1Introduction OPTIMALCONVERGENCERATESOFLANDAUEQUATIONWITHEXTERNALFORCINGINTHEWHOLESPACE

OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN

THE WHOLE SPACE

Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday Yang Tong()

Department of Mathematics, City University of Hong Kong, Hong Kong, China Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

E-mail: [email protected]

Yu Hongjun ( )

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Abstract In this paper, we combine the method of constructing the compensating func- tion introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

Key words compensating function; energy method; Landau equation; optimal conver- gence rates

2000 MR Subject Classification 35Q99; 35B40

1 Introduction

In this paper, we are concerned with the Landau equation under the influence of external forcing and source. That is, consider

∂tF+v· ∇xF−(∇xΦ +E)· ∇vF=Q(F, F) +S (1.1) with initial condition F(0, x, v) = F₀(x, v). Here, F(t, x, v) is the distribution function of particles at time t ≥0, located at x = (x₁, x₂, x₃)∈ R³ with velocity v = (v₁, v₂, v₃)∈ R³. The external force takes the form of−(∇xΦ+E) with given functions Φ = Φ(x) andE =E(t, x).

In the following discussion, E(t, x) is assumed to decay in time so that the time asymptotic

∗Received January 30, 2009. The research of the first author was supported by Strategic Research Grant of City University of Hong Kong, 7002129, and the Changjiang Scholar Program of Chinese Educational Ministry in Shanghai Jiao Tong University. The research of the second author was supported partially by NSFC (10601018) and partially by FANEDD

state is governed only by the potential Φ(x). Moreover, the source termS =S(t, x, v) is also assumed to be a given function.

Qis the usual bilinear collision operator defined by Q(F, G)(v) =∇v·

R³

φ(v−v)[F(v)∇vG(v)−G(v)∇vF(v)]dv

= 3 i,j=1

∂i

R³φ^ij(v−v)[F(v)∂jG(v)−G(v)∂jF(v)]dv, (1.2) where∂i=∂v_i. The non-negative matrix φis given by

φ^ij(v) =

δ_ij−v_iv_j

|v|²

|v|^2+γ.

Here, γ is a parameter leading to the standard classification of the hard potential (γ > 0), Maxwellian molecule (γ= 0) or soft potential (γ <0), cf. [1]. In particular,γ=−3 corresponds to the Coulomb interaction in plasma physics. The following study is restricted to the case γ≥ −2.

Throughout this paper, we consider solutions which are perturbations near a local Maxwellian.

Without loss of generality, define the perturbationf(t, x, v) byF =M+M^1/2f,where the local Maxwellian isM(v) = exp

−Φ(x)−^|v₂^|2

.Then equation (1.1) for the perturbationf(t, x, v) is

∂tf+v· ∇xf −(∇xΦ +E)· ∇vf+1

2v·Ef + e^−ΦLf= e^−Φ/2Γ(f, f) +S (1.3) with f(0, x, v) = f₀(x, v). Denote μ(v) = exp(−|v|²/2), then M =μ(v)e^−Φ. The linearized collision operatorLis defined by

Lf≡ −μ^−1/2[Q(μ, μ^1/2f) +Q(μ^1/2f, μ)] =−Af −Kf,

where Af = μ^−1/2Q(μ, μ^1/2f), Kf = μ^−1/2Q(μ^1/2f, μ), and the bilinear collision operator Γ(f, g) and the source termSare given by

Γ(f, g) =μ^−1/2Q(μ^1/2f, μ^1/2g),

S= e^Φ/2μ^−1/2S−e^−Φ/2μ^1/2v·E. (1.4) By H-theorem,L is non-negative and self-adjoint onL²(R³_v) with domain

D(L) ={f ∈L²(R³_v) Lf∈L²(R³_v)}.

Furthermore, set ψ₁(v) = 1, ψj+1(v) = vj, j = 1,2,3 and ψ₅(v) = |v|². Then the null space ofLis the five dimensional space (j= 1,2,3)

N = span{ψ₁√μ, ψ_j+1√μ, ψ₅√μ}. (1.5) For any fixed (t, x) and any functionf(t, x, v), we definePas the projection inL²(R³_v) to the null spaceN. Then decomposef(t, x, v) uniquely by

f(t, x, v) =Pf+ (I−P)f. (1.6)

Here,Pf and (I−P)f are called the macroscopic and microscopic components of the function f, respectively.

In this paper, the following notations are used. Let α and β be α = [α₁, α₂, α₃] and β= [β₁, β₂, β₃], respectively. Denote

∂_β^α≡∂_x^α₁¹∂_x^α₂²∂_x^α₃³∂_v^β₁¹∂_v^β₂²∂_v^β₃³.

If each component ofβis not greater than corresponding one ofβ, we use the standard notation β ≤β. Andβ < β means that β ≤β and|β|<|β|. C_β^β¯is the usual binomial coefficient. For the study of the optimal time decay rates, the spaceZ_q=L²(R³_v;L^q(R³_x)) is used with its norm defined by

f Z_q =

R³ R³|f(x, v)|^qdx²_q dv¹₂

.

We will use·,·to denote the standardL²inner product inR³_v, and (·,·) for the one inR³_x×R³v.

| · |2 denotes theL² norm inR³_v and · to denote theL² norms inR³_x×R³_v. For the weight function in v, we use w = w(v) = (1 +|v|)^γ+2. From now on, C denotes a generic positive constant which may vary from line to line.

We now come back to the Landau equation. Note that the collision frequency is given by σ^ij(v) =

R³

φ^ij(v−v)μ(v)dv. (1.7)

Accordingly, the following weightedL² norms will be used:

|g|²σ=

1≤i,j≤3

R³{σ^ij∂_ig∂_jg+σ^ijv_iv_jg²}dv, g ²_σ =

1≤i,j≤3

R³×R³{σ^ij∂_ig∂_jg+σ^ijv_iv_jg²}dvdx.

For the energy estimates, the instant energy functional for a solutionf(t, x, v), denoted byEl(t) has the equivalent relation with somel≥0 representing the index in the weight inv,

El(t)∼El(t) =

|α|+|β|≤N

w^l∂_β^αf(t) ². (1.8)

In addition, the corresponding dissipation functional denoted byDl(t) satisfies Dl(t)∼Dl(t) =

1≤|α|≤N

∂^αPf(t) ²+

|α|+|β|≤N

w^l∂_β^α(I−P)f(t) ²σ. (1.9)

In the following discussion, we will choose the Sobolev space index N ≥8. Notice that from [1, 9], there existsC >0 such that

C|w^1/2g|2≤ |g|σ. (1.10) Notice that El(t) and Dl(t) do not contain the temporal derivatives and Dl(t) does not contain the zero-th order derivative of the macroscopic component in the solution.

With the above preparation, the main results of this paper can be stated as follows.

Theorem 1.1 LetF₀(x, v) =M+√

M f₀(x, v)≥0 andγ≥ −1. Suppose that

(H₀) The functionsE=E(t, x),S=S(t, x, v) andf₀=f₀(x, v) satisfy

E∈C_b⁰(R⁺;H^N(R³_x)), S∈C_b⁰(R⁺;H^N(R³_x×R³_v)), f₀∈H^N(R³_x×R³_v);

(H₁) There are constantsε >0 and l≥0 such that Φ(x), E(t, x) andf₀ are bounded in the sense that ˜El(0)≤ε,and

Φ(x) _L2(R³)+

1≤|α|≤N+1

∂^αΦ(x) _L3(R³)≤ε, (1.11)

|α|≤N

(1 +|x|)∂^αE(t, x) L^∞_t,x + |x|E(t, x) _L∞

t (L²_x)≤ε. (1.12) MoreoverE(t, x) andS(t, x) decay in time like

|α|≤N

∂^αE(t, x) ²+

|α|+β|≤N

w^l−1/2∂_β^α(μ^−1/2S(t, x)) ²≤ε(1 +t)⁻¹⁻, (1.13)

whereis any positive constant. Then there is a constantε₁>0 such that for anyε≤ε₁, the Cauchy problem of the Landau equation (1.3) has a unique global classical solution f(t, x, v) withF(t, x, v) =M +√

