Digital Object Identifier (DOI) 10.1007/s00220-006-0103-4
Mathematical Physics
Global Existence of Classical Solutions to the Vlasov-Poisson-Boltzmann System
Tong Yang1, Huijiang Zhao2
1 Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. E-mail: [email protected]
2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Received: 7 November 2005 / Accepted: 4 April 2006 Published online: 7 September 2006 – © Springer-Verlag 2006
Abstract: The time evolution of the distribution function for the charged particles in a dilute gas is governed by the Vlasov–Poisson–Boltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible Navier–Stokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].
Contents
1. Introduction . . . 569
2. Preliminaries . . . 573
3. Energy Estimates . . . 582
4. Convergence Rate . . . 588
5. Appendix . . . 594
5.1 The proof of Lemma 3.3 . . . 594
5.2 The proof of Lemma 3.4 . . . 595
5.3 The proof of Lemma 3.5 . . . 598 1. Introduction
Consider the Vlasov–Poisson–Boltzmann system:
⎧⎪
⎨
⎪⎩
ft +ξ · ∇xf +∇x· ∇ξf =Q(f, f), x=ρ− ¯ρ=
R3
f dξ − ¯ρ, || →0, as|x| →+∞, (1.1)
with initial data
f(0,x, ξ)= f0(x, ξ). (1.2)
Here f(t,x, ξ)is the distribution function for the particles located at x =(x1,x2,x3)∈ R3 with velocityξ = (ξ1, ξ2, ξ3) ∈ R3 at time t ≥ 0. The self-consistent electric potential(t,x)is coupled with the distribution function f(t,x, ξ)through the Pois- son equation in (1.1). The constant background charge density is denoted byρ >¯ 0.
The short-range binary interaction between particles is given by the standard Boltzmann collision operator Q(f,g)for the hard-sphere model:
Q(f,g)(ξ)≡ 1 2
R3
S2+
f(ξ)g(ξ∗)+ f(ξ∗)g(ξ)− f(ξ)g(ξ∗)
−f(ξ∗)g(ξ)
|(ξ−ξ∗)·| dξ∗d. Here S2+ = {∈S2: (ξ−ξ∗)·≥0}, and
ξ=ξ− [(ξ−ξ∗)·], ξ∗ =ξ∗+[(ξ−ξ∗)·]
is the relation between velocitiesξ,ξ∗after and the velocitiesξ,ξ∗before the collision, which is induced by the conservation of momentum and energy.
In this paper, we will give a satisfactory global existence theory of classical solutions to the Cauchy problem (1.1) and (1.2) near a given global Maxwellian with densityρ¯ which itself is a trivial solution to the system (1.1):
M¯ =M[ ¯ρ,0,θ]¯ = ρ¯ (2πRθ)¯ 23 exp
−|ξ|2 2Rθ¯
. (1.3)
Hereθ >¯ 0 is a constant.
The Vlasov–Poisson–Boltzmann system is a classical physical model for the time evolution of charged particles, like electrons with the external force generated by the self-induced electronic field. It can also be viewed as a limiting model when the light speed tends to infinity of the Vlasov-Maxwell-Boltzmann system for the time evolu- tion of ions and electrons under the influence of the self-induced electric and magnetic fields. A lot of work has been done on these systems. In what follows, we only mention those related to this paper. The global existence of classical solutions was obtained in [14, 15] when the solution is spatially periodic. In fact, for the spatially periodic solution, the Poincaré inequality is used to control the solution by the celebrated H-theorem about the dissipation of the collision operator on the microscopic components. To be precise, the viscosity and heat conductivity in the compressible Navier–Stokes level coming from the leading order of the microscopic component give the dissipation on the spatial derivatives of the velocity and temperature, but usually not on the perturbation of these two macroscopic components themselves. This dissipation, nevertheless, yields the L2xestimate on the macroscopic components for the spatially periodic solutions because the Poincaré inequality holds.
