Semester

Summer

Date of Graduation

2010

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Harumi Hattori.

Abstract

In this thesis we study phase transition problems of conservation laws. Phase transition problems arise from various applications such as gas dynamics, mechanics and material science. Conservation laws involving phase change is an attractive field in applied mathematics. Solutions to phase transition problems are complicated for the presence of boundaries between different phases. In addition to entropy condition, criteria such as kinetic relation [1, 3] and nucleation criterion are introduced to determine the configurations of solutions.;In Chapter 1, we construct two numerical procedures to solve the Riemann problems for a system of conservation laws with phase change. We first find the solution with a stationary phase boundary by Newton iteration [ 14]. The configuration of the solution, especially the direction of the propagating phase boundary, is then determined based on the criterion suggested by Hattori [11] given that the speed of a moving phase boundary is much smaller than the speed of a shock or a rarefaction wave. One way to solve the Riemann problem with a moving phase boundary is to list all the relations and find the solution of the resulting nonlinear system. Another is to construct an iterative process to find the intersection of two projection curves.;In Chapter 2, we discuss the well posedness of the initial value problem to Euler equations related to phase transition. The solution contains two phase boundaries moving in opposite directions. Entropy condition and kinetic relationship are used as the main admissibility criteria to select the physically relevant solution. We show the existence of the entropy solution under a suitable Finiteness Condition and a Stability Condition guarantees the stability of the problem in L1∩ BV and the existence of a Lipschitz semigroup of solutions. We also discuss the well posedness of the problem given that the wave speeds do not differ significantly between different phases.

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