On the stability threshold of Couette flow for 2D Boussinesq equations

Abstract

In this paper, we prove the stability threshold $\frac{1}{2}$ for 2D Boussinesq equations around the Couette flow in $T\times R$ with Richardson number ${\gamma }^2>\frac{1}{4}$. Here the viscosity $\nu$ and thermal diffusivity $\mu$ can be different. More precisely, if ${\Vert v_{in}-(y,0)\Vert }_{H^{s+1/2}}+{\Vert {\rho }_{in}+{\gamma }^{2}y-1\Vert }_{H^{s+1/2}}\le c{(min\{\nu ,\mu \})}^{1/2}$, $\frac{\nu +\mu }{2\gamma \sqrt{\nu \mu }}<2-\varepsilon$, $s>1/2$, then the asymptotic stability holds. Compared with Zhai and Zhao (2023), the regularity assumption is weaker, and the proof is much simpler.