Dissipative Models Generalizing the 2d Navier-stokes and the Surface Quasi-geostrophic Equations

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 θ through RΛ−1P (Λ)θ where R denotes a Riesz transform, Λ = (−∆) and P (Λ) represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case P (Λ) = I while the surface quasi-geostrophic (SQG) equation to P (Λ) = Λ. We obtain the global regularity for a class of equations for which P (Λ) 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 (Λ) = (log(I −∆)) for any γ > 0 are globally regular.