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

In this article we consider the following generalized quasi-geostrophic equation \partial_tθ+ u\cdot\nabla θ+ νΛ^βθ=0, \quad u= Λ^α\mathcal{R}^\botθ, \quad x\in\mathbb{R}^2, where $ν>0$, $Λ:=\sqrt{-Δ}$, $α\in ]0,1[$ and $β\in ]0,2[$. 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 $α\in]0,1[$ and $β\in ]α+1,2]$ we obtain the global well-posedness of the system with smooth initial data; and for the case $α\in ]0,1[$ and $β\in ]2α,α+1]$ we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.