Mathematical Sciences: Nonlinear Dynamics and Variational Problems

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Loss 9500840 This project focuses on problems in analysis originating in physics that feature a strong interaction between nonlinear evolution equations and optimization problems. Specifically this project concerns establishing and applying sharp inequalities to obtain precise bounds on behavior and evolution of the solutions of hydrodynamic and kinetic equations, and equations describing collective phenomena such as the Ginzburg--Landau equation. In most of these problems, the sharp inequalities come from optimization problems for which the symmetry of the optimizers is a crucial issue. Several parts of the project are aimed at the development of methods for establishing symmetry properties of optimizers in cases, such as the Ginzburg--Landau problem, where the classical rearrangement inequalities cannot be used. This project is focused on problems in nonlinear analysis originating in physics and the applied sciences. Its primary goal will be to discover and prove new theorems by exploiting the interplay between nonlinear dynamics and optimization problems. This interaction has been the source of much progress in the investigators' recent work, and the project consists of concrete problems in which it is expected to be particularly fruitful. ***