Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$

In this paper, we consider the following modified quasi-geostrophic equations ${\mathrm{\partial }}_t\theta +{\mathrm{\Lambda }}^{\alpha }\theta +u\vec{\mathrm{\nabla }}\theta =0$, $u=\Lambda ^{\alpha-1}$\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 $C({\mathbb{R}}^{+},H^1({\mathbb{R}}^2))\mathrm{.}$