Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

TL;DR

The paper addresses the linearized 2D Euler and Navier–Stokes dynamics around mixing-layer shear flows in ${\mathbb T}\times {\mathbb R}$ by casting the problem into a Hamiltonian framework via a conjugate operator. It proves quantitative local inviscid damping uniform in small viscosity using a Mourre-type spectral analysis of a self-adjoint Hamiltonian $H$, together with a hypocoercivity-based argument for the viscous case. The main contributions are (i) a local, time-decay estimate for the transformed vorticity that translates into decay of velocity components, (ii) a uniform-in-$\nu$ extension of this decay up to the viscous time scale, and (iii) a local enhanced dissipation on time scales $t\sim \nu^{-1/3}$ after spectral localization, established through a Lyapunov-type functional. These results advance understanding of how spectral stability and Hamiltonian structure control mixing and dissipation in non-Couette shear flows, with precise decay rates and timescales.

Abstract

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order $ν^{-1/3}$, $ν$ being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.

Theorems & Definitions (16)

• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Lemma 2.4
• Corollary 2.5
• Theorem 2.1
• Corollary 2.6
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1