The global existence of classical solutions to the Cauchy problem (1.1) and (1.2) was first obtained in [30] under the condition that either the mean free path is sufficiently small or the background charge densityρ¯is sufficiently large. Notice that the uniform
boundedness on the derivatives of the solution was given only for those with temporal differentiation no more than one in [30]. The main purpose of this paper is to remove these restrictions and then give the global existence of classical solutions in the general setting. One of the main techniques used here is the elaborated energy method for the Boltzmann equation, [23, 25, 31]. Before [31], the a priori energy estimate is closed by deducing energy estimates on the microscopic component G and the macroscopic component M respectively. This was done by applying an energy method with respect to two weighted functions, that is, the local Maxwellian and an appropriately chosen global Maxwellian. Based on this method, the nonlinear stability of basic wave patterns for the Boltzmann equation with slab symmetry was obtained in [19, 24, 25, 32]. The observa- tion in [31] is that the microscopic projection of the local Maxwellian with respect to the unperturbed global Maxwellian is of quadratic order of the perturbation so that two sets of energy estimates are not necessary. Hence, it not only simplifies the analysis but also gives better understanding of the solution behavior.
Besides the classical solutions, the global existence of renormalized solutions with large initial data to the Vlasov–Poisson–Boltzmann system was proved in [22] and this result was later generalized to the case with boundary in [27]. The time asymptotic behavior of the renormalized solutions with extra regularity assumptions was studied in [6]. The decay property of the solutions to the linearized Vlasov–Poisson–Boltzmann system aroundM was studied in [9, 10]. Finally, for the perturbation around vacuum,¯ the results in [7, 16] give the global existence of smooth small-amplitude solutions for the cut-off potentials and the hard sphere model.
In what follows, the function space of solutions is the standard Sobolev space HtN,x,ξ R3×R3
:
HtN,x,ξ
R3×R3 =
⎧⎪
⎨
⎪⎩f(t,x, ξ)
∂αx∂tβ∂ξγ
f(t,x,ξ)− ¯M(ξ)
√M¯ ∈C([0,∞), L2x,ξ
R3×R3
,|α|+β+|γ| ≤N
⎫⎪
⎬
⎪⎭. (1.4) The result in this paper can be stated as follows.
Theorem 1.1. Assume that f0(x, ξ)≥0 and N ≥4. A sufficiently small constantε >0 exists such that if
E(f0)=∇x−x1
ρ0(x)− ¯ρ
L2x(R3)+
|α|+β≤N
∂xα∂ξβ(f0M(¯x(ξ),ξ)− ¯M(ξ))
L2x,ξ(R3×R3)≤ε, (1.5) then there exists a unique global classical solution f(t,x, ξ)to the Vlasov–Poisson–
Boltzmann system (1.1) and (1.2) which satisfies f(t,x, ξ)≥0 and
|α|+β≤N
∂αx∂tβ(ρ− ¯ρ,u, θ− ¯θ)(t,x)2+∇x(t,x)2
+
|α|+β+|γ|≤N
R3
R3
∂xα∂tβ∂ξγP1M¯ f(t,x, ξ)2 M¯ dξd x
+
|α|+β+|γ|≤N
t
0
R3
R3
ν(ξ)∂xα∂tβ∂ξγPM1¯ f(τ,x, ξ)2 M¯ dξd xdτ
+ t
0
⎛
⎝ρ− ¯ρ2+
|α|+β≤N−1
∇x∂xα∂tβu2+
1≤|α|+β≤N
∂xα∂tβ(ρ, θ)2
⎞
⎠dτ
≤ O(1)E(f0)2, (1.6)
whereρ0(x)is the initial density function, andν(ξ)is the collision frequency defined in (2.11).
Furthermore, we have
sup
x∈R3
⎧⎪
⎨
⎪⎩
|α|+β+|γ|≤N−3
R3 ∂αx∂tβ∂ξγ
f(t,x,ξ)− ¯M(ξ)2 M¯(ξ) dξ
12
+|∇x(t,x)|
⎫⎪
⎬
⎪⎭
≤O(1)(1 + t)−12.
