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

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.