Partial Differential Equation Methods in Kinetic Theory and Their Applications

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The project considers boundary effects for several physical models widely used to describe a hot plasma. The questions studied are motivated by, and have implications for, a broad range of applications. As in the case of a nuclear fusion device, where it is important to control and understand the plasma-wall interaction, or in Einstein's theory for general relativity, where a fundamental open question, known as cosmic censorship conjecture, is whether a gravitational collapse of a star cannot (i.e., black holes) or can (i.e., naked singularity) be observed generically. This work will construct examples of exact gravitational collapse in Einstein's theory that can be observed, and will investigate the dynamics of contact lines as well as the effect of a Coriolis force in oceans. This project provides training opportunities for graduate students.Kinetic theory provides important models for describing a confined plasma in a device. Because of the severe mathematical difficulties caused by the presence of a grazing set at the boundary, questions of well-posedness for kinetic plasma models in the presence of magnetic effect remain open. Motivated by applications such as contact-line dynamics and the effects of a constant rotation or a constant magnetic field on a fluid, the investigator will pursue several lines of research. The problems investigated are the asymptotical stability of BGK waves; the rigorous analysis of numerical evidence that indicates the existence of a relativistic Larson-Penston self-similar gravitational collapse, leading to formation of a stable naked singularity and the violation of the cosmic censorship hypothesis; and the well-posedness of a hydrodynamic model describing contact line dynamics. The project will also develop a new mechanism to construct long time (global) smooth inviscid fluid flows based on the dispersive effect induced by a constant rotation or a magnetic field.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.