(1.7)
Here and in what follows, O(1)is used to denote a generic positive constant.
Remark 1.1. The convergence rate given here is less than the convergence rate for the Boltzmann without forcing which is(1 + t)−34. However, it is better than the one for the stationary potential force obtained in [29] because of the dependence of the force on the solution. To obtain the optimal convergence rate is an interesting problem, but we will not pursue this in this paper. Another possible improvement of the above result is to study the case when the background charge densityρ¯is a function of x rather than a constant considered in this paper. In this case, the stationary solution is no longer a global Maxwellian but a local Maxwellian as a function of x andξ. For this, the existence of stationary solutions was given in [11]. Based on the existence of stationary solutions and their decay properties, the analysis in this paper could be elaborated for the study of the stability of stationary solutions.
Before concluding this section, it is worth pointing out that the energy estimate is also closed by using the decomposition with respect to the global MaxwellianM in [17]¯ for the Boltzmann equation without forcing. To be precise, based on the macroscopic projection defined by (2.6), i.e.,
PM0¯ f =
⎛
⎝a(t,x)+ 3
j=1
bj(t,x)ξj+ c(t,x)|ξ|2
⎞
⎠M,¯
one can derive a set of equations for the functions a(t,x),bj(t,x)(j = 1,2,3)and c(t,x)so that estimates on these functions can be obtained. However, the time evolu- tion of the conserved quantities
ρ,m, ρ1
2|u|2+E
, which are defined by (2.1) and governed by the conservation laws (2.8), is not clear by just analyzing these functions.
The main observation in this paper is to close the energy estimates with respect to the global MaxwellianM by estimating the conserved quantities¯
ρ, ρu, ρ1
2|u|2+E governed by the conservation laws in the form of the compressible Navier–Stokes equa- tions with the non-fluid component PM1 f defined in (2.6) appearing in the source terms.
Hence, the analytic techniques for the system of compressible Navier–Stokes equations can be used. Therefore, it not only simplifies the analysis in the previous works, but also gives better stability analysis for the Vlasov–Poisson–Boltzmann system.
The rest of the paper is organized as follows. Some preliminaries such as the refor- mulation of the problem through the decomposition and the standard estimate on the microscopic projection will be given in Sect. 2. Section 3 concerns the energy estimates for the global existence in the space HNt,x,ξ
R3×R3
. And the convergence rate in time of the solution to the global Maxwellian is proved in Sect. 4. Finally, the proofs of some technical lemmas are given in the Appendix.
2. Preliminaries
As in [30], we first reformulate the Vlasov–Poisson–Boltzmann system through the decomposition around the local Maxwellian introduced in [23] for the Boltzmann equa- tion. For the convenience of the readers, we briefly give the derivation as follows.
Let f(t,x, ξ)be the solution to the system (1.1). By the five conserved quantities, i.e., the mass densityρ(t,x), momentum density m(t,x)=ρ(t,x)u(t,x)and energy densityE(t,x)+|u(t,x)|2/2 given by
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
ρ(t,x)≡
R3
f(t,x, ξ)dξ, mi(t,x)≡
R3
ψi(ξ)f(t,x, ξ)dξ, i =1,2,3, ρ
E+ 12|u|2
(t,x)≡
R3
ψ4(ξ)f(t,x, ξ)dξ,
(2.1)
the local Maxwellian is
M≡M[ρ,u,θ](ξ)≡ ρ
(2πRθ)3exp
−|ξ−u|2 2Rθ
. (2.2)
Here θ(t,x)is the temperature related to the internal energy E(t,x)by E = 32Rθ, u(t,x) =(u1(t,x),u2(t,x),u3(t,x))is the fluid velocity, and R is the gas constant.
As usual,ψα(ξ),α=0,1, . . . ,4, are the five collision invariants:
⎧⎪
⎪⎨
⎪⎪
⎩
ψ0(ξ)≡1,
ψi(ξ)≡ξi, i=1,2,3, orψ(ξ)=ξ, ψ4(ξ)≡ 12|ξ|2.
