Hölder estimates for fractional parabolic equations with critical divergence free drifts

Abstract

This work focuses on drift-diffusion equations with fractional dissipation $(-\hat{I}”{)}^{\hat{I}}\pm$ in the regime $\hat{I}\pm \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some ${\hat{I}}^2\in (0,1)$, the $C^2$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q(BM{O}_x^{-{\hat{I}}^3})$ with $q>2$ and ${\hat{I}}^3\in (0,2\hat{I}\pm -1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.