Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

In this paper we consider the following modified quasi-geostrophic equation \partial_{t}θ+u\cdot\nablaθ+ν|D|^αθ=0, \quad u=|D|^{α-1}\mathcal{R}^{\bot}θ,\quad x\in\mathbb{R}^2 with $ν>0$ and $α\in ]0,1[\,\cup \,]1,2[$. When $α\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. As a byproduct, we also show that for every $α\in ]0,2[$, the Lipschitz norm of the solution has a uniform exponential bound.

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