ASYMPTOTIC BEHAVIOR OF AN INITIAL-BOUNDARY VALUE … · ASYMPTOTIC BEHAVIOR OF AN INITIAL-BOUNDARY...

ASYMPTOTIC BEHAVIOR OF AN INITIAL-BOUNDARY VALUEPROBLEM FOR THE VLASOV–POISSON–FOKKER–PLANCK

SYSTEM∗

L. L. BONILLA† , J. A. CARRILLO‡ , AND J. SOLER‡

SIAM J. APPL. MATH. c© 1997 Society for Industrial and Applied MathematicsVol. 57, No. 5, pp. 1343–1372, October 1997 007

Abstract. The asymptotic behavior for the Vlasov–Poisson–Fokker–Planck system in boundeddomains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered.It is proven that the distribution of particles tends for large time to a Maxwellian determined bythe solution of the Poisson–Boltzmann equation with Dirichlet boundary condition. In the proof ofthe main result, the conservation law of mass and the balance of energy and entropy identities arerigorously derived. An important argument in the proof is to use a Lyapunov-type functional relatedto these physical quantities.

Key words. boundary conditions for kinetic equations, Vlasov–Poisson–Fokker–Planck System,asymptotic behavior

AMS subject classifications. 35B40, 35Q99, 76X05, 82D10, 85A05

PII. S0036139995291544

1. Introduction and main result. The Vlasov–Poisson–Fokker–Planck(VPFP) system is a kinetic description of a physical plasma whose particles changeonly slightly their momentum during collision events. The plasma is supposed tobe in the self-consistent field created by the particles themselves. A typical examplewould be the slow momentum relaxation of a small admixture of a heavy gas in alight one (which is considered to be in equilibrium). The low density of the heavyparticles implies that their collision with one another may be neglected, whereas theirmomentum changes little when colliding with the light particles of the environment.Under these conditions the collision term in the kinetic equation may be approximatedby the Fokker–Planck form [31]. With respect to the self-consistent field created bythe particles, we may distinguish two cases: the “electrostatic” and “gravitational”potentials corresponding to heavy positively charged ions and neutral masses, respec-tively. Physical discussions on the validity of both Fokker–Planck and self-consistentfield approaches may be found in [37].

The VPFP equation for the distribution function f(t, x, v) (f dxdv is the numberof particles at time t located at a volume element dx about the position x and havingvelocities in a volume dv about the value v; from this interpretation f has to benonnegative) is

∂f

∂t+ (v · ∇x)f + divv((E − βv)f)− σ∆vf = 0(1)

on (0, T )×Ω×R3, T > 0, where Ω is a smooth enough bounded domain, for instanceΩ ∈ C2, in R3 and where β ≥ 0 and σ > 0 are constants which are related to the

∗Received by the editors September 13, 1995; accepted for publication (in revised form) May6, 1996. This research was partially supported by E.U. Human Capital and Mobility Programmecontract ERBCHRXCT930413 and DGICYT-MEC (Spain) projects PB92-0953, PB92-0248, PB92-1075, PB95-0296, and PB95-1203.

http://www.siam.org/journals/siap/57-5/29154.html†Escuela Politecnica Superior, Universidad Carlos III de Madrid, 28911 Leganes, Spain

([email protected]).‡Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071

Granada, Spain ([email protected], [email protected]).

1343

1344 L. BONILLA, J. CARRILLO, AND J. SOLER

collision (friction and diffusion) between particles. For thermal equilibrium to be apossible solution of (1), β and σ need to obey the Einstein relation, σβ = κΛ

m , where κ isthe Boltzmann constant, Λ is the temperature of the thermal bath, and m is the massof the molecules. Besides the distribution function f(t, x, v) ≥ 0, we have denoted byE(t, x) the electrostatic or gravitational vector force field, which is self-consistentlygiven by the elliptic Poisson equation

E(t, x) = −∇xΦ(t, x), −∆xΦ(t, x) = θρ(f)(t, x) on (0, T )× Ω.(2)

Here Φ(t, x) is the internal potential of the system, and ρ(f) denotes the macroscopicmass density, given by

ρ(f)(t, x) =∫R3f(t, x, v) dv.

The parameter θ is 1 in the electrostatic case and θ = −1 in the gravitational case.We do not consider the effect of an external potential on our system for the sake ofsimplicity, although our results could be extended also to that situation. See [9, 34, 40]for further physical interpretations and references about the physical meaning of theVPFP system.

We want to show that the solutions of the VPFP system tend to their stationarystates (in the appropriate functional setting; see below) as time goes to infinity. Theproof is based on deriving the mass conservation law and the balance of energy andentropy identities and exploiting the properties of a Lyapunov functional which is akind of free energy.

It is well known that the relative entropy between f and the stationary solutionsof the Vlasov–Poisson equation is the appropriate Lyapunov functional used to derivethe H-theorem; see [30, 34]. In contrast to the usual Fokker–Planck equation, which islinear in f , the VPFP system is nonlinear and the relative entropy is not a Lyapunovfunctional. Examples of nonlinear Fokker–Planck equations with known Lyapunovfunctionals include models synchronization of oscillator populations with mean-fieldcoupling (see [35, 36, 6, 4]). These examples have drift terms that depend linearly ona moment of the distribution function, which is also the case for the VPFP system.Thus, we may try to derive a Lyapunov functional for the VPFP system using thesame kind of ideas.

(i) Define a relative entropy with respect to a nonnormalized “stationary” distri-bution

η(t) =∫

Ω×R3f log

(f

f

)d(x, v),(3)

where f satisfies the equation

(v · ∇x)f + divv((E − βv)f)− σ∆v f = 0.

Here E = −∇Φ(t, x) is determined by the Poisson equation (2) with the exact distri-bution f . f is given by

f(t, x, v) = exp−βσ

(|v|22

+ Φ(t, x)− 1Mµ(t)

).(4)

Here expu = eu denotes the exponential function and M is the total mass of thesystem.

ASYMPTOTIC BEHAVIOR FOR VPFP SYSTEM 1345

(ii) Find µ(t) such that ∫Ω

∫R3f∂ log f∂t

dx dv = 0.

(iii) Show that η′(t) ≤ 0, supposing that all surface terms that may appear in thederivations below are zero.

(iv) Show that η(t) is bounded below, and justify dropping the surface integrals.To illustrate the procedure, let us derive the Lyapunov functional following the

above steps in a formal way. A more careful and rigorous computation retainingsurface terms will be performed in section 4. First, by taking the derivative of thefree energy functional we find

η′(t) =∫

Ω

∫R3

∂f

∂tlog(f

f

)dx dv +

∫Ω

∫R3

∂f

∂tdx dv −

∫Ω

∫R3f∂ log(f)∂t

dx dv.

The last two terms are zero because of mass conservation and by definition of η(t).Using the VPFP system written as

∂f

∂t= σdivv

∇vf − f∇v log(f)− f

β∇x log(f)

+σ

β∇v log(f) · ∇xf,

we find

η′(t) = −σ∫

Ω

∫R3f

∣∣∣∣∇v log(f

f

)∣∣∣∣2 dx dv ≤ 0

after integrating by parts and dropping surface integrals.To obtain µ(t) and thereby the free energy functional η(t), we write

0 =σ

β

∫Ω

∫R3f∂ log(f)∂t

dx dv =∫

Ω

∫R3

f∂Φ∂t− f

Mµ′dx dv

=∫

Ω

∫R3f∂Φ∂t

dx dv − µ′

=∂

∂t

∫Ω

∫R3fΦ dx dv − µ

−∫

Ω

∫R3

Φ∂f

∂tdx dv.

The last term is equal to∫Ω

∫R3

Φ (v · ∇x)f dx dv =∫

ΩΦ divx

(∫R3vf dv

)dx

plus surface terms that we drop in this formal calculation. We now use the continuityequation

∂

∂t

∫R3f dv + divx

(∫R3vf dv

)= 0,

and the Poisson equation so that

−∫

Ω

∫R3

Φ∂f

∂tdx dv = −

∫Ω

Φ∂ρ(f)∂t

dx


= θ

∫Ω

Φ∂∆xΦ∂t

dx = − θ

2∂

∂t

∫Ω|∇xΦ(t, x)|2 dx.

We therefore obtain the result

µ(t) =∫

Ω

Φ(t, x) ρ(f)(t, x) − θ

2|∇xΦ(t, x)|2

dx =

θ

2

∫Ω|∇xΦ(t, x)|2 dx,(5)

where we have again used Poisson equation and integration by parts.Given that η(t) is a Lyapunov functional, f will tend to a function f∞(t, x, v)

such that

η′(t) = −σ∫

Ω

∫R3

f∞(t, x, v)∣∣∣∣∇v log

(f∞

f(t, x, v)

)∣∣∣∣2 dx dv = 0.

This implies that f∞/f cannot depend on v. Hence,

f∞(t, x, v) = exp−β|v|

2

2σ

g∞(t, x) .

