This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time...This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.展开更多
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-...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.展开更多
基金partially supported by the General Research Fund(Project No.14302817)from RGC of Hong Konga Direct Grant(No.4053286)from CUHKpartly supported by NSFC Grant No.11771429,11688101,and 11671237
文摘This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.
基金supported by the National Natural Science Foundation of China(Nos.11971201,11731008)the General Research Fund from RGC of Hong Kong(No.14301719)+1 种基金a Direct Grant from CUHK(No.4053397)the Fundamental Research Funds for the Central Universities and a fellowship award from the Research Grants Council of the Hong Kong Special Administrative Region,China(No.SRF2021-1S01)。
文摘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.
