Transition-threshold conjecture for high-Reynolds-number laminar-flow stability

Determine whether the stability transition threshold exponent for high-Reynolds-number laminar flows is at most one, namely, whether there exists a threshold exponent \(\theta\leq 1\) such that perturbations of size at most \(\nu^{\theta}\) yield stability while perturbations much larger than \(\nu^{\theta}\) yield instability.

Background

The paper discusses the stability and transition to turbulence of laminar flows at high Reynolds number, with the viscosity parameter ν=Re−1\nu=Re^{-1}. For a prescribed norm, the transition threshold exponent θ\theta separates perturbation sizes associated with stability from those associated with instability. The cited conjecture asserts that this exponent should not exceed one. The paper reports this conjecture as part of the broader motivation for studying whether global existence can be obtained for initial perturbations larger than the classical viscosity-scale size.

References

Trefethen-Trefethen-Reddy-Driscoll first proposed a question: Given a norm |\cdot|_{X}, find a \theta=\theta (X) such that

\begin{aligned} &|u_0|{X}\le \nu{\theta}\Rightarrow~~ stability,\ &|u_0|{X}\gg \nu{\theta}\Rightarrow~~ instability, \end{aligned}

where the index \theta is referred to as the transition threshold in the applied literature, and it was conjectured in that \theta\le 1.

— Global solutions of compressible Navier-Stokes equations with small viscosity  (2608.17661 - Cai et al., 18 Aug 2026) in Section 1, Introduction

Theorem \ref{thm:k0} indicates that the correct conjecture is: for initial data $U_0$ with $|U_0-U{(0)}|_{Hs}\le_0$ small (in a suitable anisotropic space, uniformly in $a$), the solution of ans exists globally and converges, as $t\to\infty$, to a shear flow $(f_\infty(x_2),0)$ with $|f_\infty-a\sin x_2|$ small; moreover the non-shear part decays at the linear rate $e{-\nu k_{\min}2t}$.

ans:

{∂tU+U⋅∇U=−∇P+ν∂12U,∇⋅U=0,(x1,x2)∈TL×T2π,t>0,\begin{cases} \partial_t U+U\cdot\nabla U=-\nabla P+\nu\partial_1^2U,\\ \nabla\cdot U=0, \end{cases} \qquad (x_1,x_2)\in\mathbb T_L\times\mathbb T_{2\pi}, \quad t>0,

— Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation  (2609.34610 - Yang et al., 28 Sep 2026) in Section 5, subsection “Nonlinear stability modulo shears for $L<2\pi$”