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

该奖项将为与杰夫·劳赫教授合作的密歇根大学博士后胡安·雷东多提供支持。雷东多博士计划继续研究抛物型变分不等解中自由边界的正则性，就像经典的斯特凡问题一样。他还将研究双曲守恒律的激波解的正则性，将他的变分技巧应用到抛物型方程组中，这些抛物型方程组是通过在原始双曲型方程中添加粘性项而产生的。非线性偏微分方程组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为根本没有耗散。在这项授权中，博士后胡安·雷东多将使用变分理论中的技术，利用耗散系统和非耗散系统之间的这种关系来研究不同材料之间的边界形状和不同密度气体之间的锋面尖锐程度等问题。

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.