CONVERGENCE FROM THE TWO-SPECIESVLASOV-POISSON-BOLTZMANN SYSTEM TO THE TWO-FLUID INCOMPRESSIBLE NAVIER-STOKES-FOURIER-POISSON SYSTEM WITH OHM'S LAW*

describes the evolution of a gas of two species of oppositely charged and same mass particles(cations of q > 0 and mass m > 0, and anions of charge ?q < 0 and m > 0) under the influence of the interactions amongst themselves through collisions and their self-consistent electrostatic field.The first equation in (1.1) is the equation of Vlasov-Boltzmann for cations,the second is that of Vlasov-Boltzmann for anions,and the last is Gauss’law.Here,the particle number densities f±(t,x,v) ≥0 represent the distributions of the positively charged ions (i.e.cations)and the negatively charged ions(i.e.,anions),respectively,at time t ≥0,with position x = (x1,x2,x3) ∈T3and velocity v = (v1,v2,v3) ∈R3.The physical constant ?0> 0 is the vacuum permittivity (or electric constant) and ?xφ describes the electric field.The collision operators B(f+,f?) and B(f?,f+) have been added to the right-hand sides of the Vlasov-Boltzmann equations in (1.1) in order to account for the variations in the densities f+≥0 and f?≥0 that occur due to interspecies collisions.The self-consistent electric potential φ = φ(t,x) ∈R is coupled with the function f+?f?through the Poisson equation.The bilinear function B with hard-sphere interaction is defined by

and take the dimensionless numbers Kn,St and Ma all as ε in order to obtain the fast relaxation limit.After nondimensionalization,the scaled two-species VPB system(see Section 2.1,Section 2.2 and Section 2.4.7 of Chapter 2 in [2]) is in the form

where α>0 measures the electric repulsion according to Gauss’law, and δ >0 represents the strength of interactions.The size of the bounded parameter δ will be compared to the Knudsen number Kn=ε and divided into three cases:

? δ ～1, with strong interspecies interactions;

? δ =o(1) andδεis unbounded, with weak interspecies interactions;

? δ =O(ε), with very weak interspecies interactions.

In this paper, we consider the strong interspecies interactions, which is the most singular case.We also suppose that α = ε (as in [2]).Therefore, we have the scaled two-species VPB system

1.2 Notation and Main Results

In this paper, the symbol C(·) denotes constants which depend on certain parameters.In addition, A ?B means that there exists a constant C > 0 such that A ≤CB, and A ～B means that there exist two positive numbers C1and C2> 0, such that C1A ≤B ≤C2A.The Lpspaces are denoted by the name of the variable concerned, namely,

In this paper,we prove two main theorems.The first theorem gives a global-in-time solution(fε,gε,φε) for the two-species Vlasov-Poisson-Boltzmann system for any given the Knudsen number ε ∈(0,1]near a global equilibrium.The second theorem is the two-fluid incompressible Navier-Stoker-Fourier-Poisson system limit ε →0 taking in the solutions fε,gεof the VPB system (1.4)–(1.5), which is constructed in the following theorem.

The next theorem is about the limit to the two-fluid incompressible NSFP system with Ohm’s law.This system is a macroscopic description of a fluid based on fluctuations of mass density ρ(t,x), bulk velocity u(t,x), temperature θ(t,x), electric charge n(t,x), electric current j(t,x) and internal electric energy w(t,x) in electric filed ?xφ(t,x):

1.3 Difficulties and Ideas

Furthermore, aε, bε, cεand dεfollow the local macroscopic balance laws of mass, moment and energy from fε, and balance law of mass from gε.In fact, we multiply the first fε,gεequation (1.4)of the VPB system by the collision invariant in (1.9)and integrate by parts over v ∈R3to get that

Finally, we apply the limit from the perturbed VPB system (1.4) to the incompressible NSFP system with Ohm’s law (1.14) as ε →0, based on the global-in-time energy estimate uniformly in ε ∈(0,1].Then, we apply the Aubin-Lions-Simon Theorem to obtain enough compactness such that the limits is valid.

1.4 Historical Progress in This Field

There has been tremendous progress on the well-posedness of kinetic equations.DiPerna and Lions [11] obtained the global renormalized solutions to the Boltzmann equation for large initial data.Later, Lions applied this theory to the VPB system ([30]).For the classical solutions, Ukai [35] first considered the hard potential collision kernels.Guo developed a nonlinear energy method to prove the existence of global-in-time classical solutions to the Boltzmann equations near equilibrium[19,20].Later,there were more results on different collision kernels;we only list some of these results [12, 13, 36, 37].

