Quasi-geostrophic equation in $\mathbb{R}^2$

TL;DR

This paper investigates the solvability of the quasi-geostrophic equation in two-dimensional space, establishing solutions in subcritical and critical cases, and discusses the existence of global attractors and boundary conditions.

Contribution

It introduces a novel approach to obtain solutions in the critical case as limits of subcritical solutions when the parameter approaches a critical value.

Findings

Existence of solutions in $L^p$ and $H^s$ spaces for subcritical cases.

Construction of solutions in the critical case as limits of subcritical solutions.

Existence of global attractors in the subcritical case.

Abstract

Solvability of Cauchy's problem in $\mathbb{R}^2$ for subcritical quasi-geostrophic equation is discussed here in two phase spaces; $L^p(\mathbb{R}^2)$ with $p> \frac{2}{2\alpha-1}$ and $H^s(\mathbb{R}^2)$ with $s>1$. A solution to that equation in critical case is obtained next as a limit of the $H^s$-solutions to subcritical equations when the exponent $\alpha$ of $(-\Delta)^\alpha$ tends to $\frac{1}{2}^+$. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is discussed in the paper. In section 7 we also discuss solvability of the critical problem with Dirichlet boundary condition in bounded domain $\Omega \subset \mathbb{R}^2$, when $\| \theta_0 \|_{L^\infty(\Omega)}$ is small.