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For $\mathbb{R}^2$, the stability of smooth solutions of 2D anisotropicBoussinesq equations with horizontal dissipation is an open problem. In thiswork, 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 theregularity index $\frac 12<s<1$ is extended to an integer $1\le s$ and theelementary techniques are used.

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