Some mathematical theories on the boundary layer problem for the Boltzmann equation with mixed boundary conditions

一些關於混合邊界條件下 Boltzmann 方程邊界層問題的數學理論

Student thesis: Doctoral Thesis

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Author(s)

  • Qianzhu TIAN

Related Research Unit(s)

Detail(s)

Awarding Institution
Supervisors/Advisors
Award date15 Jul 2010

Abstract

This thesis is concerned with the mathematical theories on the boundary layer solution to the Boltzmann equation with physical boundary conditions for hard sphere model. Since the Boltzmann equation was founded, the research in this area has been one of the most important and challenging fields in mathematics not only because of many unsolved mathematical problems, but also because of its rich physical background and practical applications. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width in the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. In Chapter 2, the existence of boundary layer solutions to the Boltzmann equation with different physical boundary conditions is presented. It is divided into two parts. The first part of this chapter is focused on the existence of boundary layer solutions to the Boltzmann equation with diffuse boundary condition which means the velocity of a particle after reflection is random. An existence result is obtained under the assumption that the solution tends to a global Maxwellian in the far field. Moreover, the existence is shown to depend on the Mach number of the far field Maxwellian. In the second part, we study the existence of boundary layer solutions to the Boltzmann equation with specular reflection and reverse reflection boundary conditions respectively. In these cases, the similar results are obtained for the positive Mach number. In Chapter 3, the stability of boundary layer solutions to the Boltzmann equation with diffusive effect at the boundary is considered. When the Mach number of the far field is less than -1, the exponential decay in time is proven for linearized operator first. Then, based on this property, nonlinear stability of the boundary layers is obtained by bootstrap argument. Keywords: Boltzmann equation, Boundary layer, Mixed boundary conditions, Energy method

    Research areas

  • Mathematical models, Boundary value problems, Boundary layer