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_j of the velocity field u is determined by the scalar θ through u_j = RΛ^-1 P(Λ)θ, where 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 {double pipe}▽u{double pipe}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 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ^β with 0 ≦ β ≦ 1 is also obtained.