Inserting this expression in the VPFP system we have

∂

∂tg∞ = −v ·

∇xg∞ +

β

σg∞∇xΦ∞

,

where Φ∞ is the solution of the Poisson equation (2) associated with ρ(f∞). Asg∞(t, x) does not depend on v, we should have

∇xg∞ +β

σg∞∇xΦ∞ = 0,

∂

∂tg∞ = 0,

and, therefore,

g∞ = C∞ exp−βΦ∞

σ

,

where C∞ is a normalization constant. Thus, f∞ is a limit stationary solution of theVPFP system.

The rest of the paper makes precise and rigorous the preceding formal consider-ations. Let us start by specifying the proper boundary conditions to be considered.

Equations (1) and (2) are to be solved together with initial data fo, with Dirichletboundary conditions for the potential (∂Ω is a perfect conductor),

Φ(t, x) = 0 on ∂Ω(6)

and with boundary conditions on f defined by means of a general scattering ker-nel. From the possible kernels we will consider the ones that allow us to obtainmass conservation of the system and proper energy and entropy balance laws. Boun-dary conditions for kinetic equations are integral relations between the density ofmolecules coming out of an infinitesimal section of the boundary at a given timeand the density of the molecules impinging upon the same boundary section. Moreprecisely, let us define the sets Γx± =

v ∈ R3 such that sign(v · n(x)) = ±1

and


Γ± =

(x, v) ∈ ∂Ω× R3 such that sign(v · n(x)) = ±1

, where n(x) is the unit nor-mal outward on the boundary of the domain Ω at x ∈ ∂Ω. Denote by γ±f and γfthe traces of f on Σ± = [0,∞) × Γ± and Σ = [0,∞) × ∂Ω × R3, respectively, whenthese traces make sense (see section 2). Let us introduce the sets QT = [0, T [×Ω×R3,ΣT± = [0, T ]×Γ±, and ΣT = [0, T ]×∂Ω×R3. Given x ∈ ∂Ω and t > 0 we will assumethat

γ−f(t, x, v) =∫

Γx+

R(t, x; v, v∗) γ+f(t, x, v∗) dv∗(7)

for any v ∈ Γx−. R represents the probability that a molecule with velocity v∗ at timet striking the boundary at x reemerges in the same instant with velocity v. Specularreflection is a particular case of the above relation. We will discuss in section 2 theconcrete hypotheses in the definition of these conditions.

In order to establish the concept of solution which will be used in this work,let us introduce first the set L1(ΣT±; |v · n(x)|dSdvdt) of all integrable functions inΣT± with respect to the standard kinetic measure |v · n(x)|dSdvdt, where dS is theLebesgue measure on ∂Ω. We also define the space of functions W 1(QT ) = f ∈L1(QT ) such that ∂f

∂t + (v · ∇x)f ∈ L1(QT ). To make precise our main result andspecify the boundary conditions to be considered, we need to know a trace result, seethe following section for the statement. The trace theorem will provide us with thegood definition of the trace operator between W 1(QT ) and L1(ΣT±; |v · n(x)|dSdvdt).Let us denote by W 1,p

o (Ω) the classical Sobolev space with exponent p and zero traceon the boundary of Ω and H1

o (Ω) = W 1,2o (Ω). Now we can state the definition of weak

solution for this initial-boundary value problem.DEFINITION 1.1. Given fo ∈ L1(Ω×R3) we will say that (E, f) is a weak solution

of problem (1)–(7) with initial data fo if

f ∈ C([0,∞), L1(Ω× R3)) ∩W 1(QT ) , γ±f ∈ L1(ΣT±; |v · n(x)|dSdvdt)

for any T > 0 and (7) is verified for suitable scattering kernels. Also, for any Ψ ∈C∞o (QT ) we have∫

QT

f

(∂Ψ∂t

+ (v · ∇x)Ψ− β(v · ∇v)Ψ + (E · ∇v)Ψ + σ∆vΨ)d(t, x, v)(8)

+∫

Ω×R3foΨ(0, x, v) d(x, v) =

∫ΣT

(v · n(x)) γf Ψ dSdvdt

and E = −∇xΦ, where Φ ∈ L∞([0, T [, H1o (Ω)) is a weak solution of the Dirichlet

problem for the Poisson equation (2).The existence of weak solutions of the initial-boundary value problem for ki-

netic equations has been studied for the Boltzmann and Vlasov–Poisson systems. C.Cercignani [18] and K. Hamdache [27] proved the existence of weak and renormalizedsolutions for the Boltzmann equation with reflection-type boundary conditions. Later,R. Alexandre in [3] studied the Vlasov–Poisson equation in the case of absorbing-typeboundary conditions, but he also remarked that his method is valid to obtain existenceresults for the Vlasov–Poisson equation with the reflection-type boundary conditions.Also, N. B. Abdallah [1] deals with this problem, assuming a milder hypothesis onthe initial data than in the work of R. Alexandre. No such results seem to be knownfor the VPFP system with this kind of boundary conditions.


The VPFP system in x, v ∈ R3 has been extensively studied during the past years.Existence and uniqueness results have been obtained in several frameworks: classicalsolutions, weak solutions, renormalized solutions, and functional solutions. We referto [7, 8, 12, 13, 14, 15, 25, 33, 39, 40], and the references therein.

The study of the qualitative properties of this system is an interesting problemnowadays. In a recent work, F. Bouchut and J. Dolbeault [9] study the large-timeasymptotic for the solutions of the VPFP in the case that the particles occupy thewhole space R3. They deal with the solutions obtained in [7] assuming also that thepotential energy in the gravitational case remains bounded independently of t and thatthe system is under the action of a suitable external potential Φo. The existence ofsuch external potential is an essential point in the arguments of Bouchut and Dolbeaultin order to confine the particles in a bounded region, thus avoiding their unrestrictedspreading. Under these conditions they proved that the distribution function tendsto a Maxwellian given by the limit potential and the temperature of the surroundingbath (σβ ). Some of the techniques used in this paper are inspired by those of [9].

In another recent work J. A. Carrillo, J. Soler, and J. L. Vazquez [16] deal withVPFP system in R3 without friction (β = 0). In that work it is shown that theasymptotic behavior of some specific weak solutions of (1)–(2) is given by the funda-mental solution G of the linear part of (1). The proof relies on the self-similarity ofthe fundamental solution G and the use of the smoothing effect of the Fokker–Planckoperator proved in [8, 13]. The hypotheses assumed on the solutions are verified atleast for small initial data (see [13]).

In the case of initial-boundary value problems, we must mention the works of L.Desvillettes and J. Dolbeault [22] and L. Desvillettes [21]. They treated the Vlasov–Poisson–Boltzmann system and the Boltzmann and Bhatnager–Gross–Krook equa-tions with specular reflection boundary conditions for f and Dirichlet boundary con-ditions for the potential in [22]. They showed that the large time asymptotics ofthe corresponding systems are described by Maxwellians. In the Vlasov–Poisson–Boltzmann case, these Maxwellians are determined by the initial mass, the initialenergy, and the limit potential.

The aim of this paper is to study the asymptotic behavior of weak solutions ofthe system (1)–(2) with particles moving in a bounded domain so that the potentialand the density of particles satisfy the above boundary conditions (6) and (7). Wewill assume the following hypotheses on the solutions.

Let (E, f) be a weak solution of (1), (2), (6), and (7) with initial data fo verifying(1) f ∈ C([0,∞), L1 ∩ L∞(Ω× R3)).(2) |v|2f ∈ L∞([0, T ], L1(Ω× R3)) for any T > 0.(3) |v|2γf, γf2 ∈ C([0,∞), L1(Γ±; |v · n(x)|dSdv)).(4) Φ, ∂Φ

∂t ∈ C([0, T [, H1o (Ω)) and E ∈ L∞(]0, T ]× Ω) for any T > 0.

(5) ∇vf ∈ L2([0, T ]× Ω× R3) for any T > 0.(6) In the gravitational case, we also assume that

lim supt→∞

∫Ω|E(t, x)|2 dx < ∞.

Although these hypotheses seem reasonable from a physical point of view, wehave not found results ensuring the existence of such a solution of the VPFP problemin the previous literature. The main difficulty in proving existence of such a weaksolution (E, f) is to handle effectively the diffusion of the particles and the flow ofparticles through the boundary due to the boundary conditions. In a forthcoming


paper (see [11]), one of the authors of the present paper has proven existence of asolution of the VPFP problem with absorbing or reflection-type boundary conditionsverifying the above hypotheses except for (i) hypothesis (6) (gravitational case), (ii)E ∈ L∞(]0, T ] × Ω), and (iii) the regularity of the traces, i.e., f ∈ W 1(QT ). Toobtain these properties, a regularity result should be proved. In this regard, all theabove properties hold if we substitute an elliptic equation of greater order instead ofPoisson’s equation. For the resulting regularized problem, the regularity of E and ofthe traces is ensured (see [11]).

In order to simplify the exposition, from now on we will denote by (P) the problem(1), (2), (6), and (7). When dealing with solutions of (P), we will also assume thatthey satisfy the above hypotheses (1)–(6).