(2.3)
In both the Hilbert and Chapman–Enskog expansions, the leading order term in the expansion of the solution f(t,x, ξ)is a local Maxwellian. In particular, the solution is expanded as the summation of the local Maxwellian and an expansion of the microscopic component with respect to the Knudsen number in the Chapman–Enskog expansion. The main idea in [23] is not to expand the microscopic component into a series with respect to the Knudsen number. Instead, one can simply write the solution as the sum of the local Maxwellian and microscopic component G=G(x,t, ξ):
f(t,x, ξ)=M(t,x, ξ)+ G(t,x, ξ). (2.4)
As shown later, the Vlasov–Poisson–Boltzmann system can then be reformulated into a coupled system containing the conservation laws for the macroscopic components, the equation for microscopic component and the Poisson equation for the potential of the force.
For later use, we need to define the projection with respect to a given Maxwellian, either global or local for any function in the corresponding L2ξspace. To be precise, for any given MaxwellianM˜ = ˜M[ ˜ρ,˜u,θ]˜, we define an inner product inξ ∈R3by
h,gM˜ ≡
R3
h(ξ)g(ξ) M˜ dξ,
for functions h and g ofξsuch that the integral is well defined. By using this inner prod- uct, the subspace spanned by the collision invariants has the following set of orthogonal basis:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
χ0M˜ =χ0(ξ; ˜ρ,u,˜ θ)˜ ≡√1
˜ ρM,˜ χiM˜ =χi(ξ; ˜ρ,u,˜ θ)˜ ≡ √ξi− ˜ui
Rρ˜θ˜M,˜ i=1,2,3, χ4M˜ =χ4(ξ; ˜ρ,u,˜ θ)˜ ≡√1
6ρ˜
|ξ− ˜u|2 Rθ˜ −3
M,˜ χαM˜, χβM˜
M˜ =δαβ, forα, β =0,1,2,3,4.
(2.5)
The macroscopic projection P0M˜ and microscopic projection P1M˜ can then be defined by:
⎧⎪
⎨
⎪⎩
PM0˜h≡ 4
α=0
h, χαM˜
˜ MχαM˜, PM1˜h≡h−P0M˜h.
(2.6)
Notice that the operators P0M˜ and P1M˜ are projections, that is,
PM0˜PM0˜ =PM0˜, P1M˜P1M˜ =P1M˜, P0M˜PM1˜ =PM1˜PM0˜ =0. And the system of conservation laws
R3
ψα
ft+ξ· ∇xf +∇x· ∇ξ f
dξ =0, α=0,1, . . . ,4, (2.7) becomes
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
ρt+ divx m=0, mi t +
3 j=1
ujmi
xj
+ pxi −ρxi +
R3
ψi(ξ) ξ· ∇xG
dξ =0, i=1,2,3,
ρ
|u|2 2 +E!
t + 3 j=1
"
uj ρ
|u|2 2 +E
+ p
!#
xj −m· ∇x +
R3
ψ4(ξ) ξ· ∇xG
dξ =0.
(2.8)
Here, the equation of state for the monatomic gas is given by, p= 2
3ρE, E =θ,
where the gas constant R is chosen to be 23 without loss of generality. Notice also that the macroscopic entropy S is given by
S = −2
3lnρ+ ln 4
3πθ
+ 1.
By noticing that Gt,∇ξG, LMG, and Q(G,G)are microscopic components, the micro- scopic equation is obtained by applying the microscopic projection PM1 to the Vlasov–
Poisson–Boltzmann system (1.1):
Gt+ P1M
ξ · ∇xG +ξ · ∇xM
+∇x· ∇ξG=LMG + Q(G,G), (2.9) where LMis the linearized collision operator defined by
LMg=LM[ρ,u,θ]g≡2Q(M[ρ,u,θ],g). (2.10) The null space of LM, denoted byN, is spanned by the collision invariantsχMj , j = 0,1,2,3,4.
