Non Linear Singular Drifts and Fractional Operators

TL;DR

This paper studies parabolic PDEs with fractional operators and nonlinear singular drifts, proving Hölder continuity of solutions under boundedness conditions in Lebesgue and Besov spaces, including near-critical cases.

Contribution

It establishes Hölder regularity for solutions to fractional PDEs with nonlinear singular drifts using Besov-type estimates, covering almost critical cases in full generality.

Findings

Hölder continuity of solutions under certain boundedness conditions

Handling of almost critical cases in fractional PDEs

Development of a priori Besov-type estimates

Abstract

We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{\"o}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.