A variation on a theme of Caffarelli and Vasseur

Details A variation on a theme of Caffarelli and Vasseur A variation on a theme of Caffarelli and Vasseur

2009-08-06

arxiv.org/abs/0908.0923v2

Mathematics

Analysis of PDEs

9 pages

Scientific paper

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.

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