Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and stratified temperature

Abstract

This paper examines the stability threshold at high Reynolds numbers $\textbf{Re}$ for the three-dimensional Boussinesq equations with rotation on the domain $\Omega=\{(x,\,y,\,z)\in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\}$ around the Couette flow $(y,0,0)$ and the vertically stratified temperature $\Theta_s=1+\alpha^2 z$. For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy $\left\|u_{\mathrm{in}}\right\|_{H^{N+2}\cap W^{N+3,1}}+\left\|\theta_{\mathrm{in}}\right\|_{H^{N+1}\cap W^{N+3,1}}<\delta \mathbf{Re}^{-\frac{14}{15}}$, for any $N\geq 11$ and some $\delta>0$ independent of $\mathbf{Re}$, then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.