Dissipative Models Generalizing the 2D Navier-Stokes and Surface Quasi-Geostrophic Equations

Abstract

This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field u is determined by the active scalar theta through R Lambda P-1 (Lambda)theta, where R. denotes a Riesz transform, Lambda = (-Delta A)(1/2), and P (Lambda) represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case P (Lambda) = I, while the surface quasi-geostrophic (SQG) equation corresponds to P (Lambda) = A. We obtain the global regularity for a class of equations for which P (Lambda) and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with P (A) = (log(I - Delta))(gamma) for any gamma > 0 are globally regular.