Existence and Uniqueness of the Solution to the Anisotropic Quasi-Geostrophic Equations in the Sobolev Space

TL;DR

This paper proves the local and global existence and uniqueness of solutions to a generalized anisotropic quasi-geostrophic equation in Sobolev spaces, and analyzes their long-term behavior.

Contribution

It establishes the existence, uniqueness, and global behavior of solutions in critical Sobolev spaces for a generalized quasi-geostrophic model.

Findings

Local existence and uniqueness in Sobolev space

Global solutions for small initial data

Asymptotic behavior of solutions at infinity

Abstract

In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2\alpha}(\mathbb{R}^2)\cap H^{2-2\beta}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of our global solution in infinity.