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

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.