M f(t, x, v)≥0 for (1.1), which satisfies El(t) +

t 0

Dl(s)ds≤Cε. (1.14)

If we further assumel≥1,= ³₂ and

(H₂) f₀ Z₁+ ∇xΦ _L6/5(R³)≤ε, (1.15)

E(t) L¹_x+ μ^−1/2S _Z1 ≤ε(1 +t)^−5/4, (1.16) we have the optimal time decay of the solution to its time asymptotic state

f(t) ²= Pf(t) ²+ (I−P)f(t) ²≤Cε(1 +t)^−3/2, (1.17)

1≤|α|≤N

∂^αPf(t) ²+

|α|+|β|≤N

w^l∂_β^α(I−P)f(t) ²≤Cε(1 +t)^−5/2. (1.18) Note that if E(t, x) = 0 andS(t, x, v) = 0, equation (1.3) becomes

∂_tf+v· ∇xf− ∇xΦ· ∇vf+ e^−ΦLf= e^−Φ/2Γ(f, f) (1.19) withf(0, x, v) =f₀(x, v). Corresponding to Theorem 1.1, we have the following theorem about the existence and optimal convergence rate which holds for larger range ofγ.

Theorem 1.2 Let F₀(x, v) = M +√

Mf₀(x, v)≥ 0 and γ ≥ −2. If we assume (1.11) and ˜El(0)≤εfor some small constantε >0, then the Cauchy problem of the Landau equation (1.19) has a unique global classical solutionf(t, x, v) withF(t, x, v) =M +√

M f(t, x, v)≥0, which satisfies

El(t) + t

0

Dl(s)ds≤CEl(0). (1.20)

If we further assume (1.15) andl≥1, we have the optimal time decay:

f(t) ²= Pf(t) ²+ (I−P)f(t) ²≤Cε(1 +t)^−3/2, (1.21)

1≤|α|≤N

∂^αPf(t) ²+

|α|+|β|≤N

w^l∂_β^α(I−P)f(t) ²≤Cε(1 +t)^−5/2. (1.22) Finally in the introduction, we review some related works to the study in this paper.

Recently, there is some progress on the L² energy methods for the Boltzmann equation. One was initiated by Liu-Yu [16] and developed by Liu-Yang-Yu [15] in terms of the macro-micro decomposition around the local Maxwellian defined by the solution. The other was proposed by Guo [ 9, 10, 11] about the decomposition around a global Maxwellian. On the other hand, instead of using the spectrum analysis initiated by Grad and finished by Ukai and others, Kawashima [13] introduced the compensating function method which is based on the Fourier transform for the Botlzamnn equation. With these methods, the authors in [24] study the nonlinear stability and the optimal time decay of the solutions for the relativistic Boltzmann and Landau equations near a uniform equilibrium by combining the Kawashima’s compensating function method and the macro-micro decomposition of the solution. The main observation is that the energy estimate on the macroscopic component can be obtained by improving the Kawashima’s compensating function method [13]. And the advantage is not only to obtain the global existence of the solution without time derivative, but also to obtain the optimal time decay to the equilibrium. As a further study by using the method introduced in [24], in this paper, we consider the Landau equation with external forcing.

As for Landau equation, there have been intensive investigations, cf. [1, 2, 9, 12, 14, 17, 21] and references therein. More precisely, Desvillettes and Villani [2] proved the global existence and uniqueness of classical solutions for spatially homogeneous Landau equation for hard potentials and a large class of initial data. Degond and Lemou [1] studied the spectral properties and the dispersion relation of linearized Landau operator. Guo [9] constructed global classical solutions near a global Maxwellian in a periodic box. Hsiao and Yu extended Guo’s results to the whole space in [12]. It is shown in [14] that (1.19) has a unique global solution and it converges to the steady state with a rateO((1 +t)^−1/4).

