New results in stochastic moving boundary problems

Stochastic partial differential equations;stochastic moving boundary value problem;the Stefan boundary condition

Kim, Kun Woo

关键词: Stochastic partial differential equations;    stochastic moving boundary value problem;    the Stefan boundary condition;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/31954/Kim_KunWoo.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Moving boundary problems arise in many areas of science and engineering and they are of great importance in the areas of partial differential equations (PDEs) since they characterize phase change phenomena where a system has two phases such as solid and liquid. However, unlike other PDEs in a prescribed region such as heat equation on a bounded domain, moving boundary problems are difficult to solve theoretically or numerically since we consider partial differential equations in one or two phases and at the same time need to trace the positions of the interface. Thus, they provide deep mathematical challenges. There is a vast literature on deterministic moving boundary problems. In addition, random perturbations of partial differential equations (e.g. stochastic heat equations) have been studied extensively. However, there has not been much attention paid to random perturbations of moving boundary problems. In this thesis, we consider random perturbations of two kinds of one-dimensional moving boundary problems: the Stefan problem, which describes the melting of the ice, and a free boundary problem proposed by Ludford and Stewart and studied by Caffarelli and Vazquez.In the first part, we consider a one-dimensional Stefan problem perturbed by a multiplicative noise. The noise is Brownian in time but smoothly correlated in space. We first define a weak solution then transform this problem into a nonlinear stochastic partial differential equation (SPDE) with a fixed boundary condition. We characterize the domain of existence and prove existence and uniqueness of a solution.The second part deals with a random perturbation of a moving boundary problem proposed by Ludford and Stewart and studied by Cafferelli and Vazquez. The random perturbation is a single Brownian motion and the moving boundary condition is different from the Stefan boundary condition. We consider existence and uniqueness of a solution and focus on numerical analysis of the problem. As for the stochastic Stefan problem, we use the transformation which transforms the stochastic moving boundary problem to a nonlinear SPDE which has a fixed spatial domain. Our numerical approximations are based on the nonlinear transformed SPDE. We use the explicit finite difference method and the Euler-Maruyama scheme to discretize time and space respectively. We also investigate the convergence theory.