Linear inviscid damping and enhanced dissipation for the Kolmogorov flow

TL;DR

This work analyzes the linear dynamics around the Kolmogorov bar state on a torus, proving linear inviscid damping for the Euler equations and enhanced dissipation for the linearized Navier–Stokes equations in a unified framework. Central to the approach is the wave-operator method, which converts nonlocal, time-dependent operators into tractable local forms, enabling precise limiting-absorption and resolvent analyses for the Rayleigh equation. The authors construct explicit solution formulas and dual representations, derive sharp bounds on key kernels, and establish decay rates that confirm Bouchet–Morita predictions as well as Beck–Wayne conjectures. A notable contribution is the detailed W^{2,1} control of the integral kernels Ko and Ke, which underpins both linear damping and nonlinear enhanced dissipation arguments. Overall, the paper advances understanding of metastable, nearly inviscid flows near non-monotone shear, with implications for turbulence and long-time dynamics in 2D fluids.

Abstract

In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $ν^{\frac 23+}$, here $ν$ is the viscosity coefficient.

Theorems & Definitions (110)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Definition 3.1
• Definition 3.2