We should also mention that recently Duan-Ukai-Yang-Zhao [6] introduced a method by combining the spectrum analysis and energy method for the study on the optimal time decay rate for the Boltzmann equation and systems of fluid dynamics such as Navier-Stokes equations, cf. also [5]. Some of the ideas from this work will be used here, however, we do not need to consider the spectrum of the corresponding linearized equation because the compensating function coming from the Fourier transform will show to be enough. We believe that the method used here can be applied to the study on some more complicated systems in the kinetic theories.

2 Basic Estimates

For later use, we derive some estimates on the the bilinear collision operator Γ(f, f).

Lemma 2.1 For any g₁ = g₁(x, v) and g₂ = g₂(x, v) with the following norms well defined, we have

Γ(g1, g₂) Z₁ ≤C

|β₁|≤2

∂_β1g₁ ·

|β₂|≤2

w∂_β2g₂ . (2.1)

Proof By (1.2), Γ(g₁, g₂) can be written as Γ(g₁, g₂) =μ^−1/2(v)Q(μ^1/2g₁, μ^1/2g₂)

=μ^−1/2(v)∂i

R³φ^ij(v−v)μ^1/2(v)μ^1/2(v)

g₁(v)∂jg₂(v)−∂jg₁(v)g₂(v) dv +μ^−1/2(v)∂_i

R³

φ^ij(v−v)μ^1/2(v)μ^1/2(v)v_j 2 −vj

2

g₂(v)g₁(v)dv

=μ^−1/2(v)∂i

R³

φ^ij(v−v)μ^1/2(v)μ^1/2(v)

g₁(v)∂jg₂(v)−∂jg₁(v)g₂(v) dv

=

∂i−vi

2 _R3

φ^ij(v−v)μ^1/2(v)

g₁(v)∂jg₂(v)−∂jg₁(v)g₂(v) dv

=∂i

R³

φ^ij(v−v)μ^1/2(v)

g₁(v)∂jg₂(v)−∂jg₁(v)g₂(v) dv

−1 2

R³

φ^ij(v−v)v_iμ^1/2(v)

g₁(v)∂jg₂(v)−∂jg₁(v)g₂(v)

dv, (2.2)

where we have used the fact that 3

i=1

φ^ij(v−v)(v_i−v_i) = 3 j=1

φ^ij(v−v)(v_j −v_j) = 0.

Since|μ^1/2(v)|+|∂iμ^1/2(v)| ≤Cμ^1/4(v),the Cauchy-Schwartz inequality implies

R³

φ^ij(v−v)∂i(μ^1/2(v)g₁(v))dv

≤

R³

φ^ij(v−v)[μ^1/2(v) +|∂iμ^1/2(v)|][|g₁(v)|+|∂ig₁(v)|]dv

≤C

R³|φ^ij(v−v)|²μ^1/2(v)dv1/2

[|g₁|2+|∂ig₁|2]

≤C(1 +|v|)^γ+2[|g₁|2+|∂_ig₁|2]. (2.3) Thus,

∂_i

R³

φ^ij(v−v)μ^1/2(v)

g₁(v)∂_jg₂(v)−∂_jg₁(v)g₂(v) dv _L1

x

2

≤C

|β₁|≤2

∂β₁g₁

|β₂|≤2

(1 +|v|)^γ+2 ∂β₂g₂(v) _L2 x

2

≤C

|β₁|≤2

∂_β1g₁

|β₂|≤2

w∂_β2g₂ .

This gives the first part of (2.2). Similar argument leads to the proof of the second part of (2.2) and we skip the detail. This completes the proof of the lemma.

Lemma 2.2 For anyl≥1, it holds that

|α|≤1

∂^αΓ(f, f) ²≤CEl(t)El(t), (2.4)

where

El(t) =

1≤|α|≤N

∂^αPf(t) ²+

|α|+|β|≤N

w^l∂_β^α(I−P)f(t) ².

Note thatEl(t) is different fromDl(t) by the norms on the microscopic component.

