Fractional Laplacian with singular drift

TL;DR

This paper studies a fractional Laplacian PDE with a singular divergence-free drift, demonstrating that for small parameters, the fundamental solution remains comparable to an isotropic stable process density over time.

Contribution

It establishes global in time estimates for the fundamental solution of a fractional PDE with a non-Kato class singular drift, extending previous results to more singular vector fields.

Findings

Fundamental solution is comparable to stable process density for small r.

Results hold even when the drift is not in the Kato class.

Provides new insights into fractional PDEs with singular drifts.

Abstract

For $\alpha \in (1,2)$ we consider the equation $\partial_t u = \Delta^{\alpha/2} u - r b \cdot \nabla u$, where $b$ is a divergence free singular vector field not necessarily belonging to the Kato class. We show that for sufficiently small $r>0$ the fundamental solution is globally in time comparable with the density of the isotropic stable process