Dynamic Free Boundary Problems

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Dynamic free boundary problem refers to solving partial differential equations in a domain whose evolution is a priori unknown. One example is the problem of melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature, or solution of the heat equation, in the water region. A particular focus is given on problems where solutions are given with very little regularity. A good example is the problem of drops sliding on tilted surface. In this case the shape of the drop can change drastically when the velocity is increased, and a singularity (corner) develops at the rear of the drop when the velocity exceed a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Another example is oil drops moving up towards water in a container when initially water, the heavier fluid, is poured on top of oil. The project aims towards a better understanding of the properties of these problems and provide a framework for developing accurate computer-based numerical simulations. Graduate students will be trained by participating in the research.The problems to be discussed arise in a variety of physical phenomena, including the motion of capillary drops, crowd motion, tumor growth, and filtration of fluids in porous media. The presence of lower-dimensional structure is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. Many nonlinear PDE problems which are otherwise well-understood face significant challenges when they are coupled with a moving interface, even in seemingly simple settings. Besides the nonlinearity of the problem, the difficulty lies in the non-locality of the problem, in the sense that the behavior of solutions depends on the global geometry of the free boundary. Dynamic problems face an additional difficulty coming from the lack of a priori regularity of the free boundaries. By studying proposed problems we aim to develop general methods to deal with such difficulties. In addition to energy and integral estimates, it will be often necessary to introduce geometric methods to understand pointwise behavior of the moving interface.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.