Proof By (2.2), we have

∂^αΓ(f, f) =

α₁+α₂=α

C_α^α1μ^−1/2(v)Q(μ^1/2∂^α1f, μ^1/2∂^α2f)

=

α₁+α₂=α

C_α^α1∂i

R³

φ^ij(v−v)μ^1/2(v)

∂^α1f(v)∂_j^α2f(v)−∂_j^α1f(v)∂^α2f(v) dv

−

α₁+α₂=α

C_α^α11 2

R³

φ^ij(v−v)v_iμ^1/2(v)

·

∂^α1f(v)∂_j^α2f(v)−∂_j^α1f(v)∂^α2f(v)

dv. (2.5)

By using (2.3), we have

|∂^αΓ(f, f)|2≤C

α₁+α₂=α |β₁|≤2

|∂_β^α1

1f|2

|β₂|≤2

|(1 +|v|)^γ+2∂_β^α2

2f|2

.

Thus,

|α|=1

∂^αΓ(f, f) ²≤C

|β₁|≤2

|∂_β1f|2 ²

L^∞_x

|α₂|=1,|β₂|≤2

(1 +|v|)^γ+2∂_β^α2

2f ²

+C

|β₂|≤2

(1 +|v|)^γ+2|∂β₂f|2 ²

L^∞_x

|α₁|=1,|β₁|≤2

∂_β^α1

1f ²

≤C

|α|+|β|≤N

∂_β^αf ²

·

1≤|α|,|α|+|β|≤N

(1 +|v|)^γ+2∂_β^αf ²

≤CEl(t)El(t).

Similar argument leads to the proof of the case when |α|= 0. This completes the proof of the lemma.

We now recall two basic estimates from [9] in the following two lemmas.

Lemma 2.3 The linearized operatorLhas the properties thatLg, h=g, Lh,Lg, g ≥ 0. AndLg= 0 if and only ifg=Pg. Furthermore, there exists aδ >0 such that

Lg, g ≥δ|(I−P)g|²σ. (2.6) Lemma 2.4 For anyη >0 and l≥0, there exist constantsCη >0 andC >0 such that

w^2l∂_βLg, ∂_βg ≥ |w^l∂_βg|²σ−η

β₁≤β

|w^l∂_β1g|²σ−C_η|g|²σ, (2.7)

and

|w^2l∂_β^αΓ(f, g), ∂^α_βh|

≤C

α₁≤α,β₁+β₂≤β

|w^l∂^α_β¹

1f|2|w^l∂_β^α−α1

2 g|σ+|w^l∂_β^α1

1f|σ|w^l∂^α_β^−α1

2 g|2

|w^l∂_β^αh|σ, (2.8)

where|α|+|β| ≤N.

In the following, we prove one more weighted estimate on the nonlinear collision operator.

Lemma 2.5 Letl ≥0 and |α|+|β| ≤N. Then for anyη >0, there is some constant Cη>0 such that

w^2l∂_β^αΓ(f, f), ∂_β^α(I−P)f ≤η w^l∂_β^α(I−P)f ²_σ+C_ηEl(t)Dl(t). (2.9) Proof By using the decomposition (1.6), we have

Γ(f, f) = Γ(Pf,Pf) + Γ(Pf,(I−P)f) + Γ((I−P)f,Pf) + Γ((I−P)f,(I−P)f).

By (2.8), we obtain

w^2l∂_β^αΓ(Pf,Pf), ∂_β^α(I−P)f

≤C

α₁≤α,β₁+β₂≤β

R³

[|w^l∂_β^α1

1Pf|2|w^l∂^α_β^−α1

2 Pf|σ

+|w^l∂_β^α1

1Pf|σ|w^l∂_β^α−α1

2 Pf|2]|w^l∂_β^α(I−P)f|σdx. (2.10) Notice that there existsC >1 such that

|g|²σ≥C⁻¹|w^1/2g|2, |w^l∂_βPf|²σ≤C|Pf|². (2.11) We only consider the first term in (2.10) because the second term can be estimated similarly.

