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

本项目概述了一些问题，重点是对移动边界问题的噪声的影响。 我们特别感兴趣的数学基础的一个阶段Stefan问题的加性噪声。 噪声可以有多种形式，噪声的平滑度将影响我们希望考虑的问题。 我们提出了一些具体的问题，我们希望，将解决一些基本问题，在这一领域：存在性，逼近，光滑性，渐近性。移动边值问题是偏微分方程的关键类别之一。 最简单的例子是冰和水，两者之间边界的运动取决于界面的局部结构。 虽然这些方程很难研究，但有大量的文献。 由于噪声在物理系统中无处不在，研究噪声对这些系统的影响是很自然的。 虽然我们的研究试图发展一些随机移动边值问题的基础，一些应用领域（如燃烧和多孔介质）可能会受益于本项目中开发的理论。

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.