Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

Peter Constantin, Jiahong Wu

Research output: Contribution to journalArticle

56 Scopus citations

Abstract

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α < 1 / 2) dissipation (- Δ)α. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α = 1 / 2) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from L2 to L, from L to Hölder (Cδ, δ > 0), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be L, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1 - 2 α. Higher regularity starting from Cδ with δ > 1 - 2 α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].

Original languageEnglish (US)
Pages (from-to)159-180
Number of pages22
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume26
Issue number1
DOIs
StatePublished - Jan 1 2009

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

Keywords

  • Dissipative quasi-geostrophic equation
  • Regularity
  • Supercritical dissipation
  • Weak solutions

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