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

TL;DR

This paper establishes the nonlinear stability threshold for 2D shear flows in the Boussinesq system near Couette flow, accommodating different scaling regimes of viscosity and heat diffusion without requiring equal viscosity and thermal diffusivity.

Contribution

It proves the nonlinear stability of shear flows under more general conditions on viscosity and thermal diffusivity than previously considered.

Findings

Nonlinear stability proven for shear flows with small or fixed alpha.

Stability results hold without requiring equal viscosity and thermal diffusivity.

Analysis covers different scaling regimes of viscosity and heat diffusion.

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 $\alpha\geq0$. We separate two cases: one is $\alpha\geq 0$ 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.