Stability threshold of Couette flow for 3D Boussinesq system in Sobolev spaces
Shikun Cui, Lili Wang, Wendong Wang
TL;DR
The paper addresses nonlinear stability of Couette flow in the 3D Boussinesq system within Sobolev spaces on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$ at high Reynolds number and small thermal diffusion. It introduces a reformulated perturbation framework, a carefully designed energy functional, and a bootstrap argument to prove global-in-time solutions under small initial data with scales $\|v_{\rm in}-(y,0,0)\|_{H^{2}}\leq \varepsilon\nu$ and $\|\theta_{\rm in}\|_{H^{2}}\leq \varepsilon\nu^{2}$. A key novelty is the decomposition of zero/nonzero modes and the introduction of the good unknowns $V$ and $Q$, together with a quasi-linearization strategy to suppress the lift-up effect and capture enhanced dissipation and inviscid damping through a hierarchy of energies $E_{1},\dots,E_{7}$. The resulting estimates bound nonlinear interactions across all mode interactions (zero–zero, zero–nonzero, nonzero–nonzero) and yield global stability, highlighting how temperature coupling and shear interact to maintain stability in Sobolev norms. This advances understanding of stability thresholds for 3D Boussinesq flows and provides a rigorous framework for analyzing perturbations of shear-driven convection.
Abstract
In this paper, we investigate the nonlinear stability and transition threshold for the 3D Boussinesq system in Sobolev space under the high Reynolds number and small thermal diffusion in $\mathbb{T}\times\mathbb{R}\times\mathbb{T} $. It is proved that if the initial velocity $v_{\rm in}$ and the initial temperature $ θ_{\rm in} $ satisfy $ \|v_{\rm in}-(y,0,0)\|_{H^{2}}\leq \varepsilonν, \|θ_{\rm in}\|_{H^{2}}\leq \varepsilonν^{2} $, respectively for some $ \varepsilon>0 $ independent of the Reynolds number or thermal diffusion, then the solutions of 3D Boussinesq system are global in time.