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

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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 u j of the velocity field u is determined by the scalar θ through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\) , where \({\mathcal{R}}\) 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 \({\|\nabla u||_{L^\infty}}\) 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.

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Korea

    Dongho Chae

  2. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Peter Constantin

  3. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA

    Jiahong Wu

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  1. Dongho Chae
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  2. Peter Constantin
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Chae, D., Constantin, P. & Wu, J. Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations. Arch Rational Mech Anal 202, 35–62 (2011). https://doi.org/10.1007/s00205-011-0411-5

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