On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows

TL;DR

This paper establishes linear stability and enhanced dissipation for Kolmogorov flows in 2D Navier-Stokes equations with small viscosity, using resolvent analysis and an abstract framework, confirming numerical conjectures.

Contribution

It provides a rigorous proof of enhanced dissipation and stability estimates for Kolmogorov flows, introducing a new analytical framework for resolvent estimates in this context.

Findings

Enhanced dissipation occurs on the time scale O(ν^{-1/2})

Linear stability is confirmed for small viscosity ν

Abstract framework for resolvent estimates is developed

Abstract

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $\nu$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(\nu^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(\nu^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.