Vlasov–Poisson–Boltzmann System

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Digital Object Identifier (DOI) 10.1007/s00205-006-0009-5

Cauchy Problem for the

Vlasov–Poisson–Boltzmann System

Tong Yang, Hongjun Yu & Huijiang Zhao

Communicated byC.M. Dafermos

Abstract

The dynamics of dilute electrons can be modelled by the fundamental Vlasov–

Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in- time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [15, 17] by obtaining new entropy estimates for this physical model.

1. Introduction

The flow of dilute charged aggregate particles (e.g. electrons) in the absence of the magnetic field is governed by the fundamental Vlasov–Poisson–Boltzmann system:

⎧⎪

⎨

⎪⎩

f_t+ξ· ∇xf+∇x· ∇ξf=_κ¹Q(f, f ), x=ρ−ρ0=

R³

f dξ−ρ0, || →0, as|x| →+∞ (1.1) with initial data given by

f (0, x, ξ )=f₀(x, ξ ),

where f (t, x, ξ )is the distribution function for the particles at time t 0 lo- cated at x = (x1, x2, x3) ∈ R³ with velocity ξ = (ξ1, ξ2, ξ3) ∈ R³, and the constantκ >0 is the Knudsen number proportional to the mean free path. The self- consistent electric potential(t,x)is coupled with the distribution functionf (t,x,ξ )

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through the Poisson equation. The constant background charge density is denoted byρ₀>0. The short-range interaction between particles is accounted by the stan- dard Boltzmann collision operatorQ(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 velocities ξ, ξ_∗ after and the velocitiesξ, ξ_∗ before the collision, which is induced by conservation of momentum and energy.

Previous work has been carried out on the Vlasov–Poisson–Boltzmann system.

Global-in-time renormalized solutions with arbitrary amplitude were constructed in [14] and the result has been generalized to the case with boundary in [18]. The large-time behavior of weak solutions under extra regularity assumptions was studied in [4]. As to classical solutions, there has been some progress only recently.

That is, for any smooth periodic perturbation of a global Maxwellian that preserves mass, momentum, and total energy, the first global existence result on smooth periodic solutions was obtained in [10]. As regards the whole space, to our knowledge the only result so far is [12] where global smooth small-amplitude solutions near vacuum were constructed for a class of “soft” collision kernels. Therefore, our global existence result is new for the Vlasov–Poisson–Boltzmann system near a given global Maxwellian in the whole space.

To state the main result, we first reformulate the problem by the scaling

⎧⎪

⎪⎨

⎪⎪

⎩

f (t, x, ξ )→ ρ ρ0

f ρ

κρ0

t, ρ κρ0

x, ξ

, (t, x)→

ρ κρ₀t, ρ

κρ₀x

,

whereρis any fixed constant. It is easy to see that after scaling the solution satisfies

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

ft+ξ · ∇xf + ∇x· ∇ξf =Q(f, f ), λ_x=ρ−ρ=

R³

f dξ−ρ, ||→0, as|x|→+∞, f (0, x, ξ )=f₀(x, ξ )= _ρ^ρ₀f₀

ρ κρ0x, ξ

,

(1.2)

withλ= _κ^ρ2⁰ρ. In this paper, we will assumeλ >0 is suitably large which means that either the Knudsen numberκ(i.e. the mean free path) is sufficiently small, or that the constant background charge densityρ0is sufficiently large.

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A micro–macro decomposition of the Boltzmann equation and its solution was introduced in [15, 17]. The Boltzmann equation can be rewritten as a system akin to the equations of fluid dynamics, which henceforth will be dubbed the mac- roscopic system, coupled with an equation for a quantity describing microscopic properties of the gas, which henceforth will be referred to as the microscopic com- ponents, cf. [15]. Similarly, the quantity describing macroscopic properties of the gas will be referred to as the macroscopic components. As an illustration of this method, the global existence of classical solutions around a global Maxwellian was proved in [15] by using a simple energy method through the construction of entropy–entropy flux pairs. This method is used here for the study of the Boltzmann equation with self-induced electric field. As in fluid dynamics, the entropy–entropy flux pair plays an important role in the lower order energy estimate. Moreover, the dissipation induced by the electric field which is governed by the Poisson equation is crucial for closing the basic a priori estimate. The basic estimate in turn implies the uniform space–time integrability of the square of the perturbation of the density function. Note that the corresponding integral diverges for the Boltzmann equation or even for the Navier–Stokes equations without forcing.

