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 (9)

• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
• Eventual regularization for the slightly supercritical quasi-geostrophic equation
• Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation
• Global Regularity for the Critical Dispersive Dissipative Surface Quasi-Geostrophic Equation
• Remarks on a fractional diffusion transport equation with applications to the dissipative quasi-geostrophic equation
• 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
• Regularization for the supercritical quasi-geostrophic equation

Cited by in corpus (6)

• Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation
• Global well-posedness of slightly supercritical active scalar equations
• A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation
• Generalized surface quasi-geostrophic equations with singular velocities
• Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces
• The 2D Boussinesq equations with logarithmically supercritical velocities