For the hard sphere model, LMtakes the form, cf. [18], and [1, 3, 4, 8, 13] for the discussions on other collision kernels,
(LMh) (ξ)= −ν(ξ;ρ,u, θ)h(ξ)+
M(ξ)KM
h
√M
(ξ)
.
Here KM(·)= −K1M(·)+ K2M(·)is a symmetric compact L2ξ-operator. The collision frequencyν(ξ;ρ,u, θ)and ki M(ξ, ξ∗)which is the kernel of the operator Ki M(i =1,2) have the following expressions:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
ν(ξ;ρ,u, θ)=√22πρRθ$
Rθ
|ξ−u|+|ξ −u| |ξ−u|
0
exp
−2Ry2θ d y +Rθexp
−|ξ−2Ruθ|2# , k1M(ξ, ξ∗)=√ πρ
(2πRθ)3|ξ−ξ∗|exp
−|ξ−4Ruθ|2 −|ξ∗4R−uθ|2 , k2M(ξ, ξ∗)=√2ρ
2πRθ|ξ−ξ∗|−1exp
−|ξ−ξ8Rθ∗|2 −(|ξ|8R2θ|ξ−ξ−|ξ∗|2)2
∗|2
.
(2.11)
Moreover, for the hard sphere model,
0< ν0≤ ν(ξ;ρ,u, θ) 1 +|ξ| <∞ holds uniformly inξ for some constantν0.
Since LMis a bounded and one to one operator onN⊥, from (2.9), we have G=L−M1
PM1 (ξ· ∇xM) + L−M1
∂tG + P1M(ξ· ∇xG)+∇x· ∇ξG−Q(G,G)
=L−M1
PM1 (ξ· ∇xM)
+. (2.12)
Here
N⊥= f(ξ):
R3
χMj f(ξ)
M dξ =0, j =0,1,2,3,4.
% .
Substituting (2.12) into (2.8) yields the following conservative system for the mac- roscopic components:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
ρt+ divx m=0, mi t +
3 j=1
ujmi
xj
+ pxi −ρxi +
R3ψi(ξ)
ξ· ∇xL−M1
PM1 (ξ· ∇xM) dξ +
R3ψi(ξ)
ξ · ∇x
dξ =0, i =1,2,3, ρ
|u|2 2 +E!
t + 3 j=1
"
uj ρ
|u|2 2 +E
+ p
!#
xj
−m· ∇x +
R3
ψ4(ξ)
ξ · ∇xL−M1
PM1 (ξ· ∇xM) dξ+
R3
ψ4(ξ)
ξ· ∇x dξ=0.
(2.13)
Notice that in the above system, the viscosity and heat conductivity terms are the same as those in the compressible Navier–Stokes equations which are independent of the density gradient∇xρ.In fact, by using the Burnett functions A and B, the viscosity coefficient μ(θ)and heat conductivity coefficientκ(θ)are represented by:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
Aj(ξ)= |ξ|22−5ξj, j =1,2,3,
Bi j(ξ)=ξiξj−13δi j|ξ|2, i,j =1,2,3, μ(θ)= −Rθ
R3
Bi j
√ξ Rθ
L−M1
[1,0,θ]
Bi j
√ξ Rθ
M[1,0,θ]
dξ >0, i = j, κ(θ)= −R2θ
R3
Al
√ξ Rθ
L−M1
[1,0,θ]
Al
√ξ Rθ
M[1,0,θ]
dξ >0,
(2.14)
and the system (2.13) can be rewritten as
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
ρt+ divxm=0, mi t +
3 j=1
ujmi
xj + pxi −ρxi = 3
j=1
μ(θ)
ui xj + uj xi −23δi jdivxu
xj
−
R3
ψi(ξ) (ξ· ∇x)dξ, i =1,2,3, ρ(12|u|2+E)
t+ 3 j=1
uj
ρ1
2|u|2+E + p
xj
−m· ∇x
= 3
i,j=1
&
μ(θ)ui
ui xj + uj xi −23δi jdivxu'
xj
+ 3 j=1
κ(θ)θxj
xj −
R3ψ4(ξ) (ξ· ∇x)dξ.
