Mathematical Sciences: Penalization Techniques for the Studyof Free and Moving Boundaries

该奖项将为胡安·雷东多提供支持，他是密歇根大学的博士后，与杰夫·劳奇教授合作。雷东多博士计划继续研究抛物型变分不等式解中自由边界的规律性，就像经典的斯蒂芬问题一样。他还将研究双曲守恒定律的激波解的规律性，将他的变分技术应用于抛物线系统，这是通过在原始双曲系统中加入粘性项而得到的。非线性偏微分方程描述了许多耗散现象，如热流。如果耗散量很小，这类方程的解通常表现得好像根本没有耗散。在这项资助中，博士后Juan Redondo将使用变分理论的技术，利用耗散和非耗散系统之间的关系来研究不同材料之间边界的形状和不同密度气体之间锋面的清晰度等问题。

This award will provide support for Juan Redondo, who is a postdoc at the University of Michigan working with Professor Jeff Rauch. Dr. Redondo plans to continue his study of the regularity of free boundaries in the solutions of parabolic variational inequalities like the ones associated with the classical Stefan problem. He is also going to investigate the regularity of shock wave solutions of hyperbolic conservation laws by applying his variational techniques to the parabolic systems that come from adding viscous terms to the original hyperbolic systems. Nonlinear partial differential equations describe many dissipative phenomena like the flow of heat. If the amount of dissipation is small the solutions of such equations often behave as if there is no dissipation at all. In this grant the postdoc, Juan Redondo, will use techniques from variational theory that exploit this relationship between dissipative and non-dissipative systems in order to study things like the shape of a boundary between different materials and the sharpness of a front between gases of different densities.