Dynamic Free Boundary Problems

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Dynamic free boundary problems deal with solving partial differential equations in domain whose evolution is not known a priori. One example is the problem of water freezing into ice, where the interface of ice and water is determined by the solution of the heat equation in the water region (the distribution of temperature). This project is focused on problems whose solutions have very little regularity. For instance, when water freezes into ice, the resulting ice crystal is expected to form a fractal-like structure on its boundary. Another example is tumor growth, where the boundary of the tumor develops irregularity due to the search for regions of high nutrients. The project is aimed at providing a better understanding of the properties of the solutions to such problems and provide a framework for developing accurate numerical simulations. The project provides research training opportunities for graduate students. The presence of lower-dimensional structures is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. Many problems in nonlinear partial differential equations which are otherwise well understood pose significant challenges when they are coupled with a moving interface, even in seemingly simple settings. Besides the nonlinearity, the difficulty of the problem lies in its nonlocality, in the sense that the behavior of solutions depends on the global geometry of the free boundary. Dynamic free boundary problems face an additional difficulty coming from the lack of a priori regularity of the free boundaries. By studying the proposed problems, the Principal Investigator (PI) aims to develop general methods to deal with such difficulties. The work of the project uses geometric analysis as well as variational and probabilistic arguments. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.