AMC-SS, Collaborative Research: Explorations in Stochastic Moving Boundary Value Problems

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This project outlines some problems which focus on the effect of noise on moving boundary problems. We are in particular interested in the mathematical foundations of a one-phase Stefan problem with additive noise. The noise can take on a number of forms, and the smoothness of the noise will affect the problems we can hope to consider. We pose a number of specific problems which, we hope, will resolve some fundamental issues in this area: existence, approximation, smoothness, and asymptotics.Moving boundary value problems are one of the key classes of partial differential equations. The simplest example is ice and water, where the motion of the boundary between the two depends on the local structure of the interface. Although these equations are difficult to study, there is a large body of literature on them. Since noise is ubiquitous in physical systems, it is natural to study the effect of noise on these systems. Although our research attempts to develop some of the foundations of stochastic moving boundary value problems, a number of applied areas (e.g. combustion and porous media) may benefit from the theory developed in this project.