H\"older estimates for fractional parabolic equations with critical divergence free drifts

TL;DR

This paper establishes H"older continuity estimates for solutions to fractional parabolic equations with divergence-free drifts in critical spaces, extending regularity theory for non-local PDEs with rough coefficients.

Contribution

It provides the first a priori H"older estimates for fractional parabolic equations with critical divergence-free drifts in the regime lpha(1/2,1), using a De Giorgi type approach.

Findings

H"older norm depends only on drift size in critical spaces and initial energy.

Proves regularity for solutions with initial data in finite energy space.

Extends regularity results to fractional dissipation with critical divergence-free drifts.

Abstract

This work focuses on drift-diffusion equations with fractional dissipation $(-\Delta)^{\alpha}$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori H\"older estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta}$ norm of the solution depends only on the size of the drift in critical spaces of the form $L^{q}_{t}(BMO^{-\gamma}_{x})$ with $q>2$ and $\gamma \in (0,2\alpha-1]$, along with the $L^{2}_{x}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.