# Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations - Mathematics > Analysis of PDEs

Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: This paper is devoted to the global in time regularity problem for a familyof active scalar equations with fractional dissipation. Each component of thevelocity field $u$ is determined by the active scalar $\theta$ through$\mathcal{R} \Lambda^{-1} P\Lambda \theta$ where $\mathcal{R}$ denotes aRiesz transform, $\Lambda=-\Delta^{1-2}$ and $P\Lambda$ represents a familyof Fourier multiplier operators. The 2D Navier-Stokes vorticity equationscorrespond to the special case $P\Lambda=I$ while the surfacequasi-geostrophic SQG equation to $P\Lambda =\Lambda$. We obtain the globalregularity for a class of equations for which $P\Lambda$ and the fractionalpower of the dissipative Laplacian are required to satisfy an explicitcondition. In particular, the active scalar equations with any fractionaldissipation and with $P\Lambda= \logI-\Delta^\gamma$ for any $\gamma>0$are globally regular.

Author: ** Dongho Chae, Peter Constantin, Jiahong Wu**

Source: https://arxiv.org/