In this paper, we consider the asymptotic stability of the 2D Taylor-Couette flow in the exterior disk, with a small kinematic viscosity \( \nu \ll 1 \) and a large rotation coefficient \( |B| \) . Due to the degeneracy of the Taylor-Couette flow at infinity, we cannot expect the solution to decay exponentially in a space-time decoupled manner. As stated in a previous work (Li et al. in Linear enhanced dissipation for the 2D Taylor-Couette flow in the exterior region: A supplementary example for Gearhart-Pr \(\ddot{u}\) ss type lemma. arXiv:2501.14187), even space-time coupled exponential decay cannot be expected, and at most, we can obtain space-time coupled polynomial decay. To handle the space-time coupled decay multiplier, the previous time-independent resolvent estimate methods no longer work. Therefore, this paper introduces time-dependent resolvent estimates to deal with the space-time coupled decay multiplier \( \Lambda _k \) . We remark that the choice of \( \Lambda _k \) is not unique, here we just provide one way to construct it. Finally, as an application, we derive a transition threshold bound of \(\frac{1}{2}\) , which is the same as that for the Taylor-Couette flow in the bounded region.
