The PI proposes to continue her program on the study of solutions of the free boundary problems associated with Hele-Shaw flows and Stefan problems. The PI plans to investigate (i) the global existence in time, and uniqueness; (ii) regularity properties; (iii) the long-time behavior of solutions; and (iv) geometric properties of the free boundaries in bounded domains. She also proposes to study homogenization of the free boundaries in periodic media, moving with oscillating normal velocity caused by inhomogeneities in the media. An example is the dynamics of water droplets spreading on an irregular surface. The current goal is to prove the existence, uniqueness and regularity properties of the effective free boundary velocity. The anticipated results will provide insight into the effect of inhomogeneities in the media on the global behavior of the free boundary.
Free boundary problems arise throughout fluid mechanics and material science and involve the solution of partial differential equations in a domain whose boundary is not entirely unknown. Describing the boundary is an important part of the problem. An example is flame propagation, where it is important to track the evolution of the interface between burnt and un-burnt zones. The problems to be studied in this project include systems of equations describing solid-fluid phase transitions, fluid flow in a narrow cell, and liquid droplets on a surface. These problems have been extensively studied experimentally and computationally but very little has been proved about theequations that model these systems. This project aims to provide qualitative and quantitative information about the solutions that will explain various phenomena observed in experiments and numerical simulations.
