On the boundedness of the global solution of anisotropic quasi-geostrophic equations in Sobolev space

TL;DR

This paper proves that solutions to a specific anisotropic quasi-geostrophic equation remain bounded in Sobolev spaces over time, ensuring stability and regularity of the solutions.

Contribution

It establishes uniform Sobolev space boundedness for global solutions of an anisotropic quasi-geostrophic equation with fractional dissipation.

Findings

Solutions are bounded in Sobolev spaces uniformly over time.

The result applies to equations with fractional horizontal dissipation and vertical thermal diffusion.

Ensures stability and regularity of solutions in the studied model.

Abstract

In this paper, we show that the global solution of the surface anisotropic two-dimensional quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion established by the author in [2] is bounded in Sobolev spaces uniformly with respect to time.