Stability of the Equilibrium to the Boltzmann Equation with Large Potential Force?

Xiuhui YANG

Abstract The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian.The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space R3.Compared with the previous result[Ukai,S.,Yang,T.and Zhao,H.-J.,Global solutions to the Boltzmann equation with external forces,Anal.Appl.(Singap.),3,2005,157–193],no smallness condition on the Sobolev norm H1of the potential is needed in our arguments.The proof is based on the entropy-energy inequality and the L2?L∞estimates.

Keywords Boltzmann equation,Large potential force,Stability,Entropy-energy

1 Introduction and Formulation

The time evolution of rarefied gas in an external field can be described by the classical Boltzmann equation with additional force term

where F=F(t,x,v)is a function describing the distribution of particles at the time t≥0,at the position x=(x1,x2,x3)∈R3and with the velocity v=(v1,v2,v3)∈R3.The potential Φ=Φ(x)is independent of the time t.The collision between particles is given by the standard Boltzmann collision operator Q(F,G)with hard potential interactions:cut-off assumption,i.e.,withand u′,v′related to u,v by the usual elastic collision relations

It is easy to check that the local Maxwellian is the unique stationary state to the Boltzmann equation(1.1).In[10–11],Ukai,Yang and Zhao studied the Cauchy problem of(1.1)with γ=1 and obtained global existence and time decay rate of classical solutions near the equilibrium M.Later,in[2]they improved their previous results and obtained optimal decay rate of classical solutions.See also[9,12]for the corresponding results for the hard potential case.Lei[8]studied the non-cut off case of(1.1).Duan[1]studied the equation(1.1)in the torus of R3and obtained the stability of the stationary state.Notice that the assumption that the Sobolev norm(e.g.H4)of the external potential Φ is sufficiently small plays a crucial role in all the articles mentioned above and the methods developed there cannot be applied to the case when the Sobolev norm of Φ is large.Recently,Kim[7]studied the equation(1.1)with a large amplitude external potential in a periodic box of R3and obtained the stability of the local Maxwellian M.It should be pointed that the periodic assumption plays a crucial role in the arguments of[7]and cannot be applied to the whole space case.

The goal of this paper is to construct the classical solutions for(1.1)near the equilibrium M with large amplitude Φ on H1in the whole space R3.

As usual,we introduce the standard perturbation f(t,x,v)to M as

Here the nonlinear collision operator Γ(g1,g2)takes the form

and the linearized collision operator reads

It is well-known that K is a self-adjoint compact operator onand ν(v)is given by

for some constant C>0.Also,there exist positive constantsandsuch that

It is straightforward to verify that the number of particles and the sum of potential and kinetic energy are conserved under the evolution(1.1),thus we define the perturbation of the mass and the total energy as

Moreover,by standard arguments it follows that the H-function of the perturbation f,

does not increase in the evolution(1.1).

Noticing that the H-function H(f)does not increase during the evolution(1.1)and the energy E(f)and the total masses M(f)are constants,we define the following non-increasing entropy-energy functional:

which plays a crucial role in the study of stability of the equilibrium.

Throughout this paper the letters C and Cidenote generic constants and may change from line to line.Denote by ?x,vthe couple(?x,?v).Our main result is the following theorem.

Theorem 1.1Letfor Λ >0 andAssume that the potential Φ(x)satisfiesfor k=1,2,is sufficiently small,and

Then the initial value problem for(1.2)enjoys a unique global in time solution satisfying

Remark 1.1It turns out that G(f(0))>0 ifis sufficiently small,see Lemma 2.1 below.

Remark 1.2The two constants ? and ? in Theorem 1.1 are not necessary positive.

As pointed out in[3],due to the presence of a large amplitude potential Φ,we lose the control of the Sobolev estimate in higher order energy norms to the perturbation f.The proof of Theorem 1.1 is based on some ideas developed recently by Esposito,Guo and Marra[3]in studying the nonlinear stability of the phase state to the Vlasov-Boltzmann system of binary mixture.The strategy is to make a crucial use of the fundamental entropy-energy G(f)estimate to obtain a mixed L1?L2type of stability estimate and then bootstrap such a L1?L2stability to a L∞estimate to obtain pointwise stability estimate by following the curved trajectory induced by the force field.

The paper is organized as follows.In Section 2 we use the energy-entropy(1.7)to derive a mixed L1?L2estimate and state some results on the characteristics curves for the equation(1.1).In Section 3 we establish the nonlinear stability of the equilibrium in weighted L∞norm.

2 Entropy-Energy Estimate and Characteristics

In this section we first use the conservations of total energy and mass,and the entropy inequality to obtain a priori estimates on the deviation of the solution from the equilibrium.

