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2014 Global well-posedness of slightly supercritical active scalar equations
Michael Dabkowski, Alexander Kiselev, Luis Silvestre, Vlad Vicol
Anal. PDE 7(1): 43-72 (2014). DOI: 10.2140/apde.2014.7.43


The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasigeostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.


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Michael Dabkowski. Alexander Kiselev. Luis Silvestre. Vlad Vicol. "Global well-posedness of slightly supercritical active scalar equations." Anal. PDE 7 (1) 43 - 72, 2014.


Received: 27 February 2012; Revised: 10 April 2013; Accepted: 19 April 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1294.35092
MathSciNet: MR3219499
Digital Object Identifier: 10.2140/apde.2014.7.43

Primary: 35Q35 , 76U05

Keywords: active scalars , Burgers equation , finite time blow-up , global regularity , nonlocal dissipation , nonlocal maximum principle , SQG equation , supercritical dissipation , surface quasigeostrophic equation

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 1 • 2014
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