Dissipative models generalizing the 2D navier-Stokes and surface quasi-Geostrophic equations

Dongho Chae, Peter Constantin, Jiahong Wu

Research output: Contribution to journalArticle

18 Scopus citations

Abstract

This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field u is determined by the active scalar θ through RΛ-1P(Λ)θ, where R denotes a Riesz transform, Λ = (-Δ)1/2, and P(Λ) represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case P(Λ) = I, while the surface quasi-geostrophic (SQG) equation corresponds to P(Λ) = Λ. We obtain the global regularity for a class of equations for which P(Λ) and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with P(Λ) = (log(I - Δ))γ for any γ > 0 are globally regular.

Original languageEnglish (US)
Pages (from-to)1997-2018
Number of pages22
JournalIndiana University Mathematics Journal
Volume61
Issue number5
DOIs
StatePublished - Dec 1 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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