Jie LIAOQianrong WANG

Department of Mathematics,East China University of Science and Technology,Shanghai 200237,China

E-mail:liaojie@ecust.edu.cn;475446014@qq.com

Xiongfeng YANG

School of Mathematical Sciences;Key Laboratory of Scienti fi c and Engineering Computing(MOE),Shanghai Jiao Tong University,Shanghai,200240,China

E-mail:xf-yang@sjtu.edu.cn

Abstract In this article,we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates.The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates,which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.

Key words non-linear Vlasov–Poisson–Fokker–Planck equation;global Maxwellian;global a priori estimates;exponential convergence

1 Introduction

In this article,we consider the following initial value problem(IVP)of the nonlinear Vlasov–Poisson–Fokker–Planck(VPFP)equations

where the unknown function f=f(t,x,v)is the distribution function of particles at time t,position x and velocity v,(t,x,v)∈R+×R3×R3.

The VPFP system describes the Brownian motion of a system with large number particles in a surrounding bath,one of the applications is in electrostatic plasma,in which one considers the interactions between the electrons and a surrounding bath via the Coulomb force[1].The Brownian motion of the particles is modeled by di ff usion-in-v operator which is nonlinearly coupled with the macroscopic mass density ρ,as the di ff usion parameter.More general di ff usion parameter ρα(α ∈ [0,1])can be similarly considered,in which the parameter α is determined by the di ff usion type of a particular material[7,8].Note that in most literatures(see[2–4]and references therein),the di ff usion parameter is taken as a constant.This nonlinearity will bring new difficulties in the analysis[6].

Fokker–Planck equations were applied in various fi elds such as plasma physics,surface physics,astrophysics,the physics of polymer fl uids and particle beams,nonlinear hydrodynamics,theory of electronic circuitry and laser arrays,engineering,biophysics,population dynamics,human movement sciences,neurophysics,psychology and marketing[1,10].In mathematical view,Fokker–Planck equations can be seen as an approximation to the Boltzmann equation[8,11]for the free transport of a stochastic particle system.When certain con fi ning potential is added to the stochastic particle system,the motion of the system will be governed by the Fokker–Planck equations with an additional force term.In particular,when the Coulomb potential force is considered,one derives to the Vlasov–Poisson–Fokker–Planck model(1.1).The self-consistent potential φ is the fundamental solution of the Laplace equation thus can be represent by a singular integral operator,this extra structure gives good estimates about the density of the solution,which leads to the exponential decay of the solution towards equilibrium for this Vlasov–Poisson–Fokker–Planck model.Note that we do not have the same structure for single linear/nonlinear Fokker–Planck equations,thus we have only algebraic decay rates of the solution for linear and nonlinear Fokker–Planck equations[5,6].

There are also other Vlasov-type Fokker–Planck equations considered in literatures.For instance,when the Vlasov–Fokker–Planck equation is coupled with Maxwell equations,Euler equations or Navier–Stokes equations,respectively,we have Vlasov–Maxwell–Fokker–Planck model(see[12]and references therein),Vlasov–Euler–Fokker–Planck System(see[13]and references therein),and Vlasov–Fokker–Planck–Navier–Stokes System(see[14–16]and references therein).

1.1 Problem Considered

The global equilibrium of(1.1)is given by the Maxwellian

We will show the stability of this global Maxwellian for the nonlinear Vlasov–Poisson–Fokker–Planck equation(1.1)in the whole space by uniform-in-time energy estimates.To show it,we use the standard perturbation form

Then we get the following linearized IVP

where L is the linear Fokker–Planck operator given by

1.2 Some Properties of the Linear Fokker–Planck Operator L

Before state the properties of the operator L,we list some notations used in this article

We denote function spaces HNand HN,νwith the norm

We use c0or C to denote a generic constant which may change from line to line.The notation a.b means that there exits some constant C such that a≤Cb for a,b>0.

Now we recall some well known results of the linear Fokker–Planck operator,see[13,17].

