Regularity for the supercritical fractional Laplacian with drift

TL;DR

This paper investigates the regularity of solutions to the supercritical fractional Laplacian with drift, revealing that solutions exhibit regularity similar to the subcritical case despite the operator's non-elliptic nature.

Contribution

It proves local Sobolev regularity for solutions in the supercritical regime and analyzes the Green's kernel singularity, extending understanding of fractional operators with dominant drift.

Findings

Solutions are regular in Sobolev spaces despite non-ellipticity.

Green's kernel exhibits unusual singularity behavior in the drift direction.

Regularity matches subcritical case where diffusion dominates.

Abstract

We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, that is the case when the dominant term is given by the drift instead of the diffusion component, we prove local regularity of solutions in Sobolev spaces em- ploying tools from the theory of pseudo-differential operators. The regularity of solutions in the supercritical case is as expected in the subcritical case, when the diffusion is at least as strong as the drift component, and the operator defined by the fractional Laplacian with drift can be viewed as an elliptic operator, which is not the case in the supercritical regime. We compute the leading singularity for the Green's kernel in the supercritical range, which displays some unusual behavior: it is more singular in the half plane into which the drift vector points, than in the complementary half plane.