Persistence of the Steady Normal Shock Structure for the Unsteady Potential Flow Equation
定常正激波結構的非定常穩定性
Student thesis: Doctoral Thesis
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Award date  21 Jun 2019 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(719c5de018374961a7e40d99c399eab0).html 

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Abstract
In this thesis, we mainly investigate the dynamic stability of steady normal shock structure for unsteady potential flow equation. In the first part of this thesis, we introduce some fundamental concepts and different kinds of flow, which can be governed by different differential equations. In addition, we present the main problem of this thesis.
In the second part, with the help of the partial hodograph transformation, the main problem, which is free boundary problem, will be reduced to a initial boundary value problem in a cornered space domain with fixed boundaries. Then the main result is presented.
In the third part, we will first introduce some concepts and results concerning the mixed hyperbolic problem with constant coefficients and homogeneous initial conditions. Then we consider the wellposedness of a general initial boundary value problem in a cornered space domain governed by a linear hyperbolic equations of second order.
In the forth part, we consider the dynamic stability of steady normal structure of unsteady potential flow equation with two spacial variables in a cornered space domain. We formulate it to a mathematical problem of the wellposedness of the nonlinear hyperbolic differential equations of second order in a cornered space domain with a free boundary. To solve this nonlinear initial boundary value problem, linearised problem in the cornered space domain is considered. The linearised problem is decomposed into two auxiliary linear problems. By some proper assumptions on the coefficients, the auxiliary problems are extended to the half plane. Then we can obtain the uniqueness and existence of low regularity solution of the linearised problem by solving the two auxiliary problems in half plane. Due to the restriction of the regularity of the extended coefficients, we cannot obtain the higher order energy estimate in the half plane directly. Thanks to our extension, we can deduce an additional boundary condition on the horizontal axis for the low regularity solution. This condition enables us to find appropriate multipliers, so that the higher order energy estimate can be obtained in the cornered space domain. In our energy estimate, loss of regularity happens, so a modified NashMoser iteration is developed to obtain the existence of the nonlinear problem.
In the second part, with the help of the partial hodograph transformation, the main problem, which is free boundary problem, will be reduced to a initial boundary value problem in a cornered space domain with fixed boundaries. Then the main result is presented.
In the third part, we will first introduce some concepts and results concerning the mixed hyperbolic problem with constant coefficients and homogeneous initial conditions. Then we consider the wellposedness of a general initial boundary value problem in a cornered space domain governed by a linear hyperbolic equations of second order.
In the forth part, we consider the dynamic stability of steady normal structure of unsteady potential flow equation with two spacial variables in a cornered space domain. We formulate it to a mathematical problem of the wellposedness of the nonlinear hyperbolic differential equations of second order in a cornered space domain with a free boundary. To solve this nonlinear initial boundary value problem, linearised problem in the cornered space domain is considered. The linearised problem is decomposed into two auxiliary linear problems. By some proper assumptions on the coefficients, the auxiliary problems are extended to the half plane. Then we can obtain the uniqueness and existence of low regularity solution of the linearised problem by solving the two auxiliary problems in half plane. Due to the restriction of the regularity of the extended coefficients, we cannot obtain the higher order energy estimate in the half plane directly. Thanks to our extension, we can deduce an additional boundary condition on the horizontal axis for the low regularity solution. This condition enables us to find appropriate multipliers, so that the higher order energy estimate can be obtained in the cornered space domain. In our energy estimate, loss of regularity happens, so a modified NashMoser iteration is developed to obtain the existence of the nonlinear problem.