On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion

TL;DR

This paper investigates the regularity of solutions to a class of drift-diffusion equations with nonlocal stable-type Lévy diffusion, establishing conditions for eventual and global regularity with explicit time estimates.

Contribution

It provides new regularity results for supercritical and logarithmically supercritical cases using a nonlocal maximum principle approach.

Findings

Proves eventual regularity in supercritical cases.

Establishes global regularity in logarithmically supercritical cases.

Explicitly estimates the time after which solutions become smooth.

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 L\'evy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.