Stability threshold for two-dimensional Boussinesq systems near-Couette shear flow in a finite channel

TL;DR

The paper analyzes stability thresholds for the two-dimensional Navier–Stokes Boussinesq system in a finite channel near the Couette flow under Navier slip boundaries. It first proves linear stability (Theorem 1.1) using modewise energy functionals and a singular integral operator to obtain enhanced dissipation and inviscid damping, with decay rates tied to the dissipation coefficients $\mu$ and $\nu$. It then establishes nonlinear stability (Theorem 1.2) for full dissipation ($\sigma=1$) under anisotropic smallness of the initial data, achieving global well-posedness and uniform decay while significantly improving the temperature perturbation threshold to $1$, and demonstrating decoupled damping mechanisms for viscosity and diffusivity. The approach combines per-mode coercive energies, careful nonlinear estimates, and a bootstrap argument, extending prior bounded-domain results and clarifying the interplay between enhanced dissipation and inviscid damping in a finite-channel setting.

Abstract

In this paper, we investigate the stability threshold problem of the two-dimensional Navier-Stokes Boussinesq(NSB) equations in a finite channel $ \T \times [-1,1]$, focusing on the stability around the near Couette shear flow $ (U(y), 0)$, assuming the Navier slip boundary conditions are satisfied. In particular, when the initial data for the vorticity resides in an anisotropic Sobolev space of size $ O(\min \{ μ^{\frac{1}{2}}, ν^{\frac{1}{2}}\})$, and the initial perturbation of the temperature resides in an anisotropic Sobolev space of size $ O(\min \{ μ, ν\})$, we derive the nonlinear enhanced dissipation effect and the inviscid damping effect for the NSB system.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
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