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Mathematics > Analysis of PDEs

arXiv:0908.0923 (math)
[Submitted on 6 Aug 2009 (v1), last revised 10 Aug 2009 (this version, v2)]

Title:A variation on a theme of Caffarelli and Vasseur

Authors:Alexander Kiselev, Fedor Nazarov
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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 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.
Comments: 9 pages
Subjects: Analysis of PDEs (math.AP)
MSC classes: 35Q35; 35B65
Cite as: arXiv:0908.0923 [math.AP]
  (or arXiv:0908.0923v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.0908.0923

Submission history

From: Alexander Kiselev [view email]
[v1] Thu, 6 Aug 2009 18:09:04 UTC (13 KB)
[v2] Mon, 10 Aug 2009 09:27:51 UTC (13 KB)
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