Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations |
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Authors: | Dongho Chae Peter Constantin Jiahong Wu |
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Institution: | 1.Department of Mathematics,Sungkyunkwan University,Suwon,Korea;2.Department of Mathematics,University of Chicago,Chicago,USA;3.Department of Mathematics,Oklahoma State University,Stillwater,USA |
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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) = Λ β with 0 ≦ β ≦ 1 is also obtained. |
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