Stability threshold of the Couette flow for Navier-Stokes Boussinesq system with large Richardson number ${\gamma }^2>\frac{1}{4}$

Abstract

In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely the Richardson number ${\gamma }^2>\frac{1}{4}$. Precisely, we prove that if the initial perturbation $(u_{in},{\vartheta }_{in})$ of the Couette flow $v_s=(y,0)$ and the linear temperature ${\rho }_s=-{\gamma }^{2}y+1$ satisfies $\Vert u_{in}{\Vert }_{H^{s+1}}+\Vert {\vartheta }_{in}{\Vert }_{H^{s+2}}\le {\epsilon }_0{\nu }^{\frac{1}{2}}$, then the asymptotic stability holds.