On preservation of moduli of continuity by parabolic evolution

Abstract

In this work, we study how Lipschitz continuity propagates by a certain class of nonlinear, nonlocal parabolic equations. This work draws inspiration from ideas developed in recent years by Kiselev, Nazarov, Volberg and Shterenberg to address issues relating to the regularity of solutions of critical active scalar equations such as the the surface quasi-geostrophic equation and Burgers model. Namely, we will extend and improve on such techniques in order for them to be applicable to combustion models as well as other fluid equations such as the incompressible Navier-Stokes system and Burgers-Hilbert flow.

The main problem we address here is proving a global regularity result relating to a slight modification of the so called Michelson-Sivashinsky equation. We also give outlines of how can one use similar ideas to obtain various new regularity and partial regularity criteria for the incompressible Navier-Stokes system, as well as provide a different proof to a known criterion in terms of critical H\"older-type norms. We also outline how to extend the technique to a viscous, multi-dimensional Burgers-Hilbert problem in order to prove global regularity for this model.