To be precise, we decompose the solution of the Vlasov–Poisson–Boltzmann system f (t, x, ξ ) into the macroscopic component, i.e. the local Maxwellian M=M(t, x, ξ )=M[ρ,u,θ](ξ ), and the microscopic component, i.e. G=G(t, x, ξ ) as follows:

f (t, x, ξ )=M(t, x, ξ )+G(t, x, ξ ).

The local Maxwellian M is defined by the five conserved quantities, that is, the densityρ(t, x), the momentumm(t, x) =ρ(t, x)u(t, x), and the energy density E(t, x)+¹₂|u(t, x)|²given by:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

ρ(t, x)≡

R³

f (t, x, ξ )dξ, mⁱ(t, x)≡

R³

ψi(ξ )f (t, x, ξ )dξfori=1,2,3, ρ

E+¹₂|u|²

(t, x)≡

R³

ψ₄(ξ )f (t, x, ξ )dξ,

(1.3)

M≡M[ρ,u,θ](t, x, ξ )≡ ρ(t, x)

(2π Rθ (t, x))³exp

−|ξ−u(t, x)|² 2Rθ (t, x)

. (1.4) Here θ (t, x) is the temperature which is related to the internal energy E by E =3/2Rθ whereR is the gas constant, andu(t, x)is the fluid velocity;

ψ_α(ξ ),α=0,1,· · · ,4, are the five collision invariants, cf. [1, 2]:

⎧⎪

⎪⎨

⎪⎪

⎩

ψ₀(ξ )≡1,

ψi(ξ )≡ξi, fori=1,2,3, orψ=ξ, ψ4(ξ )≡ ¹₂|ξ|²,

(1.5)

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satisfying

R³

ψ_α(ξ )Q(h, g)dξ =0, forα=0,1,2,3,4.

In what follows, we define an inner product inξ ∈R³with respect to a given MaxwellianM as:˜

h, g_M_˜ ≡

R³

1

M˜ h(ξ )g(ξ )dξ,

for functionsh, gofξ so that the above integral is well defined. With respect to the inner product h, gM, the following functions which span the space of the macroscopic components of the solution, are pairwise orthogonal:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

χ₀(ξ;ρ, u, θ )≡ ^√¹_ρM,

χ_i(ξ;ρ, u, θ )≡ ^√^ξⁱ_Rρθ⁻^uⁱM fori=1,2,3, χ₄(ξ;ρ, u, θ )≡ ^√¹_6ρ

|ξ−u|² Rθ −3

M, χ_i, χ_j

M=δ_ij, fori, j=0,1,2,3,4.

(1.6)

By using these five functions, the macroscopic projection P0and microscopic projection P1are:

⎧⎨

⎩

P0h≡ ⁴

j=0

h, χjMχj, P1h≡h−P0h.

(1.7)

Note that the operators P0(and therefore P1) are orthogonal self-adjoint projections with respect to the inner product ·,·M.

Note thath(ξ ) is called a microscopic component if it is orthogonal to the subspace spanned by the collision invariants, that is,

R³

h(ξ )ψ_α(ξ )dξ =0, forα=0,1,2,3,4. (1.8) It is clear that such a function is in the range of the microscopic projection P1. Under the above decomposition, the solutionf (t, x, ξ )of the Vlasov–Poisson–Boltzmann system satisfies,

P0f =M, P1f =G.

Then, by using f (t, x, ξ ) = M(t, x, ξ ) + G(t, x, ξ ), the Vlasov–Poisson–

Boltzmann system (1.2)1becomes:

(M+G)t+ξ· ∇x(M+G)+ ∇x· ∇ξ(M+G)

=(2Q(G,M)+Q(G,G)) . (1.9)

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By applying P0to (1.9), we have

Mt+P0(ξ· ∇xM)+P0(ξ· ∇xG)+ ∇x· ∇ξM=0.

As usual, the system of five conservation laws can be obtained by taking the inner product of the Vlasov–Poisson–Boltzmann system (1.2)1with the collision invari- antsψ_α(ξ ):

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

ρt+divxm=0, m_it +³

j=1

u_im_j

xj +p_x_i−ρ_x_i = −

R³

ψ_i(ξ· ∇xG) dξ, i=1,2,3, ρ₁

2|u|²+E

t+ ³

j=1

u_j ρ₁

2|u|²+E +p

x_j −m· ∇x

= −

R³

ψ₄(ξ· ∇xG) dξ.

