as relabeling $\omega(t,s,\theta):=\e^{2s}\omega(t,s,\theta)$, $B(e^{s})=e^{-2s}b(e^{s})$ we obtain $$\begin{aligned}
  \label{eq:circular}
  \begin{split}
  \dt \omega + u(\e^s)\p_\theta \omega  &= B(\e^s)\p_\theta \psi , \\
  (\p_{\theta}^2 + \p_{s}^2) \psi &=\omega, 
  \end{split}\end{aligned}$$ where $B(\e^{s})=\p_{s} (\e^{-2s}\p_{s}(\e^{2s}u(\e^{s})))$.

We can reformulate both problems discussed above in a unified fashion as follows. Let $I$ be a (possibly infinite) interval, and let $U,B:I\to \R$ be given functions. We study the behavior of solutions $\omega: \T\times I\to \R$ to the linear problem $$\begin{aligned}
\label{eq:Euler}
\begin{cases}
\de_t \omega + U(y)\de_x \omega = B(y) \de_x \psi, \quad &\mbox{in } \T \times I, \\
-\Delta \psi = \omega, &\mbox{in } \T \times I,\\
\de_x \psi= 0, &\mbox{on } \T \times \de I,
\end{cases}\end{aligned}$$ with initial datum $$\begin{aligned}
\omega(0,x,y)=\omega^{in}(x,y),  \quad \mbox{in } \T \times I.\end{aligned}$$

Main result {#sec:main-results}
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Our main result consists of establishing inviscid damping, stability and scattering for flows satisfying suitable conditions, which we call *mildly degenerate flows* (c.f. Section \[sec:mildly\_degen\]). Prototypical examples here are given by shear flows of the form $U(y)=\e^{\alpha y}$ and circular flows with power law behavior as $r \downarrow 0$ and $r \uparrow \infty$.

\[thm:main\] Suppose that $U(y), B(y)$ are mildly degenerate flows on $\T_{L} \times I$. Then there exists a symmetric positive operator $A(t):L^2 \rightarrow L^2$ with $C^1$ 