Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel

Abstract

In this paper, we study the stability threshold of the two-dimensional Boussinesq equations around the Couette flow in an infinite channel $\mathbb{R}\times [-1,1]$ under no-slip boundary conditions. We prove that the Couette flow is asymptotically stable under initial perturbations satisfying $\Vert {\mathbf{v}}^{\mathrm{in}}-(y,0){\Vert }_{H^2}\le {\epsilon }_0{\nu }^{\frac{1}{2}}$, and $\Vert {\rho }^{\mathrm{in}}-1{\Vert }_{H^1}+\Vert |{\partial }_x{|}^{\frac{1}{3}}{\rho }^{\mathrm{in}}{\Vert }_{H^1}\le {\epsilon }_1{\nu }^{\frac{5}{6}}$. Compared with the work of Masmoudi, Zhai, and Zhao [J. Funct. Anal., 284 (2023), 109736], where the asymptotic stability of the 2D Navier-Stokes-Boussinesq system around Couette flow in a finite channel $\mathbb{T}\times [-1,1]$ was established, our result improves the stability threshold for the temperature from ${\nu }^{\frac{11}{12}}$ to ${\nu }^{\frac{5}{6}}$.

29pages