Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel

TL;DR

This paper proves the asymptotic stability of the two-dimensional Boussinesq system around the Couette flow in a finite channel, showing solutions remain close to the flow and temperature approaches equilibrium under small initial perturbations.

Contribution

It establishes the stability and convergence of solutions for the 2D Boussinesq system near Couette flow with small viscosity and thermal diffusion, a novel stability result in this setting.

Findings

Velocity remains close to Couette flow and converges as time goes to infinity.

Temperature stays near constant and approaches 1 over time.

Results hold for small initial perturbations proportional to viscosity and thermal diffusion.

Abstract

In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $\nu$ and small thermal diffusion $\mu$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},\rho_{in})$ satisfies $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{\nu,\mu\}^{\f12}$ and $\|\rho_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{\nu,\mu\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $\nu, \mu$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min\{\nu,\mu\}^{\f12})$ of the Couette flow, and approaches to Couette flow as $t\to\infty$; the temperature remains within $O(\min\{\nu,\mu\}^{\f{11}{12}})$ of the constant $1$, and approaches to $1$ as $t\to\infty$.