Global regularity of semi-critical case of anisotropic quasi-geostrophic equations in Sobolev spaces

In this paper, we consider the following anisotropic quasi-geostrophic equations \begin{equation}\tag*{$(AQG{)}_{\alpha ,\beta }$} \partial_tθ+ u_θ.\nablaθ+μ|\partial_1|^{2α}θ+ν|\partial_2|^{2β}θ=0,\quad u_θ=$\mathcal{R}$^{\perp}θ, \end{equation} where $\min \{\alpha ,\beta \}=\frac{1}{2}$ et $\max \{\alpha ,\beta \}\in (\frac{1}{2},1)$. This equation is a particular case of the equation introduced by Ye (2019) in \cite{YZ}. In this paper, we prove that for any initial data ${\theta }^0$ in the Sobolev space $H^s({\mathbb{R}}^2)$, $s>1$, the equation $(AQG{)}_{\alpha ,\beta }$ has a global solution $\theta$ in $C_b({\mathbb{R}}^{+},H^s({\mathbb{R}}^2))\mathrm{.}$