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

In this article we consider the following generalized quasi-geostrophic equation∂tθ+u⋅∇θ+νΛβθ=0,u=ΛαR⊥θ,x∈R2, where ν>0ν>0, Λ:=−Δ, α∈]0,1[α∈]0,1[ and β∈]0,2[β∈]0,2[. We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case α∈]0,1[α∈]0,1[ and β∈]α+1,2]β∈]α+1,2] we obtain the global well-posedness of the system with smooth initial data; and for the case α∈]0,1[α∈]0,1[ and β∈]2α,α+1]β∈]2α,α+1] we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.