We can state the main result of this work.THEOREM 1.2. Under the hypotheses (1)–(6) on the solutions, for every time

sequence tn → ∞ there exists a subsequence (that we denote with the same index)such that

f(tn, x, v)→ f∞ as n→∞

in L1(Ω× R3) with

f∞(x, v) =(

2πσβ

)− 32

λ exp−βσ

(|v|22

+ Φ∞(x))

,

where Φ∞(x) is a weak solution of

−∆Φ∞ = θ λ exp−βσ

Φ∞

(9)

in H1o (Ω) and

λ = ‖fo‖L1(Ω×R3)

(∫Ω

exp−βσ

Φ∞(x)dx

)−1

.

Moreover, in the electrostatic case the solution of (9) is unique in H1o (Ω), and we

deduce

limt→∞

‖f(t, .) − f∞‖L1(Ω×R3) = 0.

In the gravitational case, the limit problem (9) may have multiple or no solutionsdepending on λ (see the last section of this paper). Then we cannot ensure thatf(tn, x, v) converges to the same solution for every time sequence. This is probablyrelated to the hypotheses on the initial data that we may have to impose to makesure that the properties (1)–(6) on the solutions hold.

As a consequence of the above theorem, the asymptotic behavior of the phasespace distribution function is determined by studying the limit potentials which satisfythe following equation:

−∆u+ θα

(∫Ωeu dx

)−1

eu = 0,(10)

u = 0 on ∂Ω(11)


with α > 0. Equation (10) is called the Poisson–Boltzmann (PB) equation. In theelectrostatic case, K. Dressler [24] and J. Dolbeault [23] studied (10) in Ω = R3 inthe presence of an external potential. In bounded domains D. Gogny and P.L. Lions[26] showed the existence of a unique regular solution of problem (10)–(11) in theelectrostatic case in the space H1

o (Ω). In the gravitational case the PB equation hasbeen studied by A. Krzywicki and T. Nadzieja in [28, 29]. They proved that problem(10)–(11) has no classical solutions for α > 0 large enough for simply connecteddomains.

Let us point out that in Theorem 1.2 we may consider an alternative conditionto hypothesis 6 on the solutions. In fact, the condition

(6′) the potential Φ(t, x) belongs to L∞(0,∞;W 1,3o (Ω))

implies (6) but also assures us that the PB equation is well defined in L1 by meansof the Trudinger–Moser inequality.

Let us briefly comment on the techniques and main ideas in the proof of Theorem1.2. A fundamental step in the proof is based on deriving the equations for the relevantphysical quantities of the system, the balance of mass, energy, and entropy identities.These identities will allow us to understand the effects of the boundary conditions onthe fluxes of particles, energy, and entropy through the boundary of the domain. Infact, we will rigorously establish the following relations.

• Balance of mass identity:

d

dt‖f(t, ·)‖L1(Ω×R3) = −

∫∂Ω×R3

(v · n(x)) γf(t, x, v) dv dS,

for which we will prove that under the previous hypotheses on the scatteringkernel, the right-hand side of the above equality is zero, thereby obtainingthe mass conservation law.• Balance of energy identity:

d

dt

12

(∫Ω×R3

|v|2f d(x, v) + θ

∫Ω|E(t, x)|2 dx

)

= −12

∫∂Ω×R3

(v · n(x)) |v|2f dS dv − β

∫Ω×R3

|v|2f d(x, v) + 3σM.

• Balance of entropy identity:

d

dt

∫Ω×R3

f log f d(x, v) = 3βM

−∫∂Ω×R3

(v · n(x)) f(t, x, v) log f dS dv − 4σ∫

Ω×R3

∣∣∣∇v√f ∣∣∣2 d(x, v).

An important idea to determine the asymptotic behavior of solutions of the VPFPis to use the free-energy Lyapunov functional η(t). The Lyapunov functional is definedby (3), (4) and (5). The above balance laws will play a key role in proving that η(t)is indeed a Lyapunov functional. For example, by studying the fluxes of entropyand kinetic energy through the domain boundary, we will prove that η(t) is a non-increasing function. In the expression of η′(t) there appears an additional term (withrespect to the formal derivation given above) which is due to boundary contributions.


The control and analysis of such boundary terms is an essential part of our proof.Once we have proved that η(t) is a Lyapunov functional, we would like to use itto establish our large-time asymptotics. To be able to take the limit t → ∞, weneed some compactness property of the sequence fn(t, x, v) = f(tn + t, x, v). Asan intermediate step we prove the compactness of

√fn in L2([0, T ] × Ω × R3). This

allows us to obtain the compactness result of fn in C([0, T ];L1(Ω × R3)). Finallythe identification of the limiting distribution function concludes the proof of Theorem1.2.

The paper is structured as follows: In section 2 we define the boundary conditionsand establish the trace result. In section 3 we analyze the properties of solutions ofproblem (P) and derive the balance of mass, energy, and entropy identities. In section4 we show that the free energy is a Lyapunov functional. In section 5 we prove themain result of this work by using a compactness result. Finally, the last sectioncontains several final remarks and open problems.

2. Boundary conditions and trace operators. To introduce the boundaryconditions on f , we must analyze the existence of traces for f on the boundary of QTthe closure of QT . This problem was studied for the free transport operator by S.Ukai in [38] in the context of Boltzmann’s equation. Also, M. Cessenat in [19] studiedit and applied it to neutron transfer equation. Surveys of these results can be foundin [25, 20, 17].

Let us denote by Lp(Σ±; |v · n(x)|dSdvdt) the set of all the functions g on Σ±such that |g|p is integrable with respect to the kinetic measure |v · n(x)|dSdvdt.

We will recall some results concerning to the trace operators which are necessaryin our setting. For the details and proof of these results we refer to [17]. Let T bethe free transport operator which is given by

T f =∂f

∂t+ (v · ∇x)f.

Associated with each point r = (t, x, v) of the boundary of QT , we consider Rs(r)the characteristic line

Rs(r) = Rs(t, x, v) = (t+ s, x+ sv, v) with t−(r) ≤ s ≤ t+(r),

where t±(r) are the entry and exit time from QT : t+(r) is the earliest relative times > 0 at which Rs(r) belongs to the boundary of QT . Similarly, t−(r) is the latestrelative time s < 0 at which Rs(r) belongs to the boundary of QT . Strictly speaking,they are defined as

t−(r) = max −t, infs ≤ 0 such that x+ τv ∈ Ω for any τ ∈ [s, 0]

and

t+(r) = min T − t, sups ≥ 0 such that x+ τv ∈ Ω for any τ ∈ [0, s] .

The trace theorem holds between the space

W p(QT ) = f ∈ Lp(QT ) such that T f ∈ Lp(QT )

and the space Lp(ΣT±; τ(x, v)|v · n(x)|dSdvdt), where τ(x, v) = min1, t−(r) + t+(r).More exactly, we have the following result.


THEOREM 2.1. Let 1 ≤ p ≤ ∞. If we define the traces γ±f , γf in C1o (QT )

by restriction, then the trace operators have a bounded extension from W p(QT ) toLp(ΣT±; τ(x, v)|v · n(x)|dSdvdt).

We will focus our attention on the case p = 1. In this case, the idea of theproof for the above theorem is to show that f is absolutely continuous in s alongthe characteristics Rs(r) a.e. on r. In fact, if we consider f(s, r) = f(Rs(r)) and wedenote by g the L1(QT ) function such that T f = g, then for any r ∈ QT it is possibleto prove (see [17]) that

f(s′, r) = f(s, r) +∫ s′

s

g(τ, r) dτ(12)

with t−(r) ≤ s, s′ ≤ t+(r). Thus, the definition of the traces is given by

γ±f(t, x, v) = f(t±(r), r)

with (t, x, v) ∈ ΣT± and any r ∈ QT such that (t, x, v) is its exit or entry point, i.e.,that r satisfies Rs(r) = (t, x, v) with s = t±(r). Finally, let us point out that thespace L1(ΣT±; |v · n(x)|dSdvdt) is included in L1(ΣT±; τ(x, v)|v · n(x)|dSdvdt).

The following Green’s type identity can be easily deduced.LEMMA 2.2. Let f ∈ W 1(QT ) ∩ C([0, T ], L1(Ω × R3)) and γ+f ∈ L1(ΣT+; |v ·

n(x)|dSdvdt). Then γ−f ∈ L1(ΣT−; |v · n(x)|dSdvdt), and for any Ψ ∈ C∞o (QT ), itholds that ∫

QT

f T Ψ d(t, x, v) +∫QT

T f Ψ d(t, x, v)

=∫

Ω×R3[(fΨ)(T, x, v)− (fΨ)(0, x, v)] d(x, v) +

∫ΣT

(v · n(x)) γf Ψ dSdvdt.

The proof of this result is obtained by integration in (12). Finally, since we willneed to handle nonlinear functions of f and their traces on the boundary, we have tostudy for which nonlinear functions of f traces can be defined.

LEMMA 2.3. Let f ∈W 1(QT )∩C([0, T ], L1(Ω×R3)∩L∞(Ω×R3)). If ξ ∈ C2(R)with ξ′′ ∈ L∞(R) and ξ(0) = 0, then ξ(f) ∈ W 1(QT ). As a consequence, we haveγ±ξ(f) ∈ L1(ΣT±; τ(x, v)|v · n(x)|dSdvdt) and

γ±ξ(f) = ξ(γ±f).

