# Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation - Mathematics > Analysis of PDEs

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Abstract: In this paper we consider the following modified quasi-geostrophic equation\partial {t}\theta+u\cdot abla\theta+ u |D|^{\alpha}\theta=0,\quad u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta,\quad x\in\mathbb{R}^2 with$ u>0$ and $\alpha\in 0,1\,\cup \,1,2$. When $\alpha\in0,1$, theequation was firstly introduced by Constantin, Iyer and Wu in \cite{refConstanIW}. Here, by using the modulus of continuity method, we prove theglobal well-posedness of the system with the smooth initial data. As abyproduct, we also show that for every $\alpha\in 0,2$, the Lipschitz norm ofthe solution has a uniform exponential bound.

Author: ** Changxing Miao, Liutang Xue**

Source: https://arxiv.org/