(1.10)

Herepis the pressure for monatomic gases:

p=2

3ρE=Rρθ.

Moreover, the equation for the microscopic components of G is obtained by apply- ing the microscopic projection P1to (1.9):

Gt+P1(ξ· ∇xG+ξ· ∇xM)+ ∇x· ∇ξG

=L_MG+Q(G,G), (1.11)

i.e.

G = L⁻_M¹(P1(ξ· ∇xM)) +L⁻_M¹

Gt+P1(ξ· ∇xG)+ ∇x· ∇ξG−Q(G,G)

:=L⁻_M¹(P1(ξ· ∇xM))+, (1.12)

where

L_Mg=L_[ρ,u,θ]g≡Q (M+g,M+g)−Q(g, g), is the usual linearized collision operator.

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By plugging (1.12) into (1.10), we now have another representation of the Vlasov–Poisson–Boltzmann system which contains a macroscopic system

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

ρ_t+divxm=0, mit+ ³

j=1

uimj

xj+px_i−ρx_i

= −

R³

ψ_i

ξ· ∇xL⁻_M¹(P1(ξ· ∇xM))

dξ

−

R³

ψ_i(ξ· ∇x) dξ, i=1,2,3,

ρ 1

2|u|²+E

t

+ 3 j=1

uj

ρ

1

2|u|²+E

+p

xj

−m· ∇x

= −

R³

ψ₄

ξ · ∇xL⁻_M¹(P1(ξ· ∇xM))

dξ

−

R³

ψ₄(ξ· ∇x) dξ,

(1.13)

the equation (1.11) for the microscopic component G, and the Poisson equation (1.2)2for the electric potential. Note that if all the terms containingare dropped, then the above system reduces to the compressible Navier–Stokes–Poisson equations. Later in this paper, we will work on this reformulated system by applying the techniques used in the study of conservation laws together with the dissipative effect in the Boltzmann equation.

For preparation, we now recall some properties of the linearized collision oper- atorL_M. By definition,L_Mis self-adjoint with respect to the inner product h, gM, i.e.

h, LMgM= LMh, gM,

and the null spaceN ofLMis spanned by the macroscopic variables:

χ_j, j =0,· · ·,4.

For the hard-sphere model,LMtakes the form, cf. [3, 5, 9],

(LMh) (ξ )= −ν(ξ;ρ, u, θ )h(ξ )+

M(ξ )KM

h

√M

(ξ )

. (1.14)

HereKM(·)= −K1M(·)+K2M(·)is a symmetric compactL²operator, and the collision frequencyν(ξ;ρ, u, θ )andKiM(·)have the following expressions

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⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

ν(ξ;ρ, u, θ )= 2ρ

√2π Rθ

Rθ

|ξ−u|+ |ξ−u| ^|^ξ⁻^u^|

0

exp

− y² 2Rθ

dy +Rθexp

−|ξ −u|² 2Rθ

, k1M(ξ, ξ_∗)= πρ

(2π Rθ )³|ξ−ξ_∗|exp

−|ξ −u|²

4Rθ −|ξ_∗−u|² 4Rθ

, k_2M(ξ, ξ_∗)= 2ρ

√2π Rθ|ξ −ξ_∗|⁻¹exp

−|ξ −ξ_∗|²

8Rθ −(|ξ|²− |ξ_∗|²)² 8Rθ|ξ −ξ_∗|²

, wherekiM(ξ, ξ_∗)(i = 1,2)is the kernel of the operatorKiM(i = 1,2)respectively, and ν(ξ;ρ, u, θ ) ∼ (1+ |ξ|)as |ξ| → +∞. Furthermore, there exists σ₀(ρ, u, θ ) >0 such that for any functionh(ξ )∈N^⊥

h, L_MhM−σ₀(ρ, u, θ )h, hM, which implies, cf. [9]

h, L_MhM−σ (ρ, u, θ )ν(ξ )h, hM, (1.15) with some constantσ (ρ, u, θ ) >0.