Proof. Let us consider a sequence of mollifiers δt,xn = δtnδxn which approximate the

Dirac delta function. The support of δxn is located in the ball B(0, 1n ) and δxn(x) =

δxn(−x) and the support of δtn is located on the negative real axis of length 1n . Let us

denote by f the extension by zero of the function f to the whole [0, T ]× R6. We setfn = δt,xn ? f , where the symbol ? represents the usual convolution product which, inthis case, is

δt,xn ? f =∫ ∞

0

∫R3δtn(t− s)δxn(x− y)f(s, y, v)ds dy.

For each n, we define the following set:

Ωn =x ∈ Ω such that dist(x, ∂Ω) >

1n

.


Since f ∈W 1(QT ) we obtain that

∂f

∂t+ (v · ∇x)f = g

in the sense of distribution on (0, T )× Ω× R3, with g ∈ L1(QT ).Let Ψ ∈ C∞o ((0, T )× Ωm × R3). If n ≥ m, we have∫

QT

fn

(∂Ψ∂t

+ (v · ∇x)Ψ)d(t, x, v) =

∫QT

f

(∂

∂t+ (v · ∇x)

)(δt,xn ?Ψ) d(t, x, v).

Since δt,xn ?Ψ ∈ C∞o ((0, T )× Ω× R3), we can deduce∫QT

fn

(∂Ψ∂t

+ (v · ∇x)Ψ)d(t, x, v) = −

∫QT

g(δt,xn ?Ψ) d(t, x, v)

= −∫QT

gn Ψ d(t, x, v),

where gn = δt,xn ? g.Therefore, we obtain in the classical sense that

∂fn∂t

+ (v · ∇x)fn = gn

in (0, T )× Ωn × R3. Multiplying by ξ′(fn), we find

∂ξ(fn)∂t

+ (v · ∇x)ξ(fn) = ξ′(fn)gn.

Thus, if we consider Ψ ∈ C∞o ((0, T )× Ω× R3), then for large enough n we haveΨ ∈ C∞o ((0, T )× Ωn × R3)), and∫

QT

ξ(fn)(∂Ψ∂t

+ (v · ∇x)Ψ)d(t, x, v) = −

∫QT

ξ′(fn) gn Ψ d(t, x, v).

Taking into account the hypothesis on f and ξ, it is easy to show that ξ(fn)converges to ξ(f) in L∞([0, T ], L1(Ω × R3)). On the other hand, since ξ′′ ∈ L∞(R)then we have

|ξ′(fn)| ≤ a+ bfn.

Using that f ∈ C([0, T ], L∞(Ω × R3)), we obtain that ξ′(fn)Ψ is bounded inde-pendently of n. Since gn tends to g in L1(QT ), then we finally conclude that∫

QT

ξ(f)(∂Ψ∂t

+ (v · ∇x)Ψ)d(t, x, v) = −

∫QT

ξ′(f) g Ψ d(t, x, v).

Thus, we have

∂ξ(f)∂t

+ (v · ∇x)ξ(f) = gξ′(f)

in the distribution sense on (0, T )×Ω×R3, with gξ′(f) ∈ L1(QT ). As a consequence,ξ(f) ∈W 1(QT ).


The final assertion is a straightforward consequence of the definition of the traceoperator γ±.

Once we know how to deal with the traces of f , we can study the boundaryconditions introduced in section 1. In [17] C. Cercignani, R. Illner, and M. Pulvirentidiscuss the choice of suitable boundary conditions for the Boltzmann equation. Similarremarks can be done in the VPFP case. In order to define the boundary conditions,we will assume that f ∈ W 1(QT ) for any T > 0. Given x ∈ ∂Ω and t > 0 we willrequire that the phase space distribution function satisfies the boundary conditions(7) on ∂Ω:

γ−f(t, x, v) =∫

Γx+

R(t, x; v, v∗) γ+f(t, x, v∗) dv∗

for any v ∈ Γx−, where, as we have pointed out above, R represents the probabilitythat a molecule with velocity v∗ at time t striking the boundary on x reemerges at thesame instant and location with velocity v. We will assume that R has the followingproperties.

(1) R is always nonnegative.(2) R verifies the following normalization:

|v∗ · n(x)| =∫

Γx−

R(t, x; v, v∗) |v · n(x)| dv(13)

for any v∗ ∈ Γx+.(3) The so-called reciprocity principle holds; i.e., if we denote by M(v) the Max-

wellian at the wall (the Maxwellian given by the temperature of the thermal bathsurrounding the particles), R verifies

M(v) =∫

Γx+

R(t, x; v, v∗) M(v∗) dv∗(14)

for any v ∈ Γx−. In the case of the Boltzmann equation, M(v) can depend on t and x.Nevertheless, for the VPFP system the Maxwellian at the wall must be determined bythe temperature of the thermal bath which is σ

β , i.e.,

M(v) =(

2πσβ

)− 32

exp−β|v|

2

2σ

.

We can rewrite the boundary condition (7) as

γ−f = K[γ+f ](15)

in terms of an operator K defined by

(K[γ+f ])(t, x, v) =∫

Γx+

R(t, x; v, v∗) γ+f(t, x, v∗) dv∗,

for any (t, x, v) ∈ ΣT−. It is clear from (13) that K is a bounded linear operator fromL1(ΣT+; |v · n(x)|dSdvdt) into L1(ΣT−; |v · n(x)|dSdvdt).

Note that R can be even a Dirac delta measure. In fact, we are only imposing onR the minimal assumptions needed for (7) to be well-defined. As an example, R may


belong to the function space L∞t,x([0,∞) × ∂Ω;L1(Γx−,M+(Γx+))), where M+(Γx+) isthe set of positive Radon measures on Γx+.

Let us point out that this type of operators include the classical cases of specularand reversed reflection. For instance, if we consider v′ = v − 2(v · n(x))n(x), for anyv ∈ Γx−, and choose R(t, x; v, v∗) = δv′ , where δv′ is the Dirac delta function centeredon v∗ = v′, we obtain that

f(t, x, v) = f(t, x, v′) for any (x, v) ∈ Γ−.

A similar construction can be done to obtain the reversed reflection f(t, x, v) =f(t, x,−v) in Γ−.

We refer to [17, 18, 27] for further properties of these boundary conditions andphysical interpretations. In these works some important relations valid for theseboundary conditions are derived. We will review these relations in the followingsection.

3. Properties of the solution. Our first objective is to understand what arethe effects of the boundary condition (7) on the fluxes of particles, kinetic energyand entropy through the boundary of the domain. Later, we will obtain the mainbalance laws which are verified by the solution of this system. These balance lawswill be instrumental in proving that η(t) is a Lyapunov functional and then our maintheorem.

Let us recall that we will work with solutions of the problem (P), which meansweak solutions in the sense of Definition 1.1 satisfying the hypotheses 1–6 above andwith a boundary condition defined by a kernel R with the properties of the previoussection.

LEMMA 3.1. The solution of the problem (P) is such that the function

t −→ ‖f(t, ·)‖L1(Ω×R3)

is absolutely continuous in t > 0 and it obeys the continuity equation in integral form(balance of mass):

d

dt‖f(t, ·)‖L1(Ω×R3) +

∫∂Ω×R3

(v · n(x)) γf(t, x, v) dv dS = 0.

Moreover, f satisfies ∫R3

(v · n(x)) γf(t, x, v) dv = 0(16)

for any t ≥ 0 and x ∈ ∂Ω. As a consequence, the mass of f is preserved, i.e., for anyt > 0,

‖f(t, ·)‖L1(Ω×R3) = ‖fo‖L1(Ω×R3) = M.(17)

Proof. Let us prove (16). Since f verifies the boundary condition given by thescattering kernel, then multiplying (7) by |v · n(x)| and integrating in Γx− we find∫

Γx−

|v ·n(x)| γ−f(t, x, v) dv =∫

Γx−

|v ·n(x)|∫

Γx+

R(t, x; v, v∗) γ+f(t, x, v∗) dv∗ dv.

Using Fubini’s theorem we obtain∫Γx−

|v ·n(x)| γ−f(t, x, v) dv =∫

Γx+

∫Γx−

|v ·n(x)| R(t, x; v, v∗) dv γ+f(t, x, v∗) dv∗.


Then, the second property of R (13) allows us to get∫Γx−

|v · n(x)| γ−f(t, x, v) dv =∫

Γx+

|v∗ · n(x)| γ+f(t, x, v∗) dv∗.

Finally, it is easy to check that the flux of particles through ∂Ω vanishes:∫R3

(v · n(x)) γf(t, x, v) dv = 0,

due to the definition of Γx±.The balance of mass follows formally by integrating (1) in Ω×R3 and then using

the divergence theorem. To derive it rigorously we use a special test function in (8).Let h(t) ∈ C∞o ([0, T [), let χ(s) be a C∞o (R) positive function with

χ(s) = 1 on |s| ≤ 1 , χ(s) = 0 on |s| ≥ 2,

and define ψR(v) = χ( |v|R ). Then ψR(v) = 1 on |v| ≤ R, ψR(v) = 0 on |v| ≥ 2R,and ∇vψR is bounded independently of R ≥ 1.

