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UBC Theses and Dissertations

Analytical and numerical results for phase field, implicit free boundary, and fluid models Cheng, Xinyu


In this dissertation, we study analytical and numerical methods on three topics in the area of partial differential equations (PDE). These topics are: the Allen-Cahn dynamics (AC) in the study of phase field models for materials science problems, the Oxygen depletion model (OD) in the study of free boundary problems, and the stationary surface quasi-geostrophic equation (SQG) in the study of fluid dynamics. We first study the behaviour in the meta-stable regime of AC and show by computation evidence and asymptotic analysis that backward Euler method satisfies energy stability with large time steps. We also give a rigorous proof for the two-dimensional radially symmetric case. In the second project, we show several mathematical formulations of OD from the literature and give a new formulation based on a gradient flow with constraint. We prove the equivalence of all formulations and study the numerical approximations of the problem that arise from the different formulations. More general (vector, higher order) implicit free boundary value problems are discussed. In the final project, we develop a new framework of ``convex integration scheme'' and construct a non-trivial solution to the stationary SQG. We thus prove the non-uniqueness of the solutions to the stationary SQG.

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