For later use, notice that the projections P0and P1have the following properties:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

P0(ψ_jM)=ψ_jM, P1(ψ_jM)=0, j=0,1,2,3,4, L_MP1=P1L_M=L_M, P1(Q(h, h))=Q(h, h), LMP0=P0LM=0, P0(Q(h, h))=0,

ψ_jM, hM= ψ_jM,P0hM, j=0,1,2,3,4, h, LMgM= P1h, LM(P1g)M,

h, L⁻_M¹(P1g)

M=

L⁻_M¹(P1h),P1g

M=

P1h, L⁻_M¹(P1g)

M.

For a fixed temperatureθ >0, we will study the existence of classical solutions for (1.2) near the global Maxwellian

M=M_ρ,0,θ= ρ

2π Rθ3exp

−|ξ|² 2Rθ

.

As in [16], two sets of energy estimates are used, i.e. the energy estimates with respect to the local Maxwellian M[ρ,u,θ](t, x, ξ )and a suitably chosen global Max- wellian M₋ = M[ρ₋,0,θ₋](ξ ). For this, a variation of the microscopic H-theorem is needed in order to relate the dissipation estimates with different weights as in Lemma 4.2 of [16]. That is, there exists a positive constantη₀=η₀ (ρ, u, θ; ˜ρ,u.˜θ ) >˜ 0, which is not necessary small, such that if ^θ₂ <θ < θ˜ and

|u− ˜u| + |θ− ˜θ|< η0,the following microscopicH-theorem

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−

R³

GLMG M˜ dξ σ

R³

ν(ξ )G²

M˜ dξ, (1.16)

holds for some positive constantσ =σ (ρ, u, θ; ˜ρ,u,˜ θ ) >˜ 0 withM˜ =M_[_ρ,_˜_u,_˜_θ]_˜. Throughout this paper, we choose positive constantsρ₋andθ₋such that

⎧⎪

⎪⎨

⎪⎪

⎩

ρ₋=ρ,

θ

2 < θ₋< θ , θ₋−θ< η₀.

(1.17)

It is easy to see that if M(t, x, ξ )is a small perturbation of M(ξ ), (1.16) holds for such chosenρ₋andθ₋whenM˜ ≡M₋=M[ρ₋,0,θ₋].

The following are the spaces for the solution considered in this paper.

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ H^N_t,x,ξ

R³×R³

=

⎧⎨

⎩g(t, x, ξ )

∂_t^γ⁰∂_x^α∂_ξ^βg(t,xξ )

√M₋(ξ ) ∈C

[ 0,∞), L²_x,ξ

R³×R³

|γ0| + |α| + |β|N

⎫⎬

⎭,

H^N =

⎧⎨

⎩g(t, x, ξ )

∂_t^γ⁰∂_x^α∂_ξ^βg(t,x,ξ )

√M₋(ξ ) ∈C

[ 0,∞), L²_x,ξ

R³×R³ ,

|γ0|1, |γ0| + |α| + |β|N

⎫⎬

⎭.

Hereg(t, x, ξ )=f (t, x, ξ )−M(ξ ).

The main result in this paper can be stated as follows.

Theorem 1. Assume thatf₀(x, ξ ) 0 andN 4. A sufficiently small constant ε >0 and a sufficiently large constantλ0exist such that if

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

λ > λ0, λ0ε <1, E(f0)= ∇x⁻_x¹(ρ0(x)−ρ)

L²_x(R³)

+

|α|+|β|N

^∂^x^α^∂^ξ^β^(f^√⁰M^{(x,ξ )}₋(ξ )⁻^{M(ξ ))}

L²_x,ξ(^R³×R³)

ε,

(1.18)

then there exists a unique global classical solution f (t, x, ξ ) to the Vlasov–

Poisson–Boltzmann system (1.2) which satisfiesf (t, x, ξ ) 0 and which is uni- formly bounded in H^N. Furthermore,

tlim→∞ sup

x∈R³

|α|N−4

R³

∂_x^α

f (t, x, ξ )−M(ξ )²

M₋(ξ ) dξ =0. (1.19)

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Remark 1. Note that in the space H^N, the order of the differentiation onf (t, x, ξ ) with respect to time is at most one. In general, the solutions may not be uniformly bounded in the space H^N_t,x,ξ

R³×R³

. However, for any fixedT >0, there exists a positive constantC(T ) >0 such that

sup

0tT

γ₀+|α|+|β|N

R³

∂_t^γ⁰∂_x^α∂_ξ^β

f (t, x, ξ )−M(ξ )²

M₋(ξ ) dξ dx

C(T ). (1.20)

Here, we relate and contrast our work with earlier contributions by other authors.

