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

Abstract: We establish a stability threshold for the 3D Boussinesq equations around Couette flow $(y,0,0)$ with linearly stratified temperature profile $1+y$ on $\mathbb{T}\times\mathbb{R}<sup>2$, with distinct viscosity $κ$ and thermal diffusivity $μ$ and arbitrary Brunt-Väisälä frequency $β&gt;0$. Assume that $μ$ and $κ$ are comparable, and set $ν=\min{κ,μ}$. For sufficiently small (a_0>0) and (s\geq4), we prove that if the initial perturbation satisfies $$|(u_{in},θ<em>{in})|</em>{H^{s}+|(u_{0,in},θ<em>{0,in})|</em>{W^{{s,1}}\lesssimν^{{\frac{2}{3}}}}} |\lnν|<sup>{-2-2a_0}$$, 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 $ν<sup>{-1/6}$ in the high-regularity norm of the zero mode of the first component $u<sup>1_0$.