If|α₁|+|β₁| ≤N/2, for anyη >0, we easily get

R³|w^l∂^α_β¹

1Pf|2|w^l∂_β^α−α1

2 Pf|σ|w^l∂_β^α(I−P)f|σdx

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤1

w^l∇x∂^α∂_β^α1

1Pf ² w^l∂_β^α−α1

2 Pf ²_σ

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤1

∇x∂^α∂^α1Pf ² ∂^α−α1Pf ²,

which is bounded by the right hand side of (2.9). The discussion on the case when|α₁|+|β₁| ≥ N/2 is similar.

By applying Lemma 2.4 again, we have

w^2l∂_β^αΓ(Pf,(I−P)f), ∂_β^α(I−P)f

≤C

α₁≤α,β₁+β₂≤β

R³

[|w^l∂_β^α1

1Pf|2|w^l∂_β^α−α1

2 (I−P)f|σ

+|w^l∂^α_β¹

1(I−P)f|2|w^l∂_β^α−α1

2 Pf|σ]|w^l∂_β^α(I−P)f|σdx.

Again, we only consider the first term because the second term can be estimated similarly.

If|α₁|+|β₁| ≤N/2, by using the Sobolev inequality, we obtain

R³|w^l∂_β^α1

1Pf|2|w^l∂_β^α−α1

2 (I−P)f|σ|w^l∂_β^α(I−P)f|σdx

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤2

w^l∂^α∂_β^α1

1Pf ² w^l∂_β^α−α1

2 (I−P)f ²_σ

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤2

∂^α∂^α1Pf ² w^l∂_β^α−α1

2 (I−P)f ²_σ.

If|α−α₁|+|β₂| ≤N/2, we also have

R³|w^l∂_β^α1

1Pf|2|w^l∂_β^α−α1

2 (I−P)f|σ|w^l∂_β^α(I−P)f|σdx

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤2

w^l∂^α∂_β^α−α1

2 (I−P)f ²_σ w^l∂_β^α1

1Pf ²

≤η w^l∂_β^α(I−P)f ²_σ+η

|α|≤2

w^l∂^α∂_β^α−α1

2 (I−P)f ²_σ ∂^α1Pf ². Both are bounded by the right hand side of (2.9).

Since the other cases can be estimated similarly, we complete the proof of this lemma.

3 Energy Estimates and Compensating Function

In this section, we will first consider the energy estimate on the microscopic component.

For this, we have the following equation from (1.3) [∂_t+v· ∇x+ e^−ΦL](I−P)f

= e^−Φ/2Γ(f, f) + (∇xΦ +E)· ∇vf−1

2v·Ef+S−[∂t+v· ∇x]Pf. (3.1) In fact, the estimate on the microscopic component can be obtained by applying the standard energy method because of the dissipation of the linearized collision operator on the microscopic component.

Lemma 3.1 Under the assumptions (H₀) and (H₁), for anyl≥0, we have

1≤|β|,|α|+|β|≤N

d

dt w^l∂^α_β(I−P)f ²+ w^l∂_β^α(I−P)f ²_σ

≤Cε(1 +t)⁻¹⁻+C

1≤|α|≤N

∂^αPf ²+C

|α|≤N

w^l∂^α(I−P)f ²_σ+CEl(t)Dl(t). (3.2)

Proof By applying∂_β^α (β= 0) on (3.1), we obtain [∂_t+v· ∇x]∂_β^α(I−P)f+

|β₁|=1

C_β^β1∂_β1v· ∇x∂_β^α_−β

1(I−P)f + e^−Φ∂_β^αL(I−P)f

+

α₁<α

C_α^α1∂^α−α1e^−Φ∂_β^α1L(I−P)f =∂_β^α(e^−Φ/2Γ(f, f)) + (∇xΦ +E)· ∇v∂_β^αf

+

α₁<α

C_α^α1(∂^α−α1∇xΦ +∂^α−α1E)· ∇v∂_β^α1f−

α₁≤α,β₁≤β

1

2C_α^α1C_β^β1∂β₁v·∂^α1E∂_β^α₋⁻_β^α1

1f +∂_β^αS−

β₁<β

∂β−β₁C_β^β1v· ∇x∂_β^α₁Pf−[∂t+v· ∇x]∂_β^αPf. (3.3)

We take the inner product of (3.3) overR³×R³ withw^2l∂_β^α(I−P)f. The first term on the left hand side is ¹_2d^d_t w^l∂_β^α(I−P)f ².

