Regularity and asymptotics for physical evolution equations

One of the most important areas of applied analysis is in the development of robust bounds for physically motivated evolution equations. When the equations feature prominent nonlinear/nonlocal effects (which are notoriously difficult to handle), such bounds can nevertheless recover certain asymptotic properties that simplify the problem or even the equations themselves.

The focus of this lecture will be on recent results for three physical models: homogenization and asymptotics for nonlocal reaction-diffusion equations, a priori bounds for hydrodynamic equations with thermal effects, and the local well-posedness for the Landau equation. Each problem presents unique challenges arising from the nonlinearity and/or nonlocality of the equation(s), and the emphasis will be on the different methods and techniques used to treat these difficulties. The talk will touch on novelties in viscosity theory and precision in nonlocal front propagation for reaction-diffusion equations, as well as the emergence of dynamic self-regularization in the thermal hydrodynamic and Landau equations.