The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on $\mathbb{T}\times {\mathbb{R}}^2$

We establish a stability threshold for the 3D Boussinesq equations around Couette flow $(y,0,0)$ with linearly stratified temperature profile $1+y$ on $\mathbb{T}\times {\mathbb{R}}^2$, with distinct viscosity $\kappa$ and thermal diffusivity $\mu$ and arbitrary Brunt-Väisälä frequency $\beta >0$. Assume that $\mu$ and $\kappa$ are comparable, and set $\nu =\min \{\kappa ,\mu \}$. For sufficiently small $a_0>0$ and $s\ge 4$, we prove that if the initial perturbation satisfies $\mathrm{\parallel }(u_{in},{\theta }_{in}){\mathrm{\parallel }}_{H^s}+\mathrm{\parallel }(u_{0,in},{\theta }_{0,in}){\mathrm{\parallel }}_{W^{s,1}}\lesssim {\nu }^{\frac{2}{3}}\mathrm{\mid }\ln \nu {\mathrm{\mid }}^{-2-2a_0}$, then the corresponding solution exists globally in time. We also establish Strichartz-type estimates for zero mode. Moreover, our estimates further allow for a nonlinear amplification of order ${\nu }^{-1\mathrm{/}6}$ in the high-regularity norm of the zero mode of the first component $u_0^1$.