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Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations. (English) Zbl 1266.76010
Summary: 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 =Λ -1 P(Λ)θ, where 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 u|| 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 0γ1. In addition, a regularity criterion for the model corresponding to P(Λ)=Λ β with 0β1 is also obtained.
MSC:
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
35Q31Euler equations
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