  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.
