Long and Short Time Asymptotics of Systems of Nonlinear Partial Differential Equations Arising in Mean-Field Theory and Fluid-Dynamics

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This research project is an analytical study of the qualitative behavior of solutions to nonlinear systems of partial differential equations to which conventional methods of analysis do not apply. Four problem areas are under investigation: (1) long-time behavior of solutions to free-boundary problems for a nonlinear diffusion system modeling price formation in economics; (2) possible blow up for radially symmetric solutions for the incompressible Euler equation; (3) gradient flow methods for study of the quantum drift-diffusion fourth-order nonlinear parabolic system; and (4) classical and quantum kinetic models in plasma physics. The project aims to facilitate analysis of these systems by establishing links between different approaches via kinetic theory, optimal mass transportation methods, and variational techniques.This project analyzes equations that model several systems of practical importance, including price formation in economics, fluid flow, and the dynamics of plasmas. The mathematical models for these systems present substantial analytical challenges, and this work aims to improve on existing methods to enable deeper understanding of the properties of solutions to the governing equations. The results of the work will facilitate better prediction of the behavior of these complicated systems.