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

Abstract

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 $\Theta$ 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 $\Theta$.

Updated version

References in corpus (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Long time dynamics of forced critical SQG
• Decay of weak solutions to the 2D dissipative quasi-geostrophic equation
• Absence of anomalous dissipation of energy in forced two dimensional fluid equations

Cited by in corpus (1)

• On the forced surface quasi-geostrophic equation: Existence of steady states and sharp relaxation rates