Heat Transfer Problem for the Boltzmann Equation in a Channel with Diffusive Boundary Condition?

Renjun DUAN Shuangqian LIU Tong YANG Zhu ZHANG

Abstract In this paper,the authors study the 1D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperatures θ0and θ1.This problem was studied by Esposito-Lebowitz-Marra(1994,1995)where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition.However,a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary.In the regime where the Knudsen number is reasonably small,the slip phenomenon is significant near the boundary.Thus,they revisit this problem by taking into account the slip boundary conditions.Following the lines of[Coron,F.,Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation,J.Stat.Phys.,54(3–4),1989,829–857],the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points.Then they will establish a uniform L∞estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.

Keywords Boltzmann equation,Compressible Navier-Stokes approximation,Slip boundary conditions,Chapman-Enskog expansion

1 Introduction

1.1 Problem settings

In this paper,we study the steady flow of a rarefied gas in a channel which is bounded by two thermal walls located at x=0 and x=1.The walls are assumed to have a vanishing velocity and given temperatures θ0and θ1(θ0/= θ1),respectively.In the kinetic setting,the distribution function satisfies the following 1D rescaled steady Boltzmann equation:

The parameter ε>0 is the Knudsen number which is proportional to the mean free path and is assumed to be small.The Boltzmann collision term on the right-hand side of(1.1)takes the non-symmetric bilinear form of

1.2 CNS approximation

We are interested in the behavior of solution Fεin the limit ε→ 0+that is the hydrodynamic limit of the Boltzmann equation.In the absence of physical boundaries or shocks,it is well-known that the distribution function converges to a local Maxwellian with parameters satisfying the compressible Euler system,cf.[19].The Chapman-Enskog expansion yields the compressible Navier-Stokes system(CNS for short)as the first order correction.In this subsection,we give a formal derivation of CNS approximation in the setting of this paper.Before this,we define some function spaces which will be used later.Given a local Maxwellian

1.3 Slip boundary condition

In order to solve compressible Navier-Stokes system(1.10)when x∈(0,1),suitable boundary conditions are needed.If we consider the no-slip boundary condition

the approximation(1.6)matches the boundary conditions(1.3)up to O(1).However,since G contains non-Maxwellian terms,the Chapman-Enskog approximation M+εG in general does not match the boundary condition(1.3)up to O(ε),except for the case when

However,then(1.10)is overdetermined.To obtain a more accurate approximation,Coron[9]formally derived the slip boundary conditions for compressible Navier-Stokes equations,which are essentially a consequence of the analysis of the Knudsen layer.In what follows,we elaborate the derivation only in one dimensional case.We refer to[1,33—35]for the physical investigations in general cases.

As in[9],since Chapman-Enskog expansion is not valid near the boundary,we introduce Knudsen layers B0and B1around boundary points x=0 and x=1,respectively.The construction of Knudsen layers relies on the solutions to the following Milne problem:

where G is a given incoming distribution function.The well-posedness of(1.12)has been shown in[3],and is summarized in Lemma 6.1 for later use.

Now we construct the Knudsen layer B0and B1at the boundary points x=0 and x=1,respectively.Let p∈{0,1}be a boundary point.We set the boundary conditions of Navier-Stokes system as

1.4 Main result

The paper aims to justify rigorously the slip boundary conditions presented in the previous section.For this,we start with the following expansion

where α>0 is a positive constant.Here we elaborate the approximate solutions appearing in the expansion:The leading order term M=M[ρ,u,θ]is a local Maxwellian where[ρ,u,θ]satisfies the steady compressible Navier-Stokes equations with slip boundary conditions(1.18)and(1.21).It will be constructed in Subsection 3.1.The function G is a corrector at order ε which is defined in(1.11)and it satisfies(1.7).B0and B1are Knudsen layers which are defined in(1.16)and(1.19),respectively.For technical reasons,we need a high-order corrector F2which will be defined in(3.12).

Define the weight function

Theorem 1.1 Suppose|θ1? θ0|≤ δ0for small δ0.For sufficiently small ε>0 and any,there exists a unique solution Fεin the form of(1.22)to the steady Boltzmann equation(1.1)with boundary condition(1.3)and total mass condition(1.4).Moreover,there exists constant p=p(α) ∈ (2,∞),such that the remainder term FRsatisfies the following uniform-in-ε estimate:

Here the constant Cα>0 is uniform in ε.

Remark 1.1 Esposito et al.in[13—14]studied the hydrodynamic limit of(1.1)with(1.4),in the presence of a small external force.They proved that the solution converges to the steady CNS with no-slip boundary condition.In this paper,we aim to justify the more accurate CNS approximation by taking account into the slip boundary conditions.Thanks to this choice,we can avoid the higher order expansions used in[13—14].

