A variation on a~theme of Caffarelli and Vasseur
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 58-72
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Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain Hölder 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 Hölder norms using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasi-geostrophic (SQG) equation in addition to [5] and [1]. Bibl. – 6 titles.
@article{ZNSL_2009_370_a3,
author = {A. Kiselev and F. Nazarov},
title = {A variation on a~theme of {Caffarelli} and {Vasseur}},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {58--72},
publisher = {mathdoc},
volume = {370},
year = {2009},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/}
}
A. Kiselev; F. Nazarov. A variation on a~theme of Caffarelli and Vasseur. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 58-72. http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/