One of the most important features of kinetic equations (like the Boltzmann-type equation) is that they are connected to fluid equations when the Knudsen number ε is very small.Hydrodynamic limits of kinetic equations have been an active research field since the late 1970.

In the context of classical solutions,many results can be obtained by the Hilbert expansion.In [9, 33], Nishida and Caflisch used the Hilbert expansion on the compressible Euler limit.Combining the nonlinear energy method and the Hilbert expansion, Guo-Jang-Jiang justified the acoustic limit[23,24].Furthermore,Guo proved the incompressible Navier-Stokes limit[21].All of these results were based on the Hilbert expansion.In the other direction, Bardos and Ukai [5] proved the convergence for small data classical solutions from the Boltzmann equation with a hard potential to incompressible Navier-Stokes equations by using the semigroup method and the spectrum analysis method.For general collision kernels, Briant, Jiang-Xu-Zhao and Gallagher-Tristani also recently proved this incompressible Navier-Stokes limits [7, 8, 15, 26].

The purpose of the BGL program(named after the Bardos-Golse-Levermore’s work [3, 4])is to justify the weak limit from the DiPerna-Lions’renormalized solutions of the Boltzmann equations to the weak solutions of the incompressible Navier-Stokes.This program was completed by Golse and Saint-Raymond with a cutoffMaxwell collision kernel in [16].Later, this convergence result was extended to soft potentials cases,non-cutoffcases and bounded domain cases, etc.[1, 27, 31].

For the VPB system, Guo and Jang proved the limit from the scaled VPB system applied to the compressible Euler-Poisson system with hard-sphere interaction by the Hilbert expansion[22].Recently,Jiang and Zhang considered the sensitivity analysis method and energy estimates to justify the incompressible Navier-Stokes-Poisson limit and obtain the precise convergence rate for the first time [28].In [18], the authors proved the limit of applying the one-species VPB system to the incompressible Navier-Stokes-Fourier-Poisson system by using the nonlinear energy method.

2 Construction of Local Solutions

In this section, we will prove,by employing an iterative schedule, that the perturbed VPB system (1.4)–(1.5) has a unique local-in-time solution for all 0 < ε ≤1.Before doing this, we will first do some preparatory work.

2.1 Some Lemmas

In order to derive the equations of n,j and w, we introduce two functions in L2’s range as follows:

Thus, there are inverse ?Φ ∈L2(Mdv) and ?Ψ ∈L2(Mdv) such that

Proof The proof can be justified by arguments similar to those in Propositition 6.5 of[17] and Section 2.4.5 of [34].?

Then,we consider of the linearized collision operators Li(i=1,2)defined in(1.6).Ligives us the dissipative structure of the kinetic equation thanks to the coercivity of Li(see Lemma 3.3 in [21] for more details).

Lemma 2.2 For any f ∈L2(Mdv), there exists δ >0 such that

for all multi-indexes β ∈N3, where Pi(i=1,2) are defined as in (1.10).

For the bilinear symmetric operator Q defined in (1.6), we have the following estimates(relevant proofs can be found in Lemma 3.3 of [21]):

Lemma 2.3 Let gi(x,v) (i=1,2,3) be smooth functions.Then we have that

where we use the fact that P2gε= dε(t,x), (1.24) and the integral by parts.This completes the proof of (2.7) by the Poincar′e inequality in T3.?

2.2 Local-in-Time Solution

In this subsection, we will prove that the perturbed VPB System (1.4) for all 0 < ε ≤1 has a unique local-in-time solution under small initial data.The proof is divided into three steps.The first step is to construct the approximation equation.From [26], we know that the linear approximate system has a solution for the fixed ε ∈(0,1].The second step is to obtain the energy estimate of the uniform bound of ε of the approximate system.By compactness analysis,the third step is to obtain that the perturbed VPB system has a local-in-time solution under small initial data.To simplify the estimate, we introduce a new dissipative term:

ProofFor any fixed ε ∈(0,1],we consider the linear iterative approximate system(2.10)with initial data (2.11)

where Lemma 2.2 is used.Now we estimate the terms Si(1 ≤i ≤6) in (2.22).We divide the term S1into three parts, as follows:

