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arXiv · 2610.12810

Stability threshold of the Couette flow in 2D Navier-Stokes Boussinesq equations on $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$

Nangao Zhang·Gengwei Zhong·Ruizhao Zi
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Abstract

In this paper, we study the transition threshold problem for the 2D Navier-Stokes Boussinesq equations around the Couette flow with a constant background temperature in two domains: $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. It was proved that if the sizes of initial perturbations of the vorticity and the temperature in Sobolev spaces are $ν^{\fr13}$ and $ν^{\fr56}$, respectively, then this system admits a globally stable solution. Besides, we derive the enhanced dissipation estimate for the vorticity and temperature, as well as the inviscid damping estimate for the velocity. The coupling in the equations poses some difficulties to the analysis. We overcome these challenges and establish the results via the quasi-linear method and energy estimates.

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Nangao Zhang, Gengwei Zhong, Ruizhao Zi. 2026-10-08. Stability threshold of the Couette flow in 2D Navier-Stokes Boussinesq equations on $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. https://arxiv.org/abs/2610.12810

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