(2.15)
In (2.15), the structures of the compressible Euler–Poisson and the compressible Navier–
Stokes–Poisson equations are clear. For instance, when the microscopic component G is set to be zero, the system (2.15) becomes part of the compressible Euler–Poisson equations. On the other hand, when is set to be zero in (2.15), it becomes part of the compressible Navier–Stokes–Poisson equations. Both Euler–Poisson and Navier–
Stokes–Poisson systems are approximations to the Vlasov–Poisson–Boltzmann system if they are derived through the Hilbert and Chapman-Enskog expansions. Nevertheless, the above reformulation is exact to the Vlasov–Poisson–Boltzmann system and is con- sistent in spirit with those expansions where the first order approximation is a local Maxwellian.
In what follows, we will use the system (2.15) to analyze the macroscopic compo- nents and the Vlasov–Poisson–Boltzmann system itself for the microscopic component.
This approach follows from the elaborated analysis on the Boltzmann equation in [31]
which improves the previous energy method from [23] so that the analytic techniques from the theory of conservation laws and the H-theorem for the Boltzmann equation are fully and suitably used.
Finally in this section, we state some known estimates for later use. The first lemma is about some Sobolev inequalities.
Lemma 2.1. For g(x)∈H1(R3), we have
g(x)L6(R3)≤C0∇xg(x), (2.16) where C0is a uniform positive constant. Consequently, for g(x)∈ H2(R3), there exists a uniform positive constant C1such that
⎧⎨
⎩
g(x)L∞(R3)≤C1∇xg(x)1,
g(x)L4(R3)≤C1∇xg(x)34g(x)14. (2.17) Here and in what follows,·and·sdenote the standard L2(R3)and Hs(R3)norms respectively.
For the nonlinear and linearized collision operators Q(f,f)and LMG, we have the following estimates from [12].
Lemma 2.2. There exists a uniform constant C2>0 such that
R3
ν(ξ)−1Q(f,g)2
M˜ dξ ≤C2
$
R3
ν(ξ)f2 M˜ dξ·
R3
g2 M˜ dξ+
R3
f2 M˜ dξ·
R3
ν(ξ)g2 M˜ dξ
( , (2.18) whereM is any Maxwellian such that the above integrals are well defined.˜
For P1M0 f which is the microscopic projection of the solution f(t,x, ξ)with respect to a given Maxwellian M0, the microscopic version of the H-theorem states that the linearized collision operator LM0 is negative definite on PM10 f , cf. [2], i.e.,
−
R3
PM10f LM0
P1M0 f
M0
dξ ≥σ
R3
ν(ξ)PM10f2 M0
dξ,
for some positive constantσ. In fact, the Maxwellian around which the linearized colli- sion operator is defined can be different from the Maxwellian used as the weight function in L2ξ. That is, we also have the following estimate by Lemma 2.2, cf. [24].
Lemma 2.3. When θ2 < θ, there exist two positive constants˜ σ = σ(u, θ; ˜u,θ)˜ and η0=η0(u, θ; ˜u,θ)˜ such that if|u− ˜u|+|θ− ˜θ|< η0and h(ξ)∈N⊥, we have
−
R3
h LMh M˜ dξ ≥σ
R3
ν(ξ)h2
M˜ dξ. (2.19)
Here, M≡M[ρ,u,θ](ξ)andM˜ = ˜M[ ˜ρ,˜u,θ˜](ξ).
Remark 2.1.η0 in the above lemma is some positive constant depending on the first non-zero eigenvalue of the linearized collision operator LMwhich does not need to be small, cf. [24].
A direct consequence of Lemma 2.3 and the Cauchy-Schwarz inequality is the fol- lowing corollary, cf. [24].
