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

TL;DR

This paper establishes the stability threshold for the 2D Boussinesq system near Couette flow in an infinite channel, showing improved temperature stability bounds compared to previous finite channel results.

Contribution

It proves the asymptotic stability of Couette flow in an infinite channel with sharper temperature stability thresholds than prior finite channel studies.

Findings

Couette flow is asymptotically stable under specified initial perturbations.

Stability threshold for temperature improved from ν^{11/12} to ν^{5/6}.

Results extend stability analysis to infinite channel setting.

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 $\nu^{\frac56}$.