck equation, with explicit exponential rate of convergence. The proof relies on very tedious hypoellipticity techniques. It would be interesting to determine if the method applies to the relaxation operator (1.2).

A Appendix

Theorem 1.1 needs uniform in time estimates on the x, v derivatives, at any order, of the solution f. In this appendix we sketch how to obtain such bounds for the particular case of the harmonic potential Φ_{0}(x) = \frac{|x|^{2}}{2} . Let us remark that this kind of uniform estimates are not trivial at all. Even in the case of the Fokker-Planck operator, for which C^{∞}-regularity is obtained for the solution independently of the regularity of the initial data due to the regularizing effect of diffusion in velocity, this uniform in time estimate is far from being trivial [10, Section 5]. Since equation (1.1) is purely hyperbolic, we can expect this result for C^{∞} initial data with bounded derivatives of all orders. In fact, one can write f(t, x, v) as the solution of an integral equation (mild solution) in terms of the characteristics,

\frac{d}{dt}X(t; x, v) = V (t; x, v), \frac{d}{dt}V (t; x, v) = −∇_{x}Φ_{0}(X(t; x, v))

with initial conditions X(0; x, v) = x, V (0; x, v) = v. Smoothness of the solutions in (t, x, v) and time dependent L^{∞} bounds of the derivatives are proved using the regularity of the initial data, of the characteristics and of the external potential and Gronwall’s lemma. Therefore, smoothness of the solution is not an issue but uniform in time estimates are not derived in this way. The uniform in time bounds come out by a delicate study of some integrals over the characteristics of the ODE system using the explicit form of the solution for the harmonic potential heavily.

Proposition A.1 Let f ∈ C^{0}(ℝ_{t}^{+}; L^{1}(ℝ_{x}^{N} × ℝ_{v}^{N})) be a solution of equation (1.1) with Φ_{0}(x) = \frac{|x|^{2}}{2} . Suppose the initial data f_{0} has all its derivatives in x and v bounded. Then f has all its derivatives in x and v bounded uniformly for t ≥ 0.

Sketch of the proof. The proof falls naturally into 4 successive steps.

Step 1. We apply the Fourier transform with respect to the variables x, v to equation (1.1). Thus, in Fourier variables, (1.1) becomes

(\frac{∂}{∂t} + η · ∇_{ξ} − ξ · ∇_{η})f̂ = \frac{1}{τ}(f̂(t, ξ, 0) e^{−θη^{2}/2} − f̂).  (A.1)

The initial data is required to satisfy the following regularity assumption

|f̂_{0}(ξ, η)| ≤ \frac{C_{k,0}}{(1 + ξ^{2} + η^{2})^{k}}  for any k ∈ ℝ^{+}.  (A.2)

We wish to establish a similar estimate on the solution f, uniformly with respect to time.

Step 2. Let us introduce the characteristics associated to (A.1)

\{ \begin{matrix}\frac{d}{dt}Ξ(t; ξ, η) = H(t; ξ, η) & \frac{d}{dt}H(t; ξ, η) = −Ξ(t; ξ, η) \\ Ξ(0; ξ, η) = ξ & H(0; ξ, η) = η \end{matrix}

Thus, we are led to the following mild formulation for (A.1)

\begin{aligned}f̂(t, ξ, η) & = & e^{−t/τ} f̂_{0}(Ξ(−t; ξ, η), H(−t; ξ, η))+ \\ & & +\frac{1}{τ} ∫_{0}^{t} e^{(σ−t)/τ} e^{−θH^{2}(σ−t;ξ,η)/2} f̂(σ, Ξ(σ − t; ξ, η), 0) dσ. \end{aligned}  (A.3)

Let us denote by A and B respectively the terms in the right hand side (i.e. the initial data term and the collision term, respectively). With this expression, the solution f̂ can be viewed as the fixed point of the following iterative scheme

\begin{aligned}f̂_{n+1}(t, ξ, η) & = & e^{−t/τ} f̂_{0}(Ξ(−t; ξ, η), H(−t; ξ, η)) \\ & & +\frac{1}{τ} ∫_{0}^{t} e^{(σ−t)/τ} e^{−θH^{2}(σ−t;ξ,η)/2} f̂_{n}(σ, Ξ(σ − t; ξ, η), 0) dσ \\ & = & (A_{n} + B_{n})(t, ξ, η) \end{aligned}  (A.4)

Since the collision term B only involves the unknown evaluated at η = 0, one first reduces (A.4) to this situation, i. e. we are looking at the inductive relation for the macroscopic density.

