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 $\| \mathbf{v}^{\mathrm{in}} -(y,0)\|_{H^2} \le \varepsilon_0 \nu^{\frac12}$, and $\| \rho^{\mathrm{in}}-1 \|_{H^1} + \big\| |\partial_x|^{\frac13} \rho^{\mathrm{in}} \big\|_{H^1} \le \varepsilon_1 \nu^{\frac56}$. 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 $\nu^{\frac56}$.