By H¨older inequality, the second term on the left hand side of the inner product is bounded by

η w^l∂_β^α(I−P)f ²+C_η

|β₁|=1

w^l∇x∂_β^α_−β

1(I−P)f ². (3.4)

Notice that the assumption (1.11) and the Sobolev imbedding theorem imply that sup

x∈R³

|α|≤N−1

|∂^αΦ(x)| ≤Cε. (3.5)

Then by Lemma 2.4, for anyη >0, the third term on the left hand side in the inner product satisfies

(w^2le^−Φ∂_β^αL(I−P)f, ∂_β^α(I−P)f)

≥ w^l∂^α_β(I−P)f ²_σ−η

β₁≤β

w^l∂_β^α₁(I−P)f ²_σ−Cη ∂^α(I−P)f ²_σ.

In the following, we only consider the case when|α−α₁|= 1 for the last term on the left hand side of (3.3) because the other cases can be estimated similarly. By Lemma 2.4, we obtain

|(w^2l∂^α−αΦe^−Φ∂βL[(∂^α1(I−P)f], ∂_β^α(I−P)f)|

≤C

|α−α₁|=1,β₁≤β

R³|∂^α−α1Φ| · |w^l∂_β^α1

1(I−P)f|σ|w^l∂_β^α(I−P)f|σdx

≤Cε w^l∂_β^α(I−P)f ²_σ+Cε

α₁<α,β₁≤β

w^l∂_β^α1

1(I−P)f ²_σ.

For the nonlinear collision operator, by (1.11) and (3.5), Lemma 2.5 gives

|(w^2l∂_β^α(e^−Φ/2Γ(f, f)), ∂_β^α(I−P)f)| ≤η w^l∂_β^α(I−P)f ²_σ+C_ηEl(t)Dl(t).

For the second term on the right hand side of (3.3), recalling (H₀), (H₁) and|β| ≥1, we apply the decomposition onf as (1.6) to obtain

|(w^2l(∇xΦ +E)· ∇v∂^α_β(I−P)f, ∂_β^α(I−P)f)|

=|((∇xΦ +E)· ∇vw^2l∂^α_β(I−P)f, ∂_β^α(I−P)f)|

≤C( ∇xΦ L^∞+ E _L∞

t,x) w^l∂_β^α(I−P)f ²≤Cε w^l∂_β^α(I−P)f ²;

|(w^2l(∇xΦ +E)· ∇v∂^α_βPf, ∂_β^α(I−P)f)|

≤( ∇xΦ L³(R³)+ |x|E _L∞

t,x)( w^l∇v∇x∂_β^αPf ²+ w^l∂_β^α(I−P)f ²)

≤Cε

1≤|α|≤N

∂^αPf ²+Cε w^l∂^α_β(I−P)f ²,

where we have the Hardy inequality _|^g_x| ≤C ∇xg and H¨older inequality. Similarly, when

|α−α₁| ≤N/2, the third term on the right hand side of (3.3) satisfies

|(w^2l(∂^α−α1∇xΦ +∂^α−α1E)· ∇v∂_β^α1(I−P)f, ∂_β^α(I−P)f)|

≤C( ∂^α−α1∇xΦ L^∞_x + ∂^α−α1E L^∞_t,x)( w^l∇v∂_β^α1(I−P)f ²+ w^l∂_β^α(I−P)f ²)

≤Cε

|α₁|+|β₁|≤N

w^l∂_β^α1

1(I−P)f ²+Cε w^l∂_β^α(I−P)f ²;

|(w^2l(∂^α−α1∇xΦ +∂^α−α1E)· ∇v∂_β^α1Pf, ∂^α_β(I−P)f)|

≤C( ∂^α−α1∇xΦ _L3(R³)+ |x|∂^α−α1E L^∞_t,x)( w^l∇x∇v∂_β^α1Pf ²+ w^l∂^α_β(I−P)f ²)

≤Cε

1≤|α₁|≤N

∂^α1Pf ²+Cε w^l∂_β^α(I−P)f ².

