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|Title: ||Some mathematical theories on the Cauchy problem and boundary layer problem for the Boltzmann equation|
|Other Titles: ||Guan yu Boltzmann fang cheng de Kexi wen ti he bian jie ceng wen ti de yi xie shu xue li lun|
關于 Boltzmann 方程的柯西問題和邊界層問題的一些數學理論
|Authors: ||Sun, Jie ( 孫傑)|
|Department: ||Department of Mathematics|
|Degree: ||Doctor of Philosophy|
|Issue Date: ||2011|
|Publisher: ||City University of Hong Kong|
|Subjects: ||Transport theory -- Mathematical models.|
Boundary value problems.
|Notes: ||CityU Call Number: QC175.2 .S85 2011|
v, 115 leaves 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2011.
Includes bibliographical references (leaves -115)
|Abstract: ||This thesis focuses on the mathematical theories on the Cauchy problem and the boundary
layer problem of the Boltzmann equation. The Cauchy problem is well-known.
The boundary layer problem arises in the physical consideration of the condensation-evaporation
problem, and is the first order approximation of the Boltzmann equation
for small Knudsen number near a plane. The thesis is mainly divided into two parts.
In the first part, the Cauchy problems of the Boltzmann equation with potential
force in the whole space and in torus are investigated. In the whole space, we consider
the Cauchy problem with potential force with some less restrictive assumptions
compared to the previous works. We obtain the well-posedness theory and the optimal
convergence rate of the solution to the Boltzmann equation even for the hard potential
case by energy method, when the initial data is sufficiently close to a steady state. In
torus, global existence and stability of solutions to the Cauchy problem of the Boltzmann
equation with potential forces for hard potentials are considered. We prove the
stationary state is asymptotically stable with exponential rate in time for any initially
smooth, periodic, origin symmetric small perturbation which preserves the same total
mass, momentum and mechanical energy as the natural steady state and any origin
symmetric small potential force.
In the second part, the boundary layer solutions to the Boltzmann equation with
mixed boundary condition for the inverse power law are discussed. The existence of
boundary layer solutions to the Boltzmann equation with mixed boundary condition,
that is Dirichlet boundary condition weakly perturbed by diffuse reflection boundary
condition at the wall, is considered. The boundary condition is imposed on the incoming
particles, and the solution is supposed to approach to a global Maxwellian in the far field. Like the problem with Dirichlet boundary condition, the existence of a
solution depends on the Mach number of the far field Maxwellian. Furthermore, an implicit
solvability condition on the boundary data which shows the codimension of the
boundary data is related to the number of positive characteristic speeds is also given.|
|Online Catalog Link: ||http://lib.cityu.edu.hk/record=b4086812|
|Appears in Collections:||MA - Doctor of Philosophy |
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