Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space H¹

In this paper, we consider the following modified quasi-geostrophic equations $\partial_t\theta +\Lambda^\alpha\theta +u\vec\nabla\theta =0$, $u=\Lambda ^{\alpha-1}\mathcal{R}^\perp(\theta)$ where $\alpha \in ]0,1[$ is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu in \cite{CIW} as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data $\theta_\ast$ in the Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global and smooth solution $\theta $ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$