For the Vlasov–Poisson–Boltzmann system, the behavior of the solutions to the lin- earized system around a global Maxwellian M was studied in [6, 7]. When the initial data is a small perturbation of vacuum, a global existence result in the whole space was given in [12] for a class of “soft” potentials. The analysis there relies on the time decay estimate of the electric potential(t, x)which follows from theL¹_x

R³ estimate onρ(t, x)and the assumptions on the collision potentials. As pointed out in [10], this argument cannot be used for establishing existence of a global classical solution to the Vlasov–Poisson–Boltzmann system near a Maxwellian M, because it is very difficult, if not impossible, to obtain the desired L¹_x

R³

estimate on ρ(t, x)−ρ.

To construct global-in-time classical solutions around a global Maxwellian, a nonlinear energy method based on the decomposition with respect to the global Maxwellian was used in [10] for periodic data. For perturbations that preserve mass, momentum, and total energy, the existence of global smooth periodic solutions to the Vlasov–Poisson–Boltzmann system was obtained in the above reference. The analysis is based on a new estimate in the form

−

|α|N

_t

0

T³

R³

∂_x^α

f (τ, x, ξ )−M(ξ ) L_M

∂_x^α

f (τ, x, ξ )−M(ξ )

M(ξ ) dξ dxdτ

C

|α|N

_t

0

T³

R³

ν(ξ )∂_x^α

f (τ, x, ξ )−M(ξ )²

M(ξ ) dξ dxdτ. (1.21) Here T³ =[−π, π]³. In fact, for periodic solution, the zero-th order estimate of the solution in (1.21) follows directly from the Poisson equation and the Poincaré inequality which give

∇x(t, x)L²_x(T³) O(1)ρ(t, x)−ρL²_x(T³) O(1)∇xρ(t, x)L²_x(T³). This together with the conservation laws imply that both|∇x(t, x)|²and(ρ(x, t )− ρ)²are uniformly integrable over space–time. Notice that for the problem on the whole space, the above argument based on the Poincaré inequality is not valid.

In this paper, we apply the decomposition with respect to the local Maxwellian as in the case for the Boltzmann equation without forcing, which was studied in [15].

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To capture the dissipation on both the macroscopic and microscopic components of the solution, we use the macroscopic system (1.13) applying techniques from the theory of conservation laws. In this way, the behavior of the local Maxwellian is clear and the dissipative effects from the viscosity, heat conductivity and the one from the linearized collision operator on the microscopic component are clearly analyzed. Furthermore, this method would be helpful for studying the problem for the fluid dynamic limit, i.e. the behavior of the solutions when the Knudsen number tends to zero.

Besides the decomposition, note also that the a priori energy estimate is closed in the space H^N, not the usual space H_t,x,ξ^N

R³×R³

as for the Boltzmann equation without forcing. The derivatives on the macroscopic components with respect to time and space are equivalent for the Boltzmann equation without forcing because the time derivatives can be recovered from the space derivatives through the conservation laws. However, this may not be the case for the Boltzmann equation with forcing. In the momentum equation, the force may not be integrable with respect to space and time. In the system considered in this paper, there is noL²_t,x

R⁺×R³

estimate on ∇x(t, x)on the whole space. Therefore, even though the estimates on the space derivatives of the solution can be obtained through the dissipation induced by viscosity and heat conductivity, a similar estimate on the time derivatives may not be obtained. Fortunately, the estimate in the space H^N can be closed because we obtain a new space and time integrability estimate on|ρ(t, x)−ρ|²induced by the dissipative effect of the Poisson equation (1.2)2.

Finally, the energy estimates are worked out both with respect to the local Maxwellian M(t, x, ξ )and the global Maxwellian M₋(ξ )as in [16]. The energy estimate with respect to a global Maxwellian is used because some polynomials in ξ arise from the derivatives of the local Maxwellian, while the collision frequency ν(ξ )in (1.14) is only of order(1+ |ξ|)for the hard-sphere model. However, to obtain the higher order energy estimates on the macroscopic component M, there is no need to use the energy estimate with respect to M₋.

