Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations

Dongho Chae, Peter Constantin, Jiahong Wu

Research output: Contribution to journalArticlepeer-review

68 Scopus citations

Abstract

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component uj of the velocity field u is determined by the scalar θ through uj = RΛ-1 P(Λ)θ, where R is a Riesz transform and Λ = (-Δ)1/2. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound {double pipe}▽u{double pipe}L for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I - Δ)))γ with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λβ with 0 ≦ β ≦ 1 is also obtained.

Original languageEnglish (US)
Pages (from-to)35-62
Number of pages28
JournalArchive for Rational Mechanics and Analysis
Volume202
Issue number1
DOIs
StatePublished - Oct 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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