A new stability result for the 2D Boussinesq equations with horizontal dissipation

TL;DR

This paper establishes a new stability result for smooth solutions of the 2D anisotropic Boussinesq equations with horizontal dissipation in rougher function spaces, extending previous results to lower regularity levels.

Contribution

It provides a partial stability result in $H^{0,s}( ext{R}^2)$ spaces, extending prior stability results from fractional to integer regularity indices using elementary techniques.

Findings

Stability of solutions in rougher function spaces is demonstrated.

Results extend stability from fractional to integer regularity indices.

Elementary techniques are employed for the analysis.

Abstract

For $\mathbb{R}^2$, the stability of smooth solutions of 2D anisotropic Boussinesq equations with horizontal dissipation is an open problem. In this work, we present a partial answer to this problem in a rougher function space $H^{0,s}(\mathbb{R}^2)$. Moreover, the previous stability results with the regularity index $\frac 12<s<1$ is extended to an integer $1\le s$ and the elementary techniques are used.