The rest of the paper is organized as follows. The microscopic and macroscopic versions of theH-theorems will be stated in Section 2. The main energy estimates are given for the case whenN =4 in Section 3. The case whenN > 4 can be discussed in a similar fashion. The proof of Theorem 1 will be given in Section 4, and the proofs of some technical lemmas stated in Section 3 are given in the Appendix.

Notation. Throughout the paper,O(1)andC denote generic positive constants independent ofλ, andε, C(·,·)denotes a positive constant depending on the quantities in the parenthesis, andμis a sufficiently small positive constant. Note that constants may vary from line to line.

Forγ = (γ0, α, β), we use∂^γ to denote the differential operator ∂_t^γ⁰∂_x^α∂_ξ^β. Hereγ0is a non-negative integer, andα=(α1, α2, α3)andβ =(β1, β2, β3)are multi-indices with length|α|and|β|, respectively.C_b^ameans

a b

.

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In the following energy estimates, we will use the following sets of indices for different cases.

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1=!

γ =(γ0, α,0): γ01, γ0+ |α|4"

, ^j₂=!

γ =(γ₀, α, β): γ₀j, γ₀+ |α| + |β|4"

, j =1,2,3,4, 3=!

γ =(γ0, α, β): γ ∈¹₂, |β|1"

, ₄=!

γ =(γ₀, α,0): γ₀1, γ0+ |α|3"

, ₅=!

γ =(0, α, β): |α|1, |β|1, |α| + |β|4"

, ₆=!

γ =(1, α, β): |β|1, |α| + |β|3"

, 7=!

γ =(γ0, α,0): γ01, 2γ0+ |α|4"

.

2. H-theorem

The celebratedH-theorem of the Boltzmann equation is based on the bilinear structure ofQ(f, f ), that is,

R³

Q(f, f )lnf dξ 0,

where the equality holds only whenf (t, x, ξ )is a Maxwellian.

Corresponding to the macroscopic and microscopic components, the H- theorem can be seen from two viewpoints. Firstly, it is apparent from (1.15) and (1.16) that dissipation occurs due to the effect of the linearized collision operator L_Macting on the microscopic components. Secondly, as apparent from the conservation laws (1.13), dissipation occurs due to the viscosity and heat conductivity at the macroscopic level.

In the following, we derive some inequalities regarding the nonlinear and linearized collision operatorsQ(f, f )andLM. The first lemma is from [8].

Lemma 1. There exists a positive constantC >0 such that

R³

ν(ξ )⁻¹Q(f, g)² M˜ dξ C

R³

ν(ξ )f² M˜ dξ·

R³

g² M˜ dξ +

R³

f² M˜ dξ·

R³

ν(ξ )g² M˜ dξ

, (2.1)

whereM is any Maxwellian such that the above integrals are well defined.˜ Based on Lemma 1, the following result was proved in [16].

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Lemma 2. If^θ₂ <θ < θ, then there exist two positive constants˜ σ =σ (ρ, u, θ; ˜ρ,

˜

u,θ )˜ and η0 = η0(ρ, u, θ; ˜ρ,u,˜ θ )˜ such that if |u− ˜u| + |θ − ˜θ| < η0 and h(ξ )∈N^⊥, we have

−

R³

hLMh M˜ dξ σ

R³

ν(ξ )h² M˜ dξ.

Here M≡M[ρ,u,θ](t, x, ξ )andM(t, x, ξ )˜ = ˜M_[_ρ,_˜_u,_˜_θ]_˜(t, x, ξ ).

As a direct consequence of Lemma 2 and the Cauchy inequality, we have the following corollary, cf. [16].

Corollary 1. Under the assumptions in Lemma 2, forh(ξ )∈N^⊥, we have

⎧⎪

⎪⎨

⎪⎪

⎩

R³

ν(ξ ) M

L⁻_M¹h²dξσ⁻²

R³

ν(ξ )⁻¹h²(ξ ) M dξ,

R³

ν(ξ ) M₋

L⁻_M¹h²dξσ⁻²

R³

ν(ξ )⁻¹h²(ξ ) M₋ dξ.

