A variation on a theme of Caffarelli and Vasseur

TL;DR

This paper demonstrates that a uniform BMO bound on smooth velocity fields ensures solutions to certain drift diffusion equations are uniformly Hölder continuous, providing a new proof of global regularity for the critical SQG equation.

Contribution

It establishes a new link between BMO bounds on velocity and Hölder regularity of solutions, offering an alternative proof for the critical SQG equation's global regularity.

Findings

Uniform BMO bounds imply Hölder continuity of solutions.

Elementary control of Hölder norms using test functions.

Provides a third proof of global regularity for critical SQG.

Abstract

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain H\"older continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the $C^\beta$ norm of the solution for some $\beta >0.$ We use elementary tools involving control of H\"older norms using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasi-geostrophic (SQG) equation.