By using the test functions Ψ = hψR in the definition of weak solution (8), weobtain ∫

QT

f

(dh

dtψR − βh(v · ∇v)ψR + h(E · ∇v)ψR + hσ∆vψR

)d(t, x, v)

+∫

Ω×R3foh(0)ψR d(x, v) =

∫ΣT

(v · n(x)) γf hψR dSdvdt.

Given that ψR → 1 a.e. and any derivative of ψR tends to 0 a.e. as R → ∞,let us show that we can pass to the limit as R → ∞ term by term in the previousequation. Taking into account that the function (v ·∇v)ψR is bounded independentlyof R, together with the hypotheses

f ∈ C([0,∞), L1(Ω× R3))

and

E ∈ L∞(]0, T [×Ω),

we can apply the Lebesgue dominated convergence theorem to show that∫QT

fdh

dtd(t, x, v) +

∫Ω×R3

foh(0) d(x, v) =∫

ΣT(v · n(x)) γf h dSdvdt.(18)

We have therefore obtained from (18) that

d

dt‖f(t, ·)‖L1(Ω×R3) = −

∫∂Ω×R3

(v · n(x)) γf dv dS = 0

in the sense of distributions on (0, T ), which proves the assertions of the lemma.The next result will be useful in order to prove that the entropy and the flux of

entropy through the boundary are well defined. As usual, let us denote by log+ f =max0, log f and by log− f = | log f | − log+ f . The following result agrees with asimilar one in [9] for the case x ∈ R3, and we refer to [9] for a proof.


LEMMA 3.2. Assume that f ≥ 0 verifies (1+|v|2)f ∈ L1(Ω×R3). Then f log− f ∈L1(Ω× R3) and∫

Ω×R3f log− f d(x, v) ≤ C(b,Ω, fo) +

12b

∫Ω×R3

|v|2f d(x, v)

for with any b > 0. Also, if (1 + |v|2)f ∈ L1(Γ; |v · n(x)|dSdv), then f log− f ∈L1(Γ; |v · n(x)|dSdv).

The next property will allow us to handle the flux of entropy and kinetic energythrough the boundary and will be crucial to show that the free energy functional η(t)is a nonincreasing function of time.

LEMMA 3.3. The solution of the problem (P) verifies that γf log γf ∈ L1(ΣT ; |v ·n(x)|dSdvdt)) for any T > 0. Moreover, for any 0 < t < T∫

∂Ω×R3(v · n(x))

(β

2σ|v|2 + log γf

)γf dS dv ≥ 0.(19)

Proof. We first observe that x log+ x ≤ x2, for any x ≥ 0. Using the hypothesis(3) on the solutions, i.e.,

γf2 ∈ C([0,∞), L1(Γ±; |v · n(x)|dSdv)),

we can deduce that γf log+ γf ∈ L1(ΣT ; |v · n(x)|dSdvdt)). On the other hand, theprevious lemma gives us that γf log− γf ∈ L1(ΣT ; |v · n(x)|dSdvdt)).

The inequality (19) follows from the reciprocity principle (14) satisfied by R andthe Jensen inequality. This property can be found in [17] (on page 241; notice thatthe authors use the inward normal to ∂Ω as n) and [27] and references therein.

Remark 3.1. In the case of specular and reversed boundary conditions, the flux ofparticles, the flux of kinetic energy, and the flux of entropy through the boundary arenull. This fact can be easily seen because of the functions (v·n(x)) γf , |v|2(v·n(x)) γf ,and (v · n(x)) γf log γf are odd with respect to |v · n(x)|.

Let us recall a classical interpolation result (for instance, see [3]) in the theory ofkinetic equations which will be useful to deduce some properties of the density ρ(f)and of the current density j(f):

j(f)(t, x) =∫R3vf(t, x, v) dv.

LEMMA 3.4. Let

Jk(x) =∫R3|v|kfdv.

Assume that f(x, v) ∈ L1(Ω × R3) ∩ L∞(Ω × R3), with f ≥ 0 and 0 ≤ k′ ≤ k. IfJk ∈ L1(Ω), then Jk′ ∈ Lr(Ω) with r = 3+k

3+k′ .As a consequence, the solution of the problem (P) satisfies, for any T > 0,

ρ(f) ∈ L∞([0, T ], L53 (Ω)), j(f) ∈ L∞([0, T ], L

54 (Ω)).

We will now obtain the equation of balance for energy. We start with the conti-nuity equation for the density ρ(f).


LEMMA 3.5. The solution of the problem (P) satisfies the continuity equation on(0, T )× Ω in the sense of distributions

dρ(f)dt

+ divxj(f) = 0.

Proof. This property can be deduced formally by integration of (1) with respectto the variable v and the application of the divergence theorem. Using the propertiesof the solutions and ideas similar in Lemma 3.1, the result can be rigorously provenby considering in (8) a test function Ψ = ϕ(t, x)ψR(v), where ϕ ∈ C∞o (Ω× [0, T [) andψR are defined as in Lemma 3.1.

In the following lemma we obtain the balance of the potential energy.LEMMA 3.6. The solution of the problem (P) satisfies that the function

t −→∫

ΩΦρ(f) dx

is absolutely continuous and

d

dt

∫Ω

Φρ(f) dx = 2∫

Ω

∂Φ∂tρ(f) dx = −2

∫ΩE · j(f) dx.

Proof. The above relation can be deduced formally by multiplying (1) by Φ,integrating it in x and v, and using the divergence theorem. Since

Φ,∂Φ∂t∈ C([0, T ], H1

o (Ω))

and E = −∇xΦ ∈ L∞(]0, T [×Ω), then a density argument allows us to consider asequence Φn of C∞o ([0, T ] × Ω) functions that converges to Φ while their derivatives∂Φn∂t converge to ∂Φ

∂t in C([0, T ], H1o (Ω)). Moreover, we can assume that the sequence

En = −∇xΦn converges to E in L∞([0, T ], Lq(Ω)) for any 2 ≤ q <∞. Therefore, weuse arguments similar to those used in Lemma 3.1 with test function ϕ = Φn(t, x)h,where h(t) ∈ C∞o ([0, T [), to conclude, after some calculations, the announced result.Here, we use the regularity on ρ(f) and j(f) (Lemma 3.4) and the Sobolev embeddingsto take the limit n→∞.

In the next result we obtain the balance of energy identity.LEMMA 3.7. The solution of the problem (P) satisfies that the function

t −→∫

Ω×R3|v|2f d(x, v)


d

dt

(12

∫Ω×R3

|v|2f d(x, v) +θ

2

∫Ω|E|2 dx

)

= − 12

∫∂Ω×R3

(v · n(x)) |v|2γf dS dv − β∫

Ω×R3|v|2f d(x, v) + 3σM.

Proof. In an analogous way as in the previous balance laws, the balance of energycan be obtained multiplying (1) by |v|2, integrating the result with respect to x and vand then using the divergence theorem. Since the arguments are very similar to those


previously explained, we only sketch the proof. Choosing in (8) as a test functionΨ = |v|2ψR(v)h(t), where ψR and h are defined as in Lemma 3.1, we obtain∫

QT

|v|2f(dh

dtψR − βh(v · ∇v)ψR − 2βhψR

)d(t, x, v)

+∫QT

|v|2f (h(E · ∇v)ψR + hσ∆vψR) d(t, x, v)

+∫QT

(6σfh+ 4σfh(v · ∇v)ψR + 2hf(E · v)ψR) d(t, x, v)

+∫

Ω×R3fo|v|2ψR(v)h(0) d(x, v) =

∫ΣT

(v · n(x)) |v|2γf hψR dSdvdt.

Using the hypotheses 2, 3, and 4 on the solutions, i.e., taking into account that|v|2f ∈ L∞([0, T ], L1(Ω× R3)), |v|2γ±f ∈ C([0,∞), L1(Γ±; |v · n(x)|dSdv)), and E ∈L∞(]0, T [×Ω) and passing to the limit as R→∞, we easily deduce the assertions onthe lemma.

We now derive the balance of entropy identity, for which several previous resultsare needed. First, we will analyze the equation verified by a nonlinear function off (starting with f2). Later, we will show that the entropy is well defined, and fromthese facts we will get the required identity.

LEMMA 3.8. The solution of the problem (P) satisfies that the function

t −→∫

Ω×R3f2 d(x, v)


d

dt

(∫Ω×R3

f2 d(x, v))

= −∫∂Ω×R3

(v · n(x)) γf2 dS dv

+3β∫

Ω×R3f2 d(x, v) − 2σ

∫Ω×R3

|∇vf |2 d(x, v).

Proof. This relation is formally obtained from (1) by multiplying it by f , integrat-ing it in x and v, and applying the divergence theorem. Let us consider a sequence ofmollifiers δn = δtnδ

xnδvn, which approximates the Dirac delta function with support of

δxn and δvn in the ball B(0, 1n ) with δxn(x) = δxn(−x) and δvn(v) = δvn(−v), and δtn has

its support on the negative real axis of length 1n . Also, we can assume without loss

of generality that

‖∇xδn‖L1(R6) + ‖∇vδn‖L1(R6) ≤ Cn,(20)

where C is independent of n. Let us denote by f and E the extension by zero of thefunction f and the field E to the whole R6 and R3, respectively. Set fn = δn ? f . Weuse again the notation introduced in Lemma 2.3 for the domains Ωn.