Step 3. This is the more elaborate step of the proof. In this step, we focus on (A.4) with η = 0. One can show that for any k ∈ ℕ, there exists a constant C_{k} > 0 such that

sup_{t,ξ} |f̂(t, ξ, 0)| ≤ \frac{C_{k}}{(1 + ξ^{2})^{k}} .  (A.5)

One proves it by induction in n using the particular form of the characteristics and a very sharp estimate of B_{n}. The control of A_{n} is trivial from the initial data. Step 4. To finish the proof, one turns to the general case η ≠ 0. Now, we can proceed directly on f without using the iterative scheme to conclude that

sup_{t,ξ,η} |f̂(t, ξ, η)| ≤ \frac{C_{k,0} + D_{k}}{(1 + ξ^{2} + η^{2})^{k}} ,  (A.6)

which ends the proof.

B Appendix

This Appendix is devoted to the proof of the preservation of the bounds

af_{s}(x, v) ≤ f(t, x, v) ≤ Af_{s}(x, v)

for solution of (1.1), see remark 2.6. It is made by induction. We consider the characteristics for (1.1); namely,

\{ \begin{matrix}\frac{d}{dt}X(t; x, v) = V (t; x, v)  \frac{d}{dt}V (t; x, v) = −∇_{x}Φ_{0}(X(t; x, v)) \\ X(0; x, v) = x  V (0; x, v) = v \end{matrix}

We rewrite (1.1) as an ODE

\frac{d}{ds}(e^{s/τ}f(t + s, X(s), V (s))) = \frac{1}{τ} e^{s/τ}ρM_{θ}(t + s, X(s), V (s)).

Hence, we are led to the following mild formulation

\begin{matrix}f(t, x, v) = e^{−t/τ}f_{0}(X(−t), V (−t)) \\ +\frac{1}{τ(2πθ)^{N/2}} ∫_{0}^{t} e^{(σ−t)/τ} ρ(σ, X(σ − t; x, v))e^{−V^{2}(σ−t;x,v)/(2θ)} dσ. \end{matrix}

We obtain the solution f as the fixed point of the following induction scheme

\{ \begin{matrix}f_{n+1}(t, x, v) = e^{−t/τ}f_{0}(X(−t; x, v), V (−t; x, v)) \\ +\frac{1}{τ(2πθ)^{N/2}} ∫_{0}^{t} e^{(σ−t)/τ}ρ_{n}(σ, X(σ − t; x, v))e^{−V^{2}(σ−t;x,v)/(2θ)} dσ \\ ρ_{n}(t, x) = ∫_{ℝ^{N}} f_{n}(t, x, v) dv. \end{matrix}

We look only at the estimate af_{s} ≤ f_{0}, the bound from above is obtain in analogously. We note that \frac{d}{dt}(2Φ_{0}(X(t)) + V^{2}) = 0 thus energy is conserved

Φ_{0}(X(t; x, v)) + \frac{V^{2}(t; x, v)}{2} = Φ_{0}(x) + \frac{v^{2}}{2} .

Therefore, the steady state verifies

f_{s}(X(−t; x, v), V (−t; x, v)) = f_{s}(x, v).  (B.1)

Since af_{s}(x, v) ≤ f_{0}(x, v) and (B.1), we have

af_{s}(x, v) = f_{s}(X(−t; x, v), V (−t; x, v)) ≤ f_{0}(X(−t; x, v), V (−t; x, v)).

On the other hand, the induction hypothesis f_{n}(t, x, v) ≥ af_{s}(x, v) yields ρ_{n}(t, x) ≥ (2πθ)^{N/2} aze^{−Φ_{0}(x)/θ}. It follows that

\frac{1}{τ(2πθ)^{N/2}} ∫_{0}^{t} e^{(σ−t)/τ} ρ_{n}(σ, X(σ − t; x, v)) e^{−V^{2}(σ−t;x,v)/(2θ)} dσ

≥ \frac{1}{τ(2πθ)^{N/2}} ∫_{0}^{t} e^{(σ−t)/τ} (2πθ)^{N/2}az e^{−Φ_{0}(X(σ−t;x,v))/θ} e^{−V^{2}(σ−t;x,v)/(2θ)} dσ

≥ \frac{az}{τ} ∫_{0}^{t} e^{(σ−t)/τ} e^{−(Φ_{0}(X(σ−t;x,v))+V^{2}(σ−t;x,v)/2)/θ} dσ

≥ aze^{−(Φ_{0}(x)+v^{2}/2)/θ} \frac{1}{τ} ∫_{0}^{t} e^{(σ−t)/τ} dσ = af_{s}(x, v)(1 − e^{−t/τ}),

by using the energy conservation. We conclude that af_{s}(x, v) ≤ f_{n+1}(t, x, v).

Acknowledgments The authors were partially supported by the European IHP network “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications”, RNT2 2001 349. M.J.C. and J.A.C. acknowledge support from DGI-MCYT project BFM2002-01710. Part of this work was done while T.G. was visiting the Department of Applied Math. of the University of Granada. It is kindly acknowledged for its hospitality.

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