Global well-posedness of slightly supercritical active scalar equations

Michael Dabkowski, Alexander Kiselev, Luis Silvestre, Vlad Vicol

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)43-72
Number of pages30
JournalAnalysis and PDE
Volume7
Issue number1
DOIs
StatePublished - Jan 1 2014

Fingerprint

Global Well-posedness
Global Regularity
Scalar
Burgers Equation
Maximum principle
Euler equations
Nonlocal Diffusion
Quasi-geostrophic Equations
Modulus of Continuity
Maximum Principle
Euler Equations
Logarithmic
Infinity
Term

Keywords

  • Active scalars
  • Burgers equation
  • Finite time blow-up
  • Global regularity
  • Nonlocal dissipation
  • Nonlocal maximum principle
  • SQG equation
  • Supercritical dissipation
  • Surface quasigeostrophic equation

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

Global well-posedness of slightly supercritical active scalar equations. / Dabkowski, Michael; Kiselev, Alexander; Silvestre, Luis; Vicol, Vlad.

In: Analysis and PDE, Vol. 7, No. 1, 01.01.2014, p. 43-72.

Research output: Contribution to journalArticle

Dabkowski, M, Kiselev, A, Silvestre, L & Vicol, V 2014, 'Global well-posedness of slightly supercritical active scalar equations', Analysis and PDE, vol. 7, no. 1, pp. 43-72. https://doi.org/10.2140/apde.2014.7.43
Dabkowski, Michael ; Kiselev, Alexander ; Silvestre, Luis ; Vicol, Vlad. / Global well-posedness of slightly supercritical active scalar equations. In: Analysis and PDE. 2014 ; Vol. 7, No. 1. pp. 43-72.
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