Choosing a test function Ψ in (8) with compact support in (0, T )×Ωn ×R3 andusing the same arguments as in Lemma 2.3, we can easily show that

∂fn∂t

+ (v · ∇x)fn + divv((E − βv)fn)− σ∆vfn = hn(21)


holds in a classical sense in (0, T )× Ωn × R3. Here

hn = divx[vfn − δn ∗ (vf)

]+ divv

[(E − βv)fn − δn ∗ ((E − βv)f)

],

which may be simplified to obtain

hn = [(divx − βdivv)(vδn)] ? f +[E(∇vδn ? f)−∇vδn ? (Ef)

].

Multiplying (21) by fn we deduce

L[f2n] def=

∂f2n

∂t+ (v · ∇x)f2

n + divv((E − βv)f2n)− σ∆vf

2n

= 2hnfn + 3βf2n − 2σ|∇vfn|2

in (0, T ) × Ωn × R3. Therefore, for any test function Ψ with compact support in(0, T )× Ωm × R3 with n ≥ m, we have∫

QT

f2nL

t[Ψ] d(t, x, v) = −∫QT

2hnfnΨ d(t, x, v)(22)

− 3β∫QT

f2nΨ d(t, x, v) + 2σ

∫QT

|∇vfn|2Ψ d(t, x, v),

where we have denoted by Lt the adjoint operator of L.As f ∈ C([0,∞), L2(Ω× R3)), fn tends to f in C([0,∞), L2(Ω× R3)). Then the

sequences ∫QT

f2nL

t[Ψ] d(t, x, v),

∫QT

f2nΨ d(t, x, v)

converge to ∫

QT

f2Lt[Ψ] d(t, x, v),

∫QT

f2Ψ d(t, x, v).

To deal with the term hn in (21) we write hn = h(1)n + h

(2)n with

h(1)n = [(divx − βdivv)(vδn)] ? f

and

h(2)n =

[E(∇vδn ? f)−∇vδn ? (Ef)

].

Then, using the properties on the solutions and following the same steps as in [9,Proposition 2.2], we can prove that h(1)

n and h(2)n tend to zero in L∞([0, T ], L2(Ω×R3))

and in L2((0, T )× Ω× R3), respectively.Now, we can take the limit n→∞ in the identity (22) to obtain∫QT

f2Lt[Ψ] d(t, x, v) = −3β∫QT

f2Ψ d(t, x, v) + 2σ∫QT

|∇vf |2Ψ d(t, x, v)

for any test function with compact support in (0, T )× Ωm × R3.


Thus, we have proved that in the sense of distributions on (0, T )× Ω× R3

∂f2

∂t+ (v · ∇x)f2 + divv((E − βv)f2)− σ∆vf

2 = 3βf2 − 2σ|∇vf |2.

Now, we use the trace result (Lemma 2.3) to show that f2 ∈ W 1(QT ). Thus,using the hypothesis

γf2 ∈ C([0,∞), L1(Γ±; |v · n(x)|dSdv)),

the previous equation, and the Green identity (Lemma 2.2) we conclude that∫QT

f2(∂Ψ∂t

+ (v · ∇x)Ψ− β(v · ∇v)Ψ + (E · ∇v)Ψ + σ∆vΨ)d(t, x, v)

+∫QT

(3βfΨ− 2σ|∇vf |2Ψ

)d(t, x, v)

+∫

Ω×R3f2oΨ(0, x, v) d(x, v) =

∫ΣT

(v · n(x)) γf2 Ψ dSdvdt

for any Ψ ∈ C∞o (QT ).Finally, to prove the statement of the lemma we can apply reasoning similar to

that in previous lemmas: We consider the test function Ψ = h(t)ψR(v), which wasdefined in Lemma 3.1. Using that f ∈ C([0,∞), L1(Ω × R3) ∩ L∞(Ω × R3)), |v|fbelong to L∞([0, T ], L1(Ω× R3)), and ∇vf ∈ L2([0, T ]× Ω× R3) it is easy to obtainthe announced result.

We can now deduce equations of balance for nonlinear functions of f from thislemma.

LEMMA 3.9. Set ξ ∈ C2(R) with ξ′′ ∈ L∞(R) and ξ(0) = 0. The function ξ(f)belongs to L∞([0,∞), L1(Ω× R3)) and verifies, for any Ψ ∈ C∞o (QT ),∫

QT

ξ(f)(∂Ψ∂t

+ (v · ∇x)Ψ− βdivv(vΨ) + (E · ∇v)Ψ + σ∆vΨ)d(t, x, v)

+∫QT

(3βfξ′(f)Ψ− σξ′′(f)|∇vf |2Ψ

)d(t, x, v)

+∫

Ω×R3ξ(fo)Ψ(0, x, v) d(x, v) =

∫ΣT

(v · n(x)) γξ(f) Ψ dSdvdt.

Moreover, the following equation holds in the sense of distributions on (0, T )×Ω×R3:

∂ξ(f)∂t

+ (v · ∇x)ξ(f) + divv[(E − βv)ξ(f)]− σ∆vξ(f)(23)

= 3βfξ′(f)− 3βξ(f)− σξ′′(f) |∇vf |2 .

Proof. The proof follows the same steps as in Lemma 3.8. Let us keep the samenotation introduced in that lemma. By multiplying the equation

∂fn∂t

+ (v · ∇x)fn + divv((E − βv)fn)− σ∆vfn = hn in (0, T )× Ωn × R3


by ξ′(fn) we have

∂ξ(fn)∂t

+ (v · ∇x)ξ(fn) + divv(Eξ(fn))− β(v · ∇v)ξ(fn)− σ∆vξ(fn)

= 3βfnξ′(fn)− σξ′′(fn)|∇vfn|2 + hnξ′(fn)

in Ωn × R3, where hn is defined in the previous lemma.Since ξ(f) has bounded second derivative and ξ(0) = 0, then ξ(fn) ∈ L1([0, T ]×

Ω × R3) and ξ′(fn) ∈ L2([0, T ] × Ω × R3). Combining these facts and the reasoningof Lemmas 3.8 and 2.3, the result is proven.

Finally, let us deduce the balance of entropy.LEMMA 3.10. The solution of the problem (P) satisfies, for any T > 0, that

f log f ∈ L∞([0, T ], L1(Ω× R3)),

the function

t −→∫

Ω×R3f log f d(x, v)

is absolutely continuous, ∇vf = 2√f∇v√f , ∇v

√f ∈ L2([0, T ]× Ω× R3)), and

d

dt

∫Ω×R3

f log f d(x, v) = −∫∂Ω×R3

(v · n(x)) γf log γf dS dv

+ 3βM − 4σ∫

Ω×R3

∣∣∣∇v√f ∣∣∣2 d(x, v).

Proof. The balance of entropy is formally obtained by multiplying (1) by 1+log f ,integrating it in x and v, and using the divergence theorem. Since the ideas are verysimilar to those in Lemma 2.3 of [9], in order to shorten the calculations in the paper,we refer to [9] for a proof similar to our result that is based on the previous lemmas andon the choice of Ψ = ψR(v)h(t) as a test function in the definition of weak solution,which is defined in previous lemmas.

In the next section we will use all the previous identities to prove that the free-energy η(t) is a Lyapunov functional.

4. Definition and properties of the Lyapunov functional. The free-energyfunctional η(t) is defined by

η(t) =∫

Ω×R3f log

(f

f

)d(x, v),

where f is the solution of the problem (P) and f is the following functional of f :

f(t, x, v) = exp−βσ

(|v|22

+ Φ(t, x)− 1Mµ(t)

),

where M is the total mass and

µ(t) =θ

2

∫Ω|∇xΦ(t, x)|2 dx.


We now show that the functional η(t) is well defined and then obtain its derivative.LEMMA 4.1. For any t > 0 the functional η(t) is well defined and its derivative

is given by

η′(t) = −σ∫

Ω×R3

∣∣∣∣2∇v√f +β

σv√f

∣∣∣∣2 d(x, v)

−∫∂Ω×R3

(v · n(x))(β

2σ|v|2 + log f(t, x, v)

)f(t, x, v) dS dv

Proof. First, we are going to show that η is well defined. By using the definitionof f in that of η(t), we find that

η(t) =∫

Ω×R3f log f d(x, v) +

β

2σ

∫Ω×R3

|v|2f d(x, v)(24)

+β

2σθ

∫Ω|E(t, x)|2 dx.

Using this explicit formula and all the results of the previous section, we obtainthat the functional η(t) is absolutely continuous with respect to t. Moreover, bymultiplying the balance of energy identity (Lemma 3.7) by β

σ and adding it to thebalance of entropy identity (Lemma 3.10), we obtain the derivative of the functionalη by doing some easy computations. Notice that there is an additional term due tothe boundary conditions in η′(t) with respect to the formal derivation given in theintroduction.

Formula (24) shows that η(t) can be considered as a free-energy functional: it isequal to minus the entropy plus the internal energy divided by the temperature. Asimilar functional has been used in [9] to study the long-time asymptotics in Ω = R3.

