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

TL;DR

This paper proves the existence, uniqueness, and stability of finite-energy solutions to the stationary Quasi-Geostrophic equations in two-dimensional space under small forcing conditions.

Contribution

It establishes the first rigorous results on the stability and uniqueness of solutions to the stationary Quasi-Geostrophic equations in  with finite energy.

Findings

Existence of finite-energy solutions under small forcing.

Uniqueness of the solution among finite energy solutions.

Stability of the solution for the evolutionary equation.

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 $\Theta$.