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

TL;DR

This paper investigates the long-term behavior of shear flows in 2D Euler and Navier-Stokes equations, demonstrating inviscid damping and viscous dissipation using a conjugate operator approach and hypocoercivity.

Contribution

It introduces a novel application of the conjugate operator method to analyze linear inviscid damping and enhanced viscous dissipation in shear flows.

Findings

Proves uniform linear inviscid damping under spectral stability assumptions.

Establishes enhanced viscous dissipation at times of order / with small viscosity.

Employs a Hamiltonian approach combined with hypocoercivity techniques.

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 $\nu^{-1/3}$, $\nu$ being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schr\"odinger operators, combined with a hypocoercivity argument to handle the viscous case.