Holder estimates for advection fractional-diffusion equations

TL;DR

This paper investigates conditions under which solutions to advection fractional-diffusion equations are Holder continuous, providing estimates based on the order of diffusion and regularity of the drift.

Contribution

It establishes Holder continuity results for solutions of fractional-diffusion equations with drifts, depending on the diffusion order and drift regularity.

Findings

Holder estimates for diffusion order ≥ 1 with bounded drift

Holder estimates for diffusion order < 1 with Holder continuous drift

Conditions linking diffusion order and drift regularity for solution regularity

Abstract

We analyse conditions for an evolution equation with a drift and fractional diffusion to have a Holder continuous solution. In case the diffusion is of order one or more, we obtain Holder estimates for the solution for any bounded drift. In the case when the diffusion is of order less than one, we require the drift to be a Holder continuous vector field in order to obtain the same type of regularity result.