(2.2)

To construct the entropy–entropy flux pairs to the Vlasov–Poisson–Boltzmann system, we first derive the macroscopic version of theH-theorem similar to that for the Boltzmann equation without any external force [15]. We first set

−3 2ρS≡

R³

M ln Mdξ. (2.3)

As a result of direct calculation, we obtain

−3

2(ρS)_t −3

2divx(ρuS)+ ∇x

R³

(ξln M)Gdξ

=

R³

GP1(ξ· ∇xM)

M dξ, (2.4)

and

⎧⎪

⎪⎨

⎪⎪

⎩

S= −²₃lnρ+ln(2π Rθ )+1, p= ²₃ρθ=kρ⁵³exp(S), E=θ, R= ²₃.

(2.5)

Remark 2. When the macroscopic entropyS is defined as (2.3), the gas constant Ris normalized to be ²₃so that E=θ.

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A convex entropy–entropy flux pair(η, q)around the global Maxwellian M= M_[ρ,0,θ]can be given as follows, cf. [15]. We begin by writing the conservation laws (1.10) in the following form:

mt +divxn= −

⎛

⎜⎜

⎜⎝ 0

R³

ψ₁(ξ· ∇xG)dξ

R³

ψ2(ξ· ∇xG)dξ

R³

ψ₃(ξ· ∇xG)dξ

R³

ψ4(ξ· ∇xG)dξ

⎞

⎟⎟

⎟⎠ +

⎛

⎜⎜

⎜⎝ 0 ρx₁

ρ_x₂ ρ_x₃ m· ∇x

⎞

⎟⎟

⎟⎠ .

Here,

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

m=(m0, m1, m2, m3, m4)^t =

ρ, ρu1, ρu2, ρu3, ρ₁

2|u|²+θt

, n=(n1,n2,n3) ,

nj =(n^j₀, n^j₁, n^j₂, n^j₃, n^j₄)^t

=

ρu_j, u₁m_j+²₃ρθ, u₂m_j +²₃ρθ, u₃m_j+²₃ρθ, ρu_j 1

2|u|²+⁵₃θ t

, j =1,2,3.

The entropy–entropy flux pair(η, q)can then be defined by η=θ!

−³₂ρS+³₂ρS+³₂∇m(ρS)|m=m(m−m)"

, qj =θ!

−³₂ρujS+³₂∇m(ρS)|m=m(nj−nj)"

, j =1,2,3. (2.6) Since

⎧⎪

⎪⎨

⎪⎪

⎩

(ρS)m₀ =S+^|^u_2θ^|² −⁵₃, (ρS)_m_i = −^u_θⁱ, i=1,2,3, (ρS)m4 =_θ¹,

we have

η= ³₂)

ρθ−θ ρS+ρ

* S−⁵₃

θ+^|^u₂^|²+ +²₃ρθ

, , q_j =u_jη+u_j

ρθ−ρθ

, j =1,2,3.

(2.7) Note that for m in any closed bounded regionD⊂ = {m:ρ >0, θ >0}, there exists a positive constantCthat depends onD, such that the entropy–entropy flux pair in (2.7) satisfies, cf. [15, 16],

C⁻¹|m−m|²ηC|m−m|², (2.8)

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and(η, q₁, q₂, q₃)solves the following equation η_t+divxq = −∇x

R³

ξG ln M+3 2ψ₄ξG

dξ

+3

2m· ∇x +

R³

P1(ξ· ∇xM)G

M dξ. (2.9)

Integrating (2.9) with respect toxover R³gives d

dt

R³

η(t )dx = 3 2

R³

m· ∇xdx +

R³

P1(ξ· ∇xM)G

M dξ dx. (2.10)

Since

R³

m· ∇xdx= −

R³

divxmdx=

R³

(ρ−ρ)_tdx

=λ

R³

_tdx= −λ 2

d dt

R³

|∇x|²dx, (2.11) we obtain the entropy estimate

d dt

R³

η+3λ

4 |∇x|²

dx

=

R³

P1(ξ· ∇xM)G

M dξ dx, (2.12) which will be crucial in the later analysis of the macroscopic components of the solutions.

3. Energy estimates

In this section, we will give the entropy estimates used in the proof of the global existence of solutions. For this, we first assume the following a priori estimate,

N (t )²= sup

0τt

⎧⎪

⎨

⎪⎩

γ∈1

R³

|∂^γ

ρ−ρ, u, θ−θ

(τ, x)|²+λ|∇x∂^γ(τ, x)|² dx

+

γ∈¹₂

R³

|∂^γG(τ, x, ξ )|² M₋(ξ ) dξ dx

⎫⎪

⎬

⎪⎭

+

γ∈¹₂

_t

0

R³

ν(ξ )|∂^γG(τ, x, ξ )|²

M₋(ξ ) dξ dxdτ

δ₀². (3.1)

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Hereδ₀>0 is a sufficiently small constant such thatλδ₀<1.

