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

In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely the Richardson number $γ^2>\frac{1}{4}$. Precisely, we prove that if the initial perturbation $(u_{in},\vartheta_{in})$ of the Couette flow $v_s=(y,0)$ and the linear temperature $ρ_s=-γ^2y+1$ satisfies $\|u_{in}\|_{H^{s+1}}+\|\vartheta_{in}\|_{H^{s+2}}\leq ε_0ν^{\frac{1}{2}}$, then the asymptotic stability holds.

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