Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel

TL;DR

This paper proves the global stability of a time-dependent monotone shear flow in the 2D Boussinesq equations within a finite channel, establishing conditions for stability and deriving enhanced dissipation and damping estimates.

Contribution

It introduces new stability thresholds for the Boussinesq system around time-dependent shear flows using sharp resolvent and space-time estimates.

Findings

Global stability of the Boussinesq system around shear flow

Enhanced dissipation estimate of vorticity

Inviscid damping of velocity

Abstract

Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on B\'{e}nard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow $(U(t,y),0)$ with a constant background temperature $a\in\mathbb{R}$. The analysis is performed in the finite channel $\mathbb{T}\times[0,1]$ with non-slip boundary condition. By means of the sharp resolvent estimates and space-time estimates, we establish that the Boussinesq system admits a globally stable solution around the monotone shear flow, provided that the initial perturbation satisfies $\|u^{\mathrm{in}}\|_{H^2}\leq c\nu^{\frac12}, \|\langle D_x\rangle \theta^{\mathrm{in}}\|_{L^2} \leq c\nu^{\frac56}$. Moreover, we derive the enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity.