Nonlinear asymptotic stability of 2D Taylor-Couette flow in the exterior disk

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 previous work \cite{LZZ-25}, even space-time coupled exponential decay can not 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 $\frac12$, which is the same as that for the Taylor-Couette flow in the bounded region.