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

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 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). 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))) with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ with 0 ≦ β ≦ 1 is also obtained. [ABSTRACT FROM AUTHOR]