First, from the Poisson equation (1.2)2and the conservation laws (1.10), we have

N (0)O(1)E(f0), (3.2)

and⎧

⎪⎪

⎨

⎪⎪

⎩

R³

∇x∂_x^αt(t, x)²dx _λ⁴2

R³

∂_x^αm(t, x)²dx, |α|4,

R³

∇x∂_x^α(t, x)²dx _λ⁴2

|α|=|α|−1

R³

∂_x^α(ρ(t, x)−ρ)²dx, 1|α|5.

(3.3)

Sobolev’s inequality, (3.1) and (3.3) imply that

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

(ρ(t, x)−ρ, u(t, x), θ (t, x)−θ )

+

|α|1

|∂_x^α∂_t(ρ, u, θ )| + |∇x∂_x^α(ρ, u, θ )|

(t, x)O(1)δ0,

|α|3

∇x∂_x^α(t, x)+

|α|2

∇x∂_x^α_t(t, x)O(1)^δ_λ⁰ O(1)δ0,

R³

⎧⎨

⎩ 1 M₋

⎛

⎝|G|²+

|α|1

|∇x∂_x^αG|²+ |∂_x^αGt|²⎞

⎠

⎫⎬

⎭(t, x, ξ )dξ O(1)δ₀². (3.4)

TheL²estimates applied to the collision operatorsQ

∂^γG, ∂^γG

andQ(∂^γM,

∂^γG

with respect to M and M₋are given in the following lemma.

Lemma 3. Under assumption (3.1), for eachγ ∈1, γ∈1,|γ| + |γ|4, we have the following estimates with respect to the weight M,

t 0

R³

ν(ξ )⁻¹Q

∂^γG, ∂^γG²

M dξ dxdτ

O(1)δ₀² _t

0

R³

ν(ξ )

|∂^γG|²+ |∂^γG|²

M dξ dxdτ. (3.5)

In the case|γ|>0, in which a stronger estimate can be obtained _t

0

R³

ν(ξ )⁻¹Q

∂^γM, ∂^γG²

M dξ dxdτ

O(1)δ₀² t

0

R³

ν(ξ )|∂^γG|²

M dξ dxdτ

+O(1)δ₀²

|α||γ|−1

_t

0

R³

∇x∂_x^α(ρ, u, θ )²dxdτ. (3.6)

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The corresponding estimates with respect to the weight M₋ are different when

|γ|3 and are given by _t

0

R³

ν(ξ )⁻¹Q

∂^γG, ∂^γG²

M₋ dξ dxdτ

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ O(1)δ₀²

t 0

R³

ν(ξ )

|∂^γG|²+ |∂^γG|²

M₋ dξ dxdτ,

if max{|γ|,|γ|}2,

O(1)δ₀² _t

0

R³

ν(ξ ) M₋

⎛

⎝

|α|3

|∂_x^α∂^γG|²+ |∂^γG|²

⎞

⎠dξ dxdτ,

if max{|γ|,|γ|} = |γ|3,

(3.7)

and _t

0

R³

ν(ξ )⁻¹Q

∂^γM, ∂^γG²

M₋ dξ dxdτ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ O(1)δ²₀

t 0

R³

ν(ξ )|∂^γG|²

M₋ dξ dxdτ, if 0<|γ|2,

O(1)δ²₀

|α|3

_t

0

R³

ν(ξ )|∂_x^α∂^γG|²

M₋ dξ dxdτ, +O(1)δ²₀

|α||γ|−1

_t

0

R³

∇x∂_x^α(ρ, u, θ )²dxdτ, if |γ|3.

(3.8)

Proof. We only prove (3.5) and (3.8) because the proof for the other estimates is similar. Firstly, (3.4) together with the fact that ^θ₂ < θ₋< θ imply

γ∈¹₂,|γ|2

R³

ν(ξ )|∂^γG|²

M dξ

γ∈¹₂,|γ|2

R³

|∂^γG|²

M₋ dξ O(1)δ₀².