Using the relation between the flux of kinetic energy and entropy through theboundary (19) we prove that η′(t) ≤ 0. Then, η(t) is a nonincreasing function oftime. We now prove that η(t) is bounded from below under reasonable conditions.Thus, η(t) is a Lyapunov functional and it has a finite limit as t → ∞. This is thekey needed to prove the convergence of the solution to a stationary state.

LEMMA 4.2. The functional η is bounded from below. Moreover, the followingassertions hold.

(1) The following quantities are bounded for any t, with bounds which are inde-pendent of t:∫

Ω×R3f log+ f d(x, v),

∫Ω×R3

|v|2f d(x, v),∫

Ω|E(t, x)|2 dx.

(2) The functional η has a limit when t→∞.Proof. First, let us consider the electrostatic case θ = 1. In order to obtain a

bound from below we use the expression of the functional η obtained in (24) to deducethat

η(t) =∫

Ω×R3f log+ f d(x, v) +

β

2σ

∫Ω×R3

|v|2f d(x, v)

+β

2σ

∫Ω|E(t, x)|2 dx −

∫Ω×R3

f log− f d(x, v).


Taking into account that∫Ω×R3

f log+ f d(x, v) +β

2σ

∫Ω|E(t, x)|2 dx ≥ 0

and using Lemma 3.2, we show that

η(t) ≥ 12

(β

σ− 1b

) ∫Ω×R3

|v|2f d(x, v) − C,(25)

where C = C(f0, b,Ω) is the constant of Lemma 3.2. Choosing b such that b > σβ we

obtain η(t) ≥ −C.To prove assertion (1) of the lemma, it is sufficient to use the same estimates and

the nonincreasing character of the functional η to obtain

0 ≤∫

Ω×R3f log+ f d(x, v) +

β

2σ

∫Ω|E(t, x)|2 dx

+12

(β

σ− 1b

) ∫Ω×R3

|v|2f d(x, v) ≤ η(t) + C ≤ η(0) + C.

Then the assertion is easily obtained. Assertion (2) is a direct consequence of thenonincreasing character of η and its bound from below.

In the gravitational case θ = −1 all the arguments are valid except that we haveto use hypothesis (6) on the solutions, i.e., that

lim supt→∞

∫R3|E(t, x)|2 dx

is finite.In order to give another assumption which implies the boundedness of η we will

recall a classical inequality due to N. S. Trudinger and J. Moser (see [2, 32]). Also, thisresult allows us to analyze under what conditions the function expβσΦ is integrablein Ω. Although this fact is not directly needed to finish the proof of Theorem 1.2, it isinteresting in itself because it can simplify the limit process (as we shall show later),and it also can be used to study the stationary equation (10).

THEOREM 4.3 (Trudinger–Moser). Let u ∈ W 1,3o (Ω) with Ω bounded and smooth

enough. Assume that ∫Ω|∇u(x)|3 dx ≤ 1.

Then there exists a real constant C independent of u such that∫Ω

expa|u(x)| 32 dx ≤ C|Ω|,

where a ≤ ao and ao > 0 depends only on Ω. Moreover, ao is optimal; that is, for anya > ao the inequality is false.

In the next result we use this inequality to study when expβσΦ(t, x) is integrable.LEMMA 4.4. The following properties hold:(1) In the electrostatic case, exp−βσΦ(t, x) is in L1(Ω) for any t > 0. If the

solution satisfies (6′), then expβσΦ(t, x) is in L1(Ω) for any t > 0.


(2) In the gravitational case, expβσΦ(t, x) is in L1(Ω) for any t > 0. If thesolution satisfies (6′), then exp−βσΦ(t, x) is in L1(Ω) for any t > 0.

Proof. Let us show the first part of the lemma; the second part can be derivedanalogously. Since f is nonnegative, the maximum principle implies that Φ is al-ways nonnegative. Then, exp−βσΦ(t, x) is bounded by hypothesis (1). Therefore,exp−βσΦ(t, x) belongs to L1(Ω).

Let us consider a positive real number ν. Using the trivial inequality α ≤ α 32 + 1

for any α ≥ 0, we arrive at

β

σΦ =

βν

σ

Φν≤ β

σν−

12 Φ

32 +

βν

σ.(26)

Taking ν large enough to obtain

β

σν−

12 ‖Φ‖

32

L∞([0,∞),W 1,3o (Ω))

≤ ao,

we can apply the Trudinger–Moser inequality, Theorem 4.3, to conclude that∫Ω

expβ

σν−

12 |Φ(x)| 32

dx ≤ C|Ω|.

Therefore, using (26), the proof is complete.Remark 4.1. Hypothesis (6′), Φ ∈ L∞([0,∞),W 1,3

o (Ω)), clearly implies the hy-pothesis (6) on the solution in the gravitational case. Therefore, with hypothesis (6′)the functional η is bounded from below. This fact can be proved directly. Let us givea sketch of the proof. Using the inequality α log(α) ≥ α− 1 for any α > 0 we get

η(t) =∫

Ω×R3f log

(f

f

)d(x, v) ≥

∫Ω×R3

(f − f) d(x, v).

Therefore, it is enough to prove that∫Ω×R3

(f − f) d(x, v) ≥ Co,

with Co some real constant. Performing some computations and taking intoaccount the conservation of mass the last inequality is equivalent to proving thatexp−βσΦ(t, x) belongs to L1(Ω).

Thus, the previous lemma tells us that the hypothesis (6′) in the gravitationalcase implies that exp−βσΦ(t, x) ∈ L1(Ω). As a conclusion, η is bounded.

5. Large-time asymptotics. In this section we will prove Theorem 1.2. InLemma 4.2, we have shown the existence of the limit of η as t→∞. Then it is easyto deduce, for any T > 0, that

limt→∞

∫ T

0η′(t+ s)ds = 0.

Inequality (19) implies that the second term in the derivative of η in Lemma 4.1is always negative. Since the first term is obviously negative, then we deduce that theintegral of this first term in time must tend to zero when t→∞. Therefore, we haveproved that

limt→∞

‖2∇v√ft +

β

σv√ft‖L2([0,T ]×Ω×R3) = 0,(27)


where ft is defined by ft(s, x, v) = f(t + s, x, v). This will be a key relation in ourproof.

Given any sequence tn → ∞, we will denote by fn the function ftn . Analogouslywe define En and Φn. First of all, let us discuss the compactness in L1([0, T ]×Ω×R3)of the sequence fn. As the quantities∫

Ω×R3f log+ f d(x, v),

∫Ω×R3

|v|2f d(x, v),∫

Ω|E(t, x)|2 dx

are bounded independently of t (Lemma 4.2) and the mass of f is preserved, theDunford–Pettis theorem and the Banach–Alaoglu theorem show that the sequencefn is weakly relatively compact in L1([0, T ] × Ω × R3) and the sequence Enis weakly relatively compact in L2([0, T ] × Ω). To obtain the limiting distributionfunction, we have to take the limit n → ∞ in (1) satisfied by the pair (En, fn). Theabove compactness results proven for these sequences are not sufficient to pass to thelimit in the nonlinear term. For this reason, we will prove that the sequence fn isstrongly compact in L1([0, T ]×Ω×R3). This fact does not imply the convergence of thenonlinear term either. Thus we need to use a renormalized equation for

√fn, where

we will have the necessary compactness to prove the convergence of the nonlinearterm.

The above ideas were introduced in [9]. Since the same procedure can be appliedto our problem, we will recall only the main arguments and differences.

LEMMA 5.1 (Bouchut–Dolbeault). Denote by F and H two bounded subsets ofLp(R6) and Lq([0, T ], Lp(R6)), respectively, with 1 ≤ p < ∞ and 1 < q ≤ ∞. Ifg ∈ C([0, T ), Lp(R6)) is a solution of

Lo[g] def=∂g

∂t+ (v · ∇x)g − βdivv(vg)− σ∆vg = h

with initial data go ∈ F and h ∈ H, then for any τ > 0 and ω bounded open subset ofR6, f is compact in C([τ, T ), Lp(ω)).

The compactness of the sequence fn is proved in the following lemma.LEMMA 5.2. The sequence of distributions fn n ∈ N is relatively compact in

C([0, T ], L1(Ω× R3)).Proof. Due to the boundedness of∫

Ω×R3fn log+ fn d(x, v) and

∫Ω×R3

|v|2fn d(x, v),

independently of t, and conservation of mass, the proof will be carried out by showingthe existence of a subsequence of fn converging a.e. in [0, T ]×Ω×R3. It is then astraightforward consequence of the Vitali theorem that fn n ∈ N converges in L1.

We only need to prove the existence of a subsequence of

ϕε(fn) =√ε+ fn −

√ε,

which converges a.e. in [0, T ]× ω × R3. Here we take ε > 0 fixed.To avoid the problem of the boundary conditions we will multiply these functions

by some special C∞ functions of compact support. Let us consider arbitrary cutofffunctions χ1(x) ∈ C∞o (Ω) and χ2(v) ∈ C∞o (R3). We are going to apply the compact-ness lemma, Lemma 5.1, to the sequence of functions gn = χ1(x)χ2(v)ϕε(fn) withp = 1 and q = 2. The cutoff function χ1(x) allows us to work in R6 and thus to


apply Lemma 5.1. gn(0, ·) is bounded in L1(Ω × R3) independently of n due to thestraightforward inequality

ϕε(fn) ≤ 1√εfn.

