Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow

TL;DR

This work addresses the Sobolev stability of 2D dissipative fluid equations near Couette flow on $\mathbb{T}\times\mathbb{R}$ by developing a general framework that bounds nonlinear echoes through a linear stability analysis using adapted variables and time-dependent Fourier weights. The authors introduce a structured decomposition of nonlinear interactions into reaction, transport, remainder, and average terms, and prove a central nonlinear bound via precise energy estimates with $A$-type weights and resonance controls. The method yields sharp thresholds for several models: a $\mu^{1/3}$-type stability threshold (with logarithmic losses in the borderline case) for Boussinesq around Couette, and a $\;\mu^{1/3^+}$ threshold for MHD under certain viscosity/resistivity regimes, while recovering the NS threshold. The approach relies on careful frequency-set analysis, resonance weights, and enhanced dissipation to suppress echo chains, providing a unified, quantitative Sobolev-stability framework with potential applicability to broader shear-flow problems.

Abstract

We study the Sobolev stability thresholds of 2d dissipative fluid equations around Couette flow on the domain $\mathbb T\times \mathbb R$. We prove a bound for general nonlinear interactions, which, for several fluid equations, reduces the proof of nonlinear stability to a linear stability analysis. We apply this approach to the examples of Navier-Stokes, Boussinesq and magnetohydrodynamic equations around Couette flow. This improves the Sobolev stability threshold for the Boussinesq equations around Couette flow and large affine temperature to $1/3$ and for the MHD equations around Couette flow and constant magnetic field to $1/3^+$.

Theorems & Definitions (24)

• Theorem 1: Thresholds of selected fluid systems
• Theorem 2
• Remark 1
• Remark 2
• Definition 3.1
• Lemma 3.1: Properties of $m_\gamma$
• Lemma 3.2: Properties of $M_\mu$
• Lemma 3.3
• proof
• Theorem 3: Boussinesq part of Theorem \ref{['thm:thres']}