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Global well-posedness
of slightly supercritical active scalar equations
Michael Dabkowski, Alexander Kiselev, Luis Silvestre and Vlad Vicol |
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Vol. 7 (2014), No. 1, 43–72
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35Q35, 76U05
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Milestones
Received: 27 February 2012
Revised: 10 April 2013
Accepted: 19 April 2013
Published: 7 May 2014
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Authors |
| Michael Dabkowski | |
| Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109 United States |
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| Alexander Kiselev | |
| Department of Mathematics University of Wisconsin 480 Lincoln Dr. Madison, WI 53705 United States |
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| Luis Silvestre | |
| Department of Mathematics The University of Chicago Chicago, IL 60637 United States |
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| Vlad Vicol | |
| Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NJ 08544 United States |
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