On the other hand, we can calculate

Lo[gn] = χ1(x)χ2(v)Lo[ϕε(fn)] − σχ1(x) (∇vϕε(fn) · ∇vχ2)

+χ1(x) ϕε(fn) (−β(v · ∇v)χ2 − σ∆vχ2) + (v · ∇x)χ1(x)χ2(v)ϕε(fn).(28)

Conservation of mass implies again that ϕε(fn) is bounded in L2([0, T ], L1(Ω ×R3)) independently of n. Therefore, the last two terms in (28) are bounded inL2([0, T ], L1(Ω× R3)) independently of n.

Applying the equation of the nonlinear change of variables, (23) of Lemma 3.9,to ϕε(fn), we obtain

Lo[ϕε(fn)] = 3β(

fn

2√ε+ fn

− ϕε(fn))

+σ

4|∇vfn|2

(ε+ fn)32− (En · ∇v)ϕε(fn)(29)

in the sense of the distributions on [0, T ]× Ω× R3. As

∇v√f =

∇vf2√f∈ L2([0, T ]× Ω× R3),(30)

we can easily infer from Lemma 3.10 that ∇vϕε(fn) is bounded in L2([0, T ]×Ω×R3)independently of n. Hence, the second term in (28) is bounded in L2([0, T ], L1(Ω×R3))independently of n.

Equation (27) implies that

limt→∞

‖4 |∇v√fn|2 −

(β

σ

)2

|v|2 fn‖L1([0,T ]×Ω×R3) = 0.(31)

Now using the relations (27), (30), and (31) and the global boundedness of thekinetic energy (see [9] for details), one can show that χ1(x)χ2(v)Lo[ϕε(fn)] is boundedin L2([0, T ], L1(Ω× R3)) independently of n.

We can now use Lemma 5.1 to show that gn(t, x, v) is relatively compact inC([τ, T ), L2(ω)) for any ω strictly included in Ω× R3. As the cutoff functions χ1(x)and χ2(v) are arbitrary, the existence of a subsequence of fn converging a.e. in[0, T ]× Ω× R3 is easily proven.

Our task is now to identify the limiting distribution function. The previous resultproves the existence of a subsequence of tn (which we denote with the same index)and a function f∞ ∈ C([0, T ], L1(Ω× R3)) such that

fn → f∞ as n→∞

in C([0, T ], L1(Ω× R3)) for any T > 0. As a consequence√fn →

√f∞ as n→∞

in L2([0, T ] × Ω × R3). We now follow essentially the same steps as in the formalderivation of section 1, except that we must always use

√f∞ instead of f∞.


Using (27) we conclude that

2∇v√f∞ +

β

σv√f∞ = 0(32)

in the sense of distributions on [0, T ] × Ω × R3. Multiplying (32) by expβ|v|2

4σ , weobtain

∇v(

expβ|v|24σ

√f∞

)= 0

in the sense of distributions on [0, T ]×Ω×R3, which implies the existence of a functiong∞(t, x) ∈ L1

loc([0, T ]× Ω) such that√f∞ =

√g∞(t, x) exp

−β|v|

2

4σ

.(33)

This equation implies that

fn → g∞(t, x) exp−β|v|

2

2σ

as n→∞

in C([0, T ], L1(Ω × R3)), with g∞(t, x) a function to be determined. On the otherhand, using a global (in time) bound of∫

Ω|E(t, x)|2 dx

(Lemma 4.2), we can extract a subsequence En → E∞ converging weakly in L2([0, T ]×Ω).

Our next step is to describe g∞(t, x). To obtain it, we take the limit n → ∞ in(29) for ϕε(fn), thereby finding

Lo[ϕε(f∞)] = 3β(

f∞2√ε+ f∞

− ϕε(f∞))

+β2

4σ|v|2 f2

∞

(ε+ f∞)32− (E∞ · v)

β

σ

f∞√ε+ f∞

in the sense of distributions on [0, T ] × Ω × R3, where we have used (27), (31), andthe weak convergence of En. Now we can use the Lebesgue dominated convergencetheorem to take the limit ε→ 0, thereby obtaining

Lo[√f∞] = −3

2β√f∞ +

β2

4σ|v|2

√f∞ + (E∞ · v)

β

σ

√f∞

in the sense of distributions on [0, T ]× Ω× R3. Inserting

f∞ = g∞(t, x) exp−β|v|2/(2σ)

in the previous formula, we find

∂√g∞

∂t= v ·

(β

2σE∞√g∞ −∇x

√g∞

).

Thus g∞ does not depend on t, and it must satisfy

∇x√g∞(x) =

β

2σE∞

√g∞(x)(34)

in the sense of distributions on [0, T ]× Ω.


To identify g∞(x) we take the limit of the Poisson equation (2) with Dirichletboundary conditions. The convergence fn → f∞ in C([0, T ], L1(Ω × R3)) impliesthat ρ(fn) → ρ(f∞) in C([0, T ], L1(Ω)). Let us define Φ∞ as the unique solution inW 1,qo (Ω) for any 1 ≤ q < 3

2 of the equation

−∆Φ∞(x) = θρ(f∞)(x) on Ω,

with Dirichlet boundary conditions (see [10, Theorem 8]). Moreover, the continuousdependence of the solution of the Poisson equation with respect to the right-hand sidethereof in these spaces implies that Φn → Φ∞ in C([0, T ],W 1,q

o (Ω)) for any 1 ≤ q < 32 .

Furthermore, E∞ must be −∇xΦ∞, and as a consequence ∇xΦ∞ ∈ L2(Ω).We now solve (34). Formally, multiplying (34) by expβΦ∞/(2σ), we can identify

g∞ as

g∞(x) = C∞ exp− β

2σΦ∞

,

where C∞ is a constant. In the gravitational case this step can be justified rigor-ously because Lemma 4.4 proves that expβΦ∞/(2σ) is integrable. However, in theelectrostatic case this property is not known a priori. This problem can be overcomeby using the arguments given in [9] or assuming hypothesis (6′) instead of (6), whichimplies (Lemma 4.4) that expβΦ∞/(2σ) is integrable.

As the mass is preserved, we finally obtain that

f∞(x, v) = λ exp−βσ

(|v|22

+ Φ∞(x))

and that Φ∞(x) is a weak solution of

−∆Φ∞ = θ ‖fo‖L1(Ω×R6)

(∫Ω

exp−βσ

Φ∞(x)dx

)−1

exp−βσ

Φ∞

in H1

o (Ω). This completes the proof of Theorem 1.2.

6. Final remarks. (1) It is easy to see that we can prove a result analogousto our main theorem if we have Neumann or mixed Dirichlet–Neumann boundaryconditions instead of Dirichlet boundary conditions. In fact, we have to change onlyhypothesis (4) to assume that the potential belongs to the correct Sobolev spaceH1(Ω). In the potential energy identity (Lemma 3.6) the term∫

∂ΩΦ j(f) · n(x) dS

appears. This term vanishes since (j(f) · n(x)) is zero on the boundary due to thesecond property of the scattering kernel R.

(2) Our main result can be considered an existence result for the stationary prob-lem. The existence properties of the stationary equation (9) with Dirichlet boundaryconditions depend strongly on the interaction type.

(i) Electrostatic case: In this case the corresponding Euler–Lagrange functionalis strictly convex (see page 140 of [5]). As a consequence, we have a unique possible


steady state f∞. We conclude that the asymptotic behavior of the VPFP system isgiven by the unique stationary limit. Then

limt→∞

‖f(t, .) − f∞‖L1(Ω×R3) = 0.

(ii) Gravitational case: Using (9) the function u = −βσΦ∞ satisfies

−∆u = αeu∫

Ω eu dx(35)

with Dirichlet boundary conditions, where α = βσM . Equation (35) has solution(s)

depending strongly on the topology of the domain Ω (see [28, 29]). In fact, if Ω is theunit ball B(0, 1), we have the following properties:

• There exists a value of the parameter α? such that (35) with Dirichlet bound-ary conditions has at least one solution for any 0 ≤ α < α? and no solution forα > α?. In the latter case the potential energy of the system is not bounded,i.e.,

limt→∞

∫Ω|∇xΦ(t, x)|2 dx = ∞.

• For α = αo = 2 meas(∂B(0, 1)), equation (35) has infinitely many boundedsolutions (starting with a minimal one) and a unique unbounded radial solu-tion u = U(x) = −2 log |x|.• For α small enough, the solution is unique.

On the other hand, if Ω is an annulus, (35) has at least a solution for any value of theparameter α (see [28]).

When α is such that (35) has more than one solution, it is an open problem to de-termine the dynamical behavior of the distribution function f(t, x, v) as t approachesinfinity. For instance, we do not know whether it is possible for two different timesequences of the distribution function to approach different stationary states.

Acknowledgments. The authors want to express their gratitude to Jean Dol-beault for pointing out some useful references.

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