Stability of Solutions to the Quasi-Geostrophic Equations in ${\mathbb{R}}^2$

We consider the stationary Quasi-Geostrophic equation in the whole space ${\mathbb{R}}^2$ driven by a force $f$. Under certain smallness assumptions of $f$, we establish the existence of solutions with finite $L^2$ norm. This solution is unique among all solutions with finite energy. The unique solution $\mathrm{\Theta }$ is also shown to be stable in the sense: any solution of the evolutionary Quasi-Geostrophic equation driven by $f$ and starting with finite energy, will return to $\mathrm{\Theta }$.