On the regularity of a class of generalized quasi-geostrophic equations

In this article we consider the following generalized quasi-geostrophic equation \partial_t\theta + u\cdot\nabla \theta + \nu \Lambda^\beta \theta =0, \quad u= \Lambda^\alpha \mathcal{R}^\bot\theta, \quad x\in\mathbb{R}^2, where $\nu>0$, $\Lambda:=\sqrt{-\Delta}$, $\alpha\in ]0,1[$ and $\beta\in ]0,2[$. We first show a general criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case $\alpha\in]0,1[$ and $\beta\in ]\alpha+1,2]$ we obtain the global well-posedness of the system with smooth initial data; and for the case $\alpha\in ]0,1[$ and $\beta\in ]2\alpha,\alpha+1]$ we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.