The hydrodynamic limit is one of the most fundamental problems in kinetic theory.There are extensive studies on the mathematical description of relations between Boltzmann equation and various of hydrodynamic models.Now we review some of them which are most related to the topic of this paper.For more detailed references,we refer to the book by Cercignani[8]and the survey book by Saint-Raymond[32].

Let us first focus on the Euler scaling.The first mathematical proof of the compressible Euler limit was given by Nishida[31]in the analytic framework.An extension of this result has been made in[36]for the case when the solution contains initial layers.By using a truncated Hilbert expansion,Caflisch[7]justified the Euler limit for any given smooth Euler solutions;see also[24]for the result in L2-L∞framework.In the same spirit as[7],Lachowicz[28]justified the CNS approximation over the short time interval.Recently,the global-in-time CNS approximation was justified by the second and third authors in a paper with Zhao[29]for the case when the data are close to the global equilibrium.This result was extended to case of a general bounded domain in[10].On the other hand,the hydrodynamic limit to the compressible Navier-Stokes equations for the steady Boltzmann equation in a slab was studied by Esposito-Lebowitz-Marra[13—14];see also a recent survey[15].We also refer to[25,38—39]for hydrodynamic limits to some wave patterns.Very recently,the compressible Euler limit in the half-space was studied in[22]with the specular reflection boundary condition.

In diffusive scaling,there are many interesting results on the hydrodynamic limits to the incompressible fluid systems in different settings,cf.[2,5—6,12,18,20,26—27,37]and the references therein.

The rest of the paper is organized as follows.In Section 2 we will present some basic estimates on linear and nonlinear collision terms.In Section 3,the construction of approximate solutions is given.Precisely,in Subsection 3.1,we solve the steady Navier-Stokes equation with slip boundary conditions.Some properties of Knudsen layer B0and B1are given in Subsection 3.2.We construct the higher order corrector F2and give some error bounds in Subsection 3.3.In Section 4,we will study the linearized steady Boltzmann equation.In Section 5,we further construct the remainder FRand give the proof of Theorem 1.1.In Appendix,we summarize some properties of the solution to the Milne problem.

2 Estimates on Collision Operators

3 Approximate Solutions

3.1 Steady Navier-Stokes equations

In this subsection,we construct the solution to the steady Navier-Stokes equations(1.10)with slip boundary conditions(1.18)and(1.21).By(1.10)1and boundary condition u1(0)=u1(1)=0,we have u1≡0.Then by(1.10)3and boundary conditions(1.18),(1.21)for u2,u3,we have u2,u3≡0.Thus,the original problem(1.10),(1.18)and(1.21)is reduced to

which does not vanish for any ε∈ (0,ε0)with small ε0.Then by the implicit function theorem,there is a unique solution[D1,D2,P0]of(3.5)for any ε ∈ (0,ε0).The estimate(3.3)follows from the explicit formula(3.4).The proof of Lemma 3.1 is completed.

Remark 3.1 The boundary conditions(3.1)3mean that there is a temperature gap which is proportional to the normal derivatives of temperature,between fluid layer and the boundary.The proportional coefficient is of the same order as the scale of Knudsen layer.

Remark 3.2 Since the pressure P0= ρNSθNSis a positive constant,[ρNS,0,θNS]is also a solution to steady Euler equations.

3.2 Knudsen layers

In this subsection,we summarize some properties of Knudsen layers B0and B1which are defined in(1.16)and(1.19),respectively.

where Ψ0is given by(1.17).Moreover,for any ? ∈ (0,)and β >3,there exist positive constants C>0 and σ0>0,such that

Proof From the ansatz in Subsection 1.3,it is direct to check that B0satisfies(3.6).The estimate(3.7)follows from the explicit formula(1.16),(3.3)and(6.1)in Lemma 6.1.We omit the details for brevity.

Similarly,for B1,we have the following lemma.

3.3 Error terms

4 Linear Problem

4.1 L2-estimate

4.2 Lp-estimates on PMFR

4.3 Weighted L∞estimate

5 Justification of the Expansion

In this section,we will solve the remainder system(3.9)with boundary condition(3.13)and then give the proof of Theorem 1.1.

6 Appendix

The following lemma summarizes the well-posedness of Milne problem in L∞space that was proved in[3,37].

Acknowledgement The authors would like to thank Professor Kazuo Aoki for introducing the problem as well as pointing out Coron’s paper[9]in 2018.