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

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 $\Vert {\theta }_0{\Vert }_{L^{\infty }(\Omega )}$ is small.