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

Chinese Annals of Mathematics,Series B 2022年第6期43卷 1071-1100页

作者： Renjun DUAN Shuangqian LIU Tong YANG Zhu ZHANG Department of Mathematics The Chinese University of Hong KongShatinHong KongChina School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences Central China Normal UniversityWuhan 430079China Department of Applied Mathematics The Hong Kong Polytechnic UniversityHong KongChina.

In this paper,the authors study the 1 D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.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.

关键词： Boltzmann equation Compressible Navier-Stokes approximation Slip boundary conditions Chapman-Enskog expansion

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