Inviscid Models Generalizing the 2d Euler and the Surface Quasi-geostrophic Equations

Any classical solution of the 2D 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−1P (Λ)θ where R is a Riesz transform and Λ = (−∆). The 2D Euler vorticity equation corresponds to the special case P (Λ) = I while the SQG equation 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−∆))) with 0 ≤ γ ≤ 1. In addition, a regularity criterion for the model corresponding to P (Λ) = Λ with 0 ≤ β ≤ 1 is also obtained.