Global solution of anisotropic Quasi-Geostrophic Equations in Sobolev Space

TL;DR

This paper proves the existence of global solutions for anisotropic quasi-geostrophic equations in Sobolev spaces with specific regularity, extending previous results by utilizing Gevrey-class regularity techniques.

Contribution

It extends the known global existence results for these equations to a broader Sobolev space range with additional parameter conditions.

Findings

Global solutions exist in Sobolev spaces with 2-2α< s < 2.

Solution regularity is shown to be Gevrey-class near zero.

Results depend on specific conditions on parameters α and β.

Abstract

In \cite{YZ}, the author proved the global existence of the two-dimensional anisotropic quasi-geostrophic equations with condition on the parameters $\alpha,$ $\beta$ in the Sobolev spaces $H^s( \R^2)$; $s\geq 2$. In this paper, we show that this equations has a global solution in the spaces $H^s(\R^2)$, where $\max\{2-2\alpha,2-2\beta\}< s<2$, with additional condition over $\alpha$ and $\beta$. The proof is based on the Gevrey-class regularity of the solution in neighborhood of zero.