Vol. 7, No. 1, 2014

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Global well-posedness of slightly supercritical active scalar equations

Michael Dabkowski, Alexander Kiselev, Luis Silvestre and Vlad Vicol

Vol. 7 (2014), No. 1, 43–72
Abstract

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.

Keywords
surface quasigeostrophic equation, active scalars, global regularity, finite time blow-up, supercritical dissipation, nonlocal maximum principle, nonlocal dissipation, SQG equation, Burgers equation
Mathematical Subject Classification 2010
Primary: 35Q35, 76U05
Milestones
Received: 27 February 2012
Revised: 10 April 2013
Accepted: 19 April 2013
Published: 7 May 2014
Authors
Michael Dabkowski
Department of Mathematics
University of Michigan
2074 East Hall
530 Church Street
Ann Arbor, MI 48109
United States
Alexander Kiselev
Department of Mathematics
University of Wisconsin
480 Lincoln Dr.
Madison, WI 53705
United States
Luis Silvestre
Department of Mathematics
The University of Chicago
Chicago, IL 60637
United States
Vlad Vicol
Department of Mathematics
Princeton University
Fine Hall, Washington Road
Princeton, NJ 08544
United States