flows and introduce our notion of *mildly degenerate* flows. Our main results are then stated in Subsection \[sec:main-results\].

The linearized Euler equations {#sec:linEuler}
------------------------------

We consider the linearization of the 2D incompressible Euler equations $$\begin{aligned}
  \partial_{t} \omega + v \cdot \nabla \omega =0\end{aligned}$$ around

1.  A shear flow $v=(U(y), 0), \omega=-U'(y)$ on $\T \times I$, where $I$ is a possible infinite interval.

2.  A circular flow $v=u(r) r e_\theta, \omega =\frac{1}{r}\partial_r (r^{2}u(r))$ on $\R^2\setminus
      \{0\}$ or $\R^2\setminus B_{r_1}(0)$ or $B_{r_2}(0)\setminus B_{r_1}(0)$.

The linearized Euler equations around the shear flow are given by $$\begin{aligned}
  \label{eq:shear}
  \begin{split}
  \partial_{t} \omega + U(y)\partial_x \omega &= U''(y)\partial_x \psi, \\
  \Delta \psi &=\omega,
  \end{split}\end{aligned}$$ where for an infinite interval we prescribe $\nabla \psi \in L^2$ and on the boundary require that $\partial_x \psi=0$ (impermeable walls).

For the circular flow, we first consider the linearized problem in polar coordinates $$\begin{aligned}
  \dt \omega + u(r)\p_{\theta} \omega &= b(r) \p_{\theta} \omega, \\
  (\p_{r}^{2}+ \frac{1}{r}\p_{r} + \frac{1}{r^{2}}\p_{\theta}^{2})\phi&=f,\end{aligned}$$ where $b(r)= -\frac{1}{r}\p_{r} (\frac{1}{r}\p_{r}(r^{2} u(r)))$. Here, the distinguished cases with $b(r)=0$ for $r>0$ are given by the Taylor-Couette flow $u(r)=C_{1} +
\frac{C_{2}}{r^{2}}$ corresponding to constant angular velocity for $C_{2}=0$ and a point vortex for $C_{1}=0$, respectively.

Introducing log-polar coordinates $(x,y)=\e^s(\cos\theta, \sin\theta)$ as well 