Dissipative models generalizing the 2D Navier-Stokes and the Surface Quasi-Geostrophic Equations Dongho ChaePeter ConstantinJiahong Wu 35Q5335B3535B6576D03generalized surface quasi-geostrophic equationsglobal existence 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 corresponds 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. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4756 10.1512/iumj.2012.61.4756 en Indiana Univ. Math. J. 61 (2012) 1997 - 2018 state-of-the-art mathematics http://iumj.org/access/