Stability for the Two‐Dimensional Boussinesq Equations With Fractional Horizontal Dissipation and Thermal Diffusion

In this paper, we focus on the stability of perturbation near a steady state of the two‐dimensional (2D) Boussinesq system with fractional horizontal dissipation and thermal diffusion. In 2D full space R2$\mathbb {R}^2$, by virtue of the absence of vertical dissipation, the stability remains open in the Sobolev framework. When the domain is T×R$\mathbb {T}\times \mathbb {R}$, where T=[0,1]$\mathbb {T}=[0,1]$ is a periodic interval, we obtain the stability in the Sobolev space H2$H^2$. Furthermore, in view of the presence of fractional operators, some of the standard energy estimate techniques no longer work. To overcome these difficulties, we resort to some new anisotropic interpolation inequalities and the strong Poincaré‐type inequalities involving fractional derivatives. Moreover, the oscillatory part of the solution is proven to converge exponentially to zero in H1$H^1$ as time goes to infinity. Our results extend some known ones and provide support for numerical simulation of Boussinesq equations with partial fractional dissipation.