? Basic properties of the linearized Fokker–Planck operatorL.The linearized Fokker–Planck operator L is self adjoint and

?Decomposition of the solution.We use the projections onto

thus we have the decomposition

Moreover,the norm of g can be given by

?Coercivity of the operator?L.The coercivity of the operator?L can be written in terms of the above projections such that,there exists a positive constant λ0>0 that

The Fokker-Planck operator is a well-known hypoelliptic operator.Di ff usion(dissipative)in v-variable together with transport(conservative)v·?xhas a regularizing e ff ect not only in v but also in t and x.Note that this is nontrivial as the di ff usion only acts on the velocity.This phenomenon can be obtained by applying H?rmander commutator estimates to the linear Fokker-Planck operator[18].For more details,we refer to[19].On the other hand,the Fokker-Planck operator is also known as a hypocoercive operator[17],which is an very important issue in kinetic theory concerning the convergence rate to an equilibrium.This technique is hard to be applied to nonlinear model.However,we get the exponential convergence of the solution towards equilibrium,by observing the dissipation of macroscopic part σ,see(2.17),from the extra structure of the Poisson equation for the Coulomb potential.

1.3 Main Results

The main results of this article are stated in the following theorem.

Theorem 1.1(Uniform energy estimates and exponential convergence to the equilibrium)Let N ≥ 4 be an integer.Assume that there is a sufficiently small positive constant δ0such that the initial data g0satis fi es

Then the solution(g,φ)to the IVP(1.3)satisfy the uniform energy estimate

for some positive constants c0and C,for all time t>0.Moreover,there exit C>0 and η>0 such that

Remark 1.21.The existence of global solution can be obtained by standard continuity argument,based on above uniform a priori energy estimates and local solution constructed by a Picard type iteration scheme,following the steps in[13,15].

2.Other than the algebraic decay rates of the solution for linear/nonlinear Fokker–Planck equations[5,6],we have obtained the exponential decay rate for our model due to the good structure of the Poisson equation[2].

3.Our uniform energy estimates and decay estimates include that of v-derivatives,which is not included in the result of[2].

The main difficulty of the analysis is from the nonlinearity on the right hand side of the perturbed equation(1.3),which is the coupling of macroscopic density and the Fokker–Planck operator.Note that when we take derivativeto the equation as usual,this nonlinear term gives

The plan of this article is the following.We will prove the uniform energy estimate(1.7)by constructing proper energy functional and dissipation functional in Section 2.Section 3 will be devoted to the proof of the main results.

2 Uniform a Priori Estimates

De fi ne the energy functional by

Note that v-derivatives are included in our energy functional and dissipation functional,which is di ff erent from that of[2].In this section,we will show the following result.

Proposition 2.1(Uniform energy estimates) There exist small parameters κ1and κ2in the de fi nition of ENsuch that

for some positive constant c0.

In the following,we will estimate the unknown perturbation function under the a priori assumption that

for any given T,N≥4 and 0<ε0?1 sufficiently small.By Sobolev inequality,we know that

or equivalently,

2.1Basic Energy Estimates of?EN(t)

2.2 Estimate of FN(t)

Because the di ff erential operator L involves the v-derivatives,we need the control the functional FN(t)to close the energy estimate.The result is the following.

Lemma 2.3There exists a constant C>0 independent of t such that the solution to the perturbed equation(1.3)satisfy

2.3 Energy Estimates from Macroscopic Equations

Note that there is no dissipation of the macroscopic part P0g(i.e.,σ)in(2.9)and(2.13).To get the dissipation of this part,we use the extra “damping” in the macroscopic equation.Now we derive the macroscopic equations about σ and u.By using(1.4),we have

Combining(2.18)–(2.22),we have(2.17)thus the proof of the proposition is completed.?

2.4 Proof of Proposition 2.1

In summary,we have shown(2.9),(2.13)and(2.17),which are recalled here for convenience that

Then Proposition 2.1 is proved by computing(2.23)+κ1(2.24)+κ2(2.25)with suitably chosen small parameters κ1and κ2. ?

3 The Proof of Main Results

Proof of Theorem 1.1Let ε0in(2.8)be small enough such that C(EN(t))12≤c0in(2.7).Thus,we have uniform energy estimate for all 0≤t≤T,

In particular,it holds

Then we have closed the a priori assumption in(2.8)for any given T.Thus(1.7)is proved.

Next,we proof the decay estimates in(1.8).Note from the Poisson equation of φ,for|α|=N,

we readily note from(2.5)–(2.6)that

Let ε0in(2.8)be small enough as above such that C(EN(t))≤ c0in(2.7),then,there exists a constant η>0 such that

thus we readily have the exponential decay estimate in(1.8).The proof of Theorem 1.1 is completed. ?