Improved stability threshold of the Two-Dimensional Couette flow for Navier-Stokes-Boussinesq Systems via quasi-linearization

TL;DR

This paper enhances the stability threshold for the 2D Couette flow in stratified fluids governed by Navier-Stokes-Boussinesq equations, using quasi-linearization to improve perturbation size requirements.

Contribution

It introduces a quasi-linearization approach to improve the perturbation size threshold for asymptotic stability of Couette flow in stratified fluids.

Findings

Perturbation size requirement improved to 9/3 from 9/6.

Decomposition of the main system into linear and nonlinear/quasi-linear equations.

Application of quasi-linearization method to stability analysis.

Abstract

In this paper, we improve the size requirement of the perturbations for the asymptotic stability of the Couette flow in stratified fluids governed by the two-dimensional Navier-Stokes-Boussinesq system. More precisely, the size of perturbed temperature is improved to $\nu^{2/3}$ from $\nu^{5/6}$ in the paper of Zhang and Zi [J. Math. Pure. Anal. 179:123-182 (2023)]. The idea is the quasi-linearization. The main system is decomposed into two or more equations: a good equation (might be linear) that carries the regularity and size of the initial data and some quasi-linear and nonlinear equations that contain the nonlinear part, which start from zero initial data.