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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Award date  15 Jul 2010 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(dfaa8c014df547e5a0ce673d913487b0).html 

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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
 Mathematical models, Boundary value problems, Boundary layer