Enhanced dissipation and nonlinear asymptotic stability of the Taylor-Couette flow for the 2D Navier-Stokes equations

TL;DR

This paper proves the nonlinear asymptotic stability of 2D Taylor-Couette flow under large perturbations, utilizing enhanced dissipation effects and weighted norms to improve transition thresholds and handle unbounded initial energy.

Contribution

It introduces new weighted $L^2$ norms and analysis techniques to demonstrate enhanced dissipation and stability for Taylor-Couette flow with unbounded initial energy.

Findings

Proved asymptotic stability of Taylor-Couette flow for large perturbations.

Established sharp resolvent and decay estimates for the flow.

Allowed the outer cylinder to tend to infinity, handling unbounded initial energy.

Abstract

In this paper, we study the nonlinear stability of a steady circular flow created between two rotating concentric cylinders. The dynamics of the viscous fluid are described by 2D Navier-Stokes equations. We adopt scaling variables. For the rescaled equations, we prove that the steady flow (Taylor-Couette flow) is asymptotically stable up to a large perturbation of initial data. Back to the original 2D Navier-Stokes equations, this implies an improved transition threshold for the Taylor-Couette flow. The improvement is due to enhanced dissipation and new observations and constructions of weighted $L^2$ norms, which capture a hidden structure between the viscosity constant $\nu$ and (different) rotating speeds and locations of two coaxial cylinders. In particular, we allow the location of the outer cylinder to tend to infinity, which renders the initial fluid kinetic energy not uniformly bounded. Due to enhanced-dissipation effect, we also establish a sharp resolvent estimate, desired space-time bounds and optimal decaying estimates, which lead to the proof of nonlinear asymptotic stability of 2D Taylor-Couette flow.