Dissipative models generalizing the 2D Navier-Stokes and the 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 $\mathcal{R}{\Lambda }^{-1}P(\Lambda )\theta$ where $\mathcal{R}$ denotes a Riesz transform, $\Lambda =(-\Delta {)}^{1/2}$ 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 to $P(\Lambda )=\Lambda$. 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 $P(\Lambda )=(\log (I-\Delta ){)}^{\gamma }$ for any $\gamma >0$ are globally regular.

References in corpus (3)

• Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$
• Global well-posedness for the 2 D quasi-geostrophic equation in a critical Besov space
• Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation

Cited by in corpus (1)

• Generalized surface quasi-geostrophic equations with singular velocities