Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG)

$$\begin{aligned} \theta _t \;\!+\, (-\Delta )^{\alpha }\,\theta \,+\, u_{\theta } \cdot \nabla \theta \;=\; 0, \quad \hbox {with } u_{\theta }\;=\;(\partial _2(-\Delta )^{\frac{\gamma -2}{2}}\theta , -\partial _1(-\Delta )^{\frac{\gamma -2}{2}}\theta ), \end{aligned}$$

in the non-homogenous Sobolev space \(H^{1+\gamma -2\alpha }(\mathbb {R}^2)\), where \(\alpha \in (0,\frac{1}{2})\) and \(\gamma \in (1,2\alpha +1)\). To this end, we need consider that the initial data for this equation are small. More precisely, we assume that \(\Vert \theta _0\Vert _{H^{1+\gamma -2\alpha }}\) is small enough in order to obtain a unique \(\theta \in C([0,\infty );H^{1+\gamma -2\alpha }(\mathbb {R}^2))\) that solves (MQG) and satisfies the following limit:

$$\begin{aligned} \lim _{t\rightarrow \infty } \Vert \theta (t)\Vert _{H^{1+\gamma -2\alpha }}=0. \end{aligned}$$