Transition threshold of Couette flow for 2D Boussinesq equations

2025/06/04 by Xiaoxia Ren, Dongyi Wei, Ren, Xiaoxia +1 · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2506.03679

Abstract

In this paper, we prove the stability threshold of α≤ \frac13 for 2D Boussinesq equations around the Couette flow in \mathbbT× ℝ with Richardson number γ²>\frac14 and different viscosity ν and thermal diffusivity μ. More precisely, if ‖v_in-(y,0)‖_H^s+1/2+ ‖ρ_in+γ² y-1‖_H^s+1/2≤ c(min\ν,μ\)^1/3, (ν+μ)/(2γ√(νμ) )< 2-ε, s>3/2, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp.

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