Nonlinear asymptotic stability and transition threshold for 2D Taylor-Couette flows in Sobolev spaces

TL;DR

This paper proves the asymptotic stability of 2D Taylor-Couette flow in Sobolev spaces at high Reynolds numbers, revealing optimal decay rates and the influence of rotation on enhanced dissipation.

Contribution

It establishes the first rigorous proof of nonlinear asymptotic stability for 2D Taylor-Couette flow with explicit decay rates and introduces new eigenfunction bases for analyzing inviscid damping.

Findings

Proves exponential decay of perturbations with optimal rates.

Shows rotation accelerates dissipation depending on the rotational coefficient.

Constructs explicit eigenfunction bases for circular flow Laplacians.

Abstract

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor-Couette (TC) flow for the incompressible Navier-Stokes equations. The explicit form of velocity for 2D TC flow is given by $u=(Ar+\frac{B}{r})(-\sin \theta, \cos \theta)^T$ with $(r, \theta)\in [1, R]\times \mathbb{S}^1$ being an annulus and $A, B$ being constants. Here, $A, B$ encode the rotational effect and $R$ is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor-Couette flow is asymptotically stable, even at high Reynolds number ($Re\sim \nu^{-1}$), with a sharp exponential decay rate of $\exp(-\nu^{\frac13}|B|^{\frac23}R^{-2}t)$ as long as the initial perturbation is less than or equal to $\nu^\frac12 |B|^{\frac12}R^{-2}$ in Sobolev space. The powers of $\nu$ and $B$ in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient $B$. The larger the $|B|$, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows.