标题： 二维Boussinesq方程的库特流转换阈值 显示英文标题

标题： Transition threshold of Couette flow for 2D Boussinesq equations

摘要： 本文中，我们证明了在带有理查森数为$\gamma^2>\frac14$的库特流$\mathbb{T}\times \mathbb{R}$上，二维Boussinesq方程关于$\alpha\leq \frac13$的稳定性阈值，其中粘性系数为$\nu$且热扩散率为$\mu$。 更精确地说，如果 $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|\rho_{in}+\gamma^2 y-1\|_{H^{s+1/2}}\leq c(\min\{\nu,\mu\})^{1/3}$, $\frac{\nu+\mu}{2\gamma\sqrt{\nu \mu} }< 2-\varepsilon$, $s>3/2$，则渐近稳定性成立。 这一稳定性阈值与二维Sobolev空间中Navier-Stokes方程的最佳稳定性阈值一致。 并且在无粘阻尼效应的意义下，初始数据的正则性假设应是尖锐的。 显示英文摘要

摘要： In this paper, we prove the stability threshold of $\alpha\leq \frac13$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number $\gamma^2>\frac14$ and different viscosity $\nu$ and thermal diffusivity $\mu$. More precisely, if $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|\rho_{in}+\gamma^2 y-1\|_{H^{s+1/2}}\leq c(\min\{\nu,\mu\})^{1/3}$, $\frac{\nu+\mu}{2\gamma\sqrt{\nu \mu} }< 2-\varepsilon$, $s>3/2$, 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.