Remark on Stability of Scale-Critical Stationary Flows in a Two-Dimensional Exterior Disk

Abstract

We prove the asymptotic stability of stationary flows in a two-dimensional exterior disk under suitable decay and smallness conditions on the stationary flows and the initial perturbations. The class of stationary flows considered in this paper includes some typical circular flows which decay in the scale-critical order $O(|x|^{-1})$ as $|x|\rightarrow \infty$.

Appendix

Here, we state some basic estimates for the modified Bessel functions. For the proof we refer to [1]; see also [7, Lemmas A.2, A.3].

Lemma 6.

(i) Let $\mathfrak {R}(\mu ) >0$. Then for any $M>0$ there is a constant $C\,{=}\,C(M,\mu )>0$ such that if $|z|\le M$ then

$$\begin{aligned} | I_\mu (z) | \le C |z|^{\mathfrak {R}(\mu )}\,,~~~~~~~~~~ | K_\mu (z) |\le C |z|^{-\mathfrak {R}(\mu )}\,. \end{aligned}$$

(96)

(ii) Let $\varepsilon \in (0, \frac{\pi }{2})$. Then there is a constant $C=C (\varepsilon ,\mu )>0$ such that if $z\in \Sigma _{\frac{\pi }{2}-\varepsilon }$ and $|z|\ge 1$ then

$$\begin{aligned} |I_\mu (z) | \le C |\mathfrak {R}(z)|^{-\frac{1}{2}} e^{\mathfrak {R}(z)}\,,~~~~~~~ |K_\mu (z)|\le C |\mathfrak {R}(z)|^{-\frac{1}{2}} e^{-\mathfrak {R}(z)}\,. \end{aligned}$$

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Moreover, for $|z|\gg 1$, the following expansion holds.

$$\begin{aligned} {\begin{matrix} &{} I_\mu (z) = \frac{1}{\sqrt{2\pi z}} e^{z} \big (1 + l_\mu (z) \big )\,,~~~~~~~ K_\mu (z) = \sqrt{\frac{\pi }{2z}} e^{-z} \big ( 1+ k_\mu (z) \big )\,,\\ &{} | l_\mu (z) | + | k_\mu (z) | \le C |z|^{-1}\,. \end{matrix}} \end{aligned}$$

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