Transition threshold of Couette flow for 2D Boussinesq equations

Abstract

In this paper, we prove the stability threshold of $\alpha \le \frac{1}{3}$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number ${\gamma }^2>\frac{1}{4}$ and different viscosity $\nu$ and thermal diffusivity $\mu$. More precisely, if $\Vert v_{in}-(y,0){\Vert }_{H^{s+1/2}}+\Vert {\rho }_{in}+{\gamma }^{2}y-1{\Vert }_{H^{s+1/2}}\le c(\min \{\nu ,\mu \}{)}^{1/3}$, $\frac{\nu +\mu }{2\gamma \sqrt{\nu \mu }}<2-\epsilon$, $s>3/2$, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp.

32 pages