a finite periodic channel $\T
      \times [0,1]$ with impermeable walls. Here, boundary effects impose strong limits on the achievable regularity of the linearized problem to fractional Sobolev spaces and as consequence also on the nonlinear problem. In [@Zill5], these results are extended to weighted Sobolev spaces, following the works of [@Zhang2015inviscid] and the setting of circular flows. Here, in order to avoid degeneracies results are limited to annuli $B_{R_{2}}\setminus B_{R_{1}}$ with $0<R_{1}<R_{2}<\infty$.

-   In [@Zhang2015inviscid], a very different spectral approach is used instead to establish linear inviscid damping under different/weaker conditions. Recently, these methods have further been able to establish similar results for flows close to Poiseuille flow [@WZZvorticitydepl] and for the Kolmogorov flow [@WZZkolmogorov].

The main motivating example of this work is given by perturbations of the velocity profile $\frac{1}{r}e_{\theta}$ on $\R^{2}\setminus \{0\}$ of a point vortex at the origin. While for the unperturbed problem, solutions are explicit, for perturbations the degenerate behavior as $r$ tends to $0$ or $\infty$ makes the question of (asymptotic) stability and damping mathematically very challenging.

As the main results of this article, we establish linear inviscid damping, stability and scattering for a class of *mildly degenerate* flows (c.f. Section \[sec:mildly\_degen\]). This class includes 