Global well-posedness for a modified critical dissipative quasi-geostrophic equation

In this paper we consider the following modified quasi-geostrophic equation∂tθ+u⋅∇θ+ν|D|αθ=0,u=|D|α−1R⊥θ,x∈R2 with ν>0ν>0 and α∈]0,1[∪]1,2[α∈]0,1[∪]1,2[. When α∈]0,1[α∈]0,1[, the equation was firstly introduced by Constantin, Iyer and Wu (2008) in [11]. Here, by using the modulus of continuity method, we prove the global well-posedness of the system. As a byproduct, we also show that for every α∈]0,2[α∈]0,2[, the Lipschitz norm of the solution has a uniform exponential upper bound.