Corollary 2.1. Under the assumptions in Lemma 2.3, for h(ξ)∈N⊥, we have
R3
ν(ξ) M˜
L−M1h2dξ ≤σ−2
R3
ν(ξ)−1h2(ξ)
M˜ dξ. (2.20)
The following estimates are based on the following a priori estimate which we will prove by energy method in Sect. 3. Since the analysis for the case when N >4 is similar to the one when N =4. In what follows, we only give the estimates when N=4.
Set
N(t)2= sup
0≤τ≤t
|α|+β≤4
∂xα∂tβ(ρ− ¯ρ,u, θ− ¯θ)(τ,x)2+∇x(τ,x)2
+
|α|+β+|γ|≤4
R3
R3
∂xα∂βt∂ξγPM1¯ f(τ,x,ξ)2 M¯ dξd x
%
< ε2.
(2.21)
By the Sobolev inequality, for x ∈R3and 0≤τ ≤t , the a priori estimate (2.21) implies that
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
|α|+β≤2
∂xα∂tβ(ρ− ¯ρ,u, θ− ¯θ)(τ,x)+∇x∂xα∂tβ(τ,x)
≤O(1)ε,
|α|+β+|γ|≤2
R3
∂xα∂tβ∂ξγP1M¯ f(τ,x,ξ)2
¯
M dξ ≤O(1)ε2.
(2.22)
To perform the energy estimates, we first give the estimate on the difference between the microscopic projections of f(t,x, ξ)with respect to the local Maxwellian M and the global MaxwellianM. Note that¯
PM1¯ f =G + PM1¯M=PM1 f + PM1¯M.
Since P1M¯M is a smooth function ofρ,u,θandξsatisfying
⎧⎪
⎨
⎪⎩
PM1¯M
(u,θ)=(0,θ)¯ =0,
∇(u,θ)PM1¯M
(u,θ)=(0,θ)¯ =PM1¯
∇(u,θ)M
(u,θ)=(0,θ)¯ =0,
P1M¯M is of the quadratic order of u andθ− ¯θwhich are the perturbations in the velocity and temperature variables. Hence, Lemma 2.1 and the a priori estimate (2.21) give the following lemma, cf. [31].
Lemma 2.4. For|α|+β+|γ| ≤4, we have
R3
R3
∂xα∂tβ∂ξγG2
M¯ dξd x≤ O(1)
R3
R3
∂xα∂tβ∂ξγPM1¯ f2 M¯ dξd x + O(1)ε2
|α|+β≤3
∇x∂xα∂tβ(u, θ)2. (2.23)
The next lemma concerns the estimate on the nonlinear collision operator Q(f,f). Lemma 2.5. Under the a priori estimate (2.21), we have
|α|+β+|γ|≤4
R3
R3
ν(ξ)−1∂xα∂tβ∂ξγQ
PM1¯ f,PM1¯ f2
M¯ dξd x
≤O(1)ε2
|α|+β+|γ|≤4,β≤β
R3
R3
ν(ξ)∂xα∂tβ∂ξγPM1¯ f2
M¯ dξd x, (2.24)
|α|+β+|γ|≤4
R3
R3
ν(ξ)−1∂xα∂tβ∂ξγQ
PM1¯ f,PM0¯(M− ¯M)2
M¯ dξd x
≤O(1)ε2
|α|+β+|γ|≤4,β≤β
R3
R3
ν(ξ)∂xα∂tβ∂ξγPM1¯ f2
M¯ dξd x, (2.25)
and
|α|+β+|γ|≤4
R3
R3
ν(ξ)−1∂xα∂tβ∂ξγQ
P0M¯(M− ¯M),PM0¯(M− ¯M)2
M¯ dξd x
≤O(1)ε2
|α|+β≤3
∇x∂xα∂tβ(ρ,u, θ)2. (2.26)
Proof. We only prove (2.24) because the proof of (2.25) and (2.26) is similar. Since
∂xα∂tβ∂ξγQ
P1M¯ f,PM1¯ f
=
(α,β,γ)≤(α,β,γ )
Cα,β,γα,β,γQ
∂xα∂tβ∂ξγPM1¯ f,
∂xα−α∂tβ−β∂ξγ−γP1M¯ f ,