Similarly, when|α−α₁| ≥N/2, this term has the same bound. Thus, we have

|(w^2l(∂^α−α1∇xΦ +∂^α−α1E)· ∇v∂_β^α1f, ∂_β^α(I−P)f)|

≤Cε

|α₁|+|β₁|≤N

w^l∂_β^α1

1(I−P)f ²+Cε

1≤|α₁|≤N

∂^α1Pf ²+Cε w^l∂_β^α(I−P)f ².

For the fourth term on the right hand side of (3.3), by using the decomposition (1.6) again, the condition (H₁) and the relation|g|σ ≥C|(1 +|v|)^1/2g|2, we get

(w^2l∂_β1v·∂^α1E∂_β^α₋⁻_β^α1

1(I−P)f, ∂_β^α(I−P)f)

≤C ∂^α1E _L∞

t,x( w^l(1 +|v|)^1/2∂_β^α₋⁻_β^α1

1(I−P)f ²+ w^l(1 +|v|)^1/2∂_β^α(I−P)f ²)

≤Cε w^l∂^α_β−⁻_β^α1

1(I−P)f ²_σ+Cε w^l∂^α_β(I−P)f ²_σ; (w^2l∂β₁v·∂^α1E∂_β^α₋⁻_β^α1

1Pf, ∂_β^α(I−P)f)

≤C |x|∂^α1E _L∞

t,x ∇x∂^α−α1Pf ²+Cε w^l∂_β^α(I−P)f ²_σ

≤Cε ∇x∂^α−α1Pf ²+Cε w^l∂_β^α(I−P)f ²_σ, where we have used|β| ≥1 and the Hardy inequality.

By using (1.4), (3.5) and the condition (H₁), we have (w^2l∂_β^αS, ∂ _β^α(I−P)f)

≤Cη w^l−1/2∂_β^α(e^Φ/2μ^−1/2S) ²+Cη ∂^α(e^−Φ/2E) ²+ 2η w^l∂_β^α(I−P)f ²_σ

≤2Cηε(1 +t)⁻¹⁻+ 2η w^l∂_β^α(I−P)f ²_σ. Since|β| ≥1, we have

(w^2l∂_β−β1v· ∇x∂_β^α₁Pf, ∂_β^α(I−P)f) ≤C ∇x∂^αPf ²+η w^l∂_β^α(I−P)f ²_σ.

And for anyη >0, the last term on the right hand side of (3.3) satisfies

|(w^2l[∂_t+v· ∇x]∂_β^αPf, ∂_β^α(I−P)f)| ≤η w^l∂_β^α(I−P)f ²_σ+C_η[ ∂_t∂^αPf ²+ ∇x∂^αPf ²].

Hence, based on the above inequality, the last term on the right hand side of (3.3) satisfies (w^2l[∂t+v· ∇x]∂_β^αPf, ∂_β^α(I−P)f)

≤η w^l∂_β^α(I−P)f ²_σ+Cη[ε(1 +t)⁻¹⁻+

1≤|α|≤N

∂^αPf ²+

1≤|α|≤N

∂^α(I−P)f ²].(3.6)

On the other hand, applying ∂^α (|α| ≤N −1) on (3.1) gives [∂t+v· ∇x]∂^αPf =∂^α(e^−Φ/2Γ(f, f))−∂^α((∇xΦ +E)· ∇vf)−1

2∂^α(v·Ef) +∂^αS

−[∂t+v· ∇x]∂^α(I−P)f −

α₁≤α

∂^α1e^−ΦL[∂^α−α1(I−P)f]. (3.7)

Since

P∂t∂^αf, ∂^αΓ(f, f)−[∂_t+L]∂^α1(I−P)f= 0,