Stability threshold for 2D shear flows of the Boussinesq system near Couette

Abstract

In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain $\mathbb{T}\times \mathbb{R}$. The main goal is to prove the nonlinear stability of the shear flow $(U^S,{\Theta }^S)=((e^{\nu t{\partial }_{yy}}U(y),0{)}^{\top },\alpha y)$ with $U(y)$ close to $y$ and $\alpha \ge 0$. We separate two cases: one is $\alpha \ge 0$ small scaling with the viscosity coefficients and the case without smallness of $\alpha$ and fixed heat diffusion coefficient. The novelty here is that we don't require $\mu =\nu$ and only need to assume that $\mu$ is scaled with $\nu$ or fixed, where $\mu$ is the inverse of the Reynolds number and $\nu$ is the heat diffusion coefficient.

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