Stability threshold of Couette flow for 3D Boussinesq system in Sobolev spaces

In this paper, we investigate the nonlinear stability and transition threshold for the 3D Boussinesq system in Sobolev space under the high Reynolds number and small thermal diffusion in $\mathbb{T}\times \mathbb{R}\times \mathbb{T}$. It is proved that if the initial velocity $v_{\mathrm{i}\mathrm{n}}$ and the initial temperature ${\theta }_{\mathrm{i}\mathrm{n}}$ satisfy $\mathrm{\parallel }{v}_{\mathrm{i}\mathrm{n}}-(y,0,0){\mathrm{\parallel }}_{H^2}\le \epsilon \nu ,\mathrm{\parallel }{\theta }_{\mathrm{i}\mathrm{n}}{\mathrm{\parallel }}_{H^2}\le \epsilon {\nu }^2$, respectively for some $\epsilon >0$ independent of the Reynolds number or thermal diffusion, then the solutions of 3D Boussinesq system are global in time.