Long Term Regularity of Solutions of Fluid Models

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This research project studies a class of partial differential equations that describe physical phenomena including water waves, plasmas, and gravitation. The complicated nature of the solutions to such equations often precludes complete mathematical analysis, and applications rely on numerical approximations. This project studies the solutions of these equations rigorously and recovers quantitative and qualitative information about their behavior as mathematical theorems. The project involves graduate students and postdoctoral associates in the research and fosters collaborations with researchers in related fields, including physics and engineering. This project studies the solutions of several quasilinear dispersive partial differential equations, including water-wave models in two and three spatial dimensions, the more general case of two-fluid interfaces, and the Einstein-Klein-Gordon equations of general relativity. The main problems under study concern the global stability of the equilibrium solutions of these equations and the construction of solutions that become singular in finite time. The project aims to contribute new methods and tools to this dynamic field.