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

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 theta through u(j) = R Lambda P-1(Lambda)theta, where R is a Riesz transform and Lambda = (-Delta)(1/2). The two-dimensional Euler vorticity equation corresponds to the special case P(Lambda) = I while the SQG equation corresponds to the case P(Lambda) = Lambda. We develop tools to bound parallel to del u parallel to(L infinity) for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Lambda) = (log(I + log(I - Delta)))(gamma) with 0 <= gamma <= 1. In addition, a regularity criterion for the model corresponding to P(Lambda) = Lambda(beta) with 0 <= beta <= 1 is also obtained.