On the Regularity Issues of a Class of Drift-Diffusion Equations with Nonlocal Diffusion

Abstract

In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by some zero-order pseudo-differential operators. Through using the method of nonlocal maximum principle in a unified way, we prove the global well-posedness result in some slightly supercritical cases, and show the eventual regularity result in the supercritical type cases. The time after which the solution is smoothly regular in the supercritical type cases can be evaluated appropriately, so that we can prove a type of global result recently obtained by [17] and also show the global regularity of vanishing viscosity solution at some logarithmically supercritical cases.2010 Mathematics Subject Classification. 76D03, 35Q35, 35Q86.

Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

“…This approach was used to study the forced critical SQG in [2], and developed in a supercritical context (with an additional decay factor which allows to exploit "eventual regularization" properties of the equation) in [4]. Some results related to this approach were also obtained in [14,12,16].…”