Lemma 2.1There exist κ >0 and Cκ>0 such that

ProofThe proof is similar to that of Lemma 2.2 in[3],so we present it here for completeness.First,we can construct solutions(see[4])such that

We expand G(f)at the equilibrium M and use(1.5)to cancel the linear part of the expansion,which takes the form

For some small number 0< κ <1,we introduce the indicator functionsandand split the integral into

In the case of F(t)≤ (1?κ)M,we havethus

Combining these two cases and noticingfor|F(t)?M|≤ κM,we conclude

In order to study the curved trajectory to the Boltzmann equation(1.1),we define the characteristics curves[X(s;t,x,v),V(s;t,x,v)]for(1.1)passing through(t,x,v)at s=t,such that

Lemma 2.2Fix N>0.Let|v|≤N.Then there exists T1>0 and 0≤s

ProofMultiplying(2.2)by V(s;t,x,v)and noticing(2.1),we obtain the conservation of particle energy

For given T1>0 and fixed N>0,noticing?? ≤ Φ(x)≤ ??,we obtain from(2.4)and(2.1)that

From(2.1)–(2.2),we have

and we deduce that for 0≤s

The Taylor expansion for X(s;t,x,v)=(X1(s;t,x,v),X2(s;t,x,v),X3(s;t,x,v))around t reads

for some suitable T1>0.Therefore,the estimate(2.3)holds.

3 Weighted L∞Stability

In this section we shall use the entropy-energy inequality and the estimates on the characteristics to show the stability of the equilibrium.In fact,we obtain that the perturbation f is arbitrarily small at any positive time in a suitable weighted L∞norm provided that it is initially sufficiently small.We use the weight functionwithand Λ a positive constant to be chosen later.

Lemma 3.1Let h=wf.Under the assumptions of Theorem 1.1,there exist 00,0< Π <1,and CT0>0 such that,if khkL∞ < δ,then

ProofWe first write the equation for h=wf from(1.2)as

the solution to the following transport equation

can be written as

We note that,for Λ≥1,

Fix a small constant ?>0.We can choose Λ sufficiently large such that

Since ν(τ)≥ ν0>0 and Φ ≤ ??,the third term in(3.2)is bounded by

For the last term in(3.2),by[5,Lemma 5],it follows

Noticing Φ ≤ ??,we get the bound for the last term by

Thanks to

we obtain,by integrating by parts,that

We shall mainly concentrate on the second term in(3.2).Let k(v,v′)be the corresponding kernel associated with K in(3.1).Then the Grad’s estimate implies that

uniformly in Λ for some constant

Denote

We now use(3.2)for h(s,X(s),v′)again to evaluate

In fact,we can bound the third term in(3.2)by

Since ν(τ) ≥ ν0,by taking L∞norm for h and using the estimates(3.6)and(1.8),we bound the first term in(3.7)by.Similarly,noticing the fact Φ ≤ ?? andand using the estimate(3.4),the third term can be bounded by

and the last nonlinear term can be bounded by

We now concentrate on the second term in(3.7)and we follow the same spirit of the proof of Theorem 6 in[5].Since(2.2)implies that,for any T>0 and for fixed N>0 large enough,we have

Thanks to the estimate

we divide the above integral into three cases according to the size of v,v′,v′′and for each case,an upper bound of the second term in(3.7)will be obtained.

Case 1|v|≥ N.In this case,sincethe estimate(3.8)implies that

We can find an upper bound for the second term in(3.7)by

Case 2Observe thatthus we have eitherTherefore,either one of the following is valid correspondingly for some σ>0,

By(3.5),we have

We use this bound to combine the cases ofas

We first integrate v′for the first integral and use(3.8)to integrate kwover v′′.We then integrate v′′for the second integral and use(3.8)to integrate kwover v′.Noticingwe thus find an upper bound

Case 3a|v|≤ N,|v′|≤ 2N,|v′′|≤ 3N.This is the last remaining case because if|v′|>2N,it is included in Case 2;while if|v′′|>3N,either|v′|≤ 2N or|v′|≥ 2N is also included in Case 2.We further assume that t?s≤ ?.We can bound the second term in(3.7)by

Case 3b|v|≤ N,|v′|≤ 2N,|v′′|≤ 3N,and t? s ≥ ?.We can bound the second term in(3.7)by

Splitting

We then integrate the first term above in v′′and the second term above in v′.By(3.6),we can use such an approximation(3.12)to bound the s1,s integration by

The first term in(3.13)is further bounded by

Since kN(V(s),v′)andare bounded,the second term in(3.13)is controlled by

To estimate this term,we introduce a new variable

and apply Lemma 2.2 to X(s1;s,X(s;t,x,v),v′)with s=s1,t=s,x=X(s;t,x,v),and v=v′.Noticing 0 ≤ s≤ t? ?

for|v′|≤ 2N.By integrating over v′(bounded)and using the change of variable(3.15),we get

By collecting all the above terms,we conclude that,for H(g(0))small,

then N sufficiently large,and finally ? sufficiently small to conclude our lemma.

Proof of Theorem 1.1Assume thatis sufficiently small.We first establish(1.9).Choose any n=1,2,3,···and use Lemma 3.1 repeatedly to get

For any t,we can find n such that nT0≤t≤(n+1)T0,and from L∞estimate from[0,T0],we conclude(1.9)by

To prove(1.10),we take x and v derivatives to(1.2)to get

By[6,Lemma 2.2],we have

Since L=ν?K ≥0,by multiplying(3.16)with?xf and(3.17)with?vf and then integrating them overrespectively,we can follow the procedures in[4]to get

Hence(1.10)follows from the Gronwall Lemma sinceis bounded by(1.9).With such an estimate,we easily obtain the uniqueness by taking L2estimate for the difference for(1.2).Therefore,we complete our proof of Theorem 1.1.