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3 Uniform Estimates with ε and Global-in-Time Solutions

In this section, we will extend the local solution in Lemma 2.6 to a global-in-time solution of (1.4)–(1.5) by deriving energy estimates uniformly in ε ∈(0,1] under small initial data.For simplicity, we will drop the lower index ε of fε,gεand φεin the perturbed VPB system (1.4)so that

3.1 Pure Spatial Derivative Estimates: Kinetic Dissipations

In this subsection, we will consider the energy estimates on the pure spatial derivative of G.We will prove the following lemma:

Consequently, we have the bound (3.2) by plugging all estimates (3.7), (3.8), (3.11) and(3.12) into (3.3) and summing up for all |α|≤N.This completes the proof of Lemma 3.1.?

3.2 Macroscopic Energy Estimates: Fluid Dissipations

where all terms on the right-hand side are the coefficients of l, h and m, defined as in (1.23),in terms of the basis (3.13).

As for g, we have that

where ?l,?h and ?m are defined as in (1.22).

Our goal is to find a macroscopic dissipation.The high order derivatives of the fluid coefficients (a(t,x),b(t,x),c(t,x)) and d(t,x) are dissipative from the balance laws (1.24) and(3.14)–(3.18).This is similar to the case of the Boltzmann equation.More specifically, we give the following lemma:

3.3 Energy Estimates for (x,v)-Mixed Derivatives

In this subsection, we will derive a closed energy estimate.Because of the above two subsections, we only need to estimate the energy of (x,v)-mixed derivatives of the kinetic part(I ?P)G.We give the following lemma:

We also introduce the instant energy dissipative rate functional

where we use Lemma 2.4 and (1.8).Furthermore, we estimate W25,W26by using arguments similar to those above to get that

for all |α|+|β|≤N with β /=0 and for all 0<ε ≤1.

Let η > 0 be a sufficiently small number to be given later.Plug (3.2) in Lemma 3.1 and η times of (3.23) in Remark 3.3 into the η2times of the above inequality (3.59).Then there exists a small positive number η0>0, independent of ε, such that, for all 0<η ≤η0,|β|, so we can apply an induction over |β| = k, which ranges between 0 and N, to obtain the inequality (3.49).For simplicity, we omit the details of the induction, and the proof of Lemma 3.4 is complete.?

3.4 Global Classical Solutions: Proof of Theorem 1.1

From the differential inequality(3.53)in Remark 3.5 and the energy bound(2.9)in Lemma 2.6, it is easy to deduce that, for any [t1,t2]?[0,T] and 0<ε ≤1,

Therefore,the continuity of EN,η1(Gε,φε)(t)and the definition of T?imply that T?=+∞.In other words, we can extend the local-in-time solution (g+ε(t,x,v),g?ε(t,x,v),φε(t,x)) constructed in Lemma 2.6 to a global-in-time solution.Moreover,the uniform energy bound(1.13)can be derived from(3.52)and(3.62).The proof of Theorem 1.1 is complete.

4 Limit to Two-Fluid Incompressible NSFP Equations with Ohm’s Law

In this section, our goal is to derive the two-fluid incompressible NSFP system with Ohm’s law (1.14) from the perturbed VPB system (1.4)–(1.5) as ε →0, based on the uniform global energy bound (1.13) in Theorem 1.1.

4.1 Limits from the Global Energy Estimate

In order to reduce the equations regrading nε,jεand wε, we take the inner products with the second g-equation of the perturbed VPB equation (1.4) in L2vby 1, ?Φ(v) and ?Ψ(v), which are defined in (2.2), to get that

4.2 Convergences to Limit Equations

In this subsection,our goal is to deduce the two-fluid incompressible NSFP equations(1.14)with Ohm’s law from the local conservation laws (4.7) and (4.8) and the convergence obtained in the previous subsection.

4.2.1 Incompressibility and Boussinesq Relation

The first equation of (4.7) and the uniform bound (4.10) yields that

in the sense of distribution as ε →0.In summary, the limits (4.44) and (4.45) show the convergence (4.43).The limits (4.41), (4.42) and (4.43) imply that

in the sense of distribution as ε →0, where we make use of the bound (4.1), (4.2) and Lemma 2.1.As a result, we have that