Dissipative models generalizing the 2D navier-Stokes and 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 = (-Δ)1/2, 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 corresponds 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 - Δ))γ for any γ > 0 are globally regular.