dependence on $t$ such that any solution $\omega$ of satisfies $$\begin{aligned}
    \langle \omega(t), A(t) \omega(t) \rangle_{L^2} &\approx \|\omega(t)\|_{L^2}^2, \\
    \ddt \langle \omega(t), A(t) \omega(t) \rangle_{L^2} &\leq -c \int |U'(y)| |\nabla \psi (t)|^{2} \dd x \dd y \leq 0.
  \end{aligned}$$ In particular, this implies $L^2$ stability and $$\begin{aligned}
    \int |U'(y)| |\nabla \psi (t)|^{2} \dd x \dd y  \in L^1_t.
  \end{aligned}$$ If further, the initial data is in $H^{1}$ or $H^{2}$, then also $$\begin{aligned}
    \|\omega(t, x- t U(y), y)\|_{H^{1}} \leq C \|\omega^{in}\|_{H^{1}}, \\
    \|\min(1, \dist(y, \p I)) \de_{yy}\omega(t, x- t U(y), y)\|_{L^{2}} \leq C \|\omega^{in}\|_{H^{2}},
  \end{aligned}$$ and $$\begin{aligned}
    \int |U'(y)| |\nabla \psi (t)|^{2} \dd x \dd y \leq \frac{C}{\l t\r^{2}} \|\omega^{in}\|^2_{H^{1}}, \\
    \int |U'(y)| |\psi (t)|^{2} \dd x \dd y \leq \frac{C}{\l t\r^4}\|\omega^{in}\|^2_{H^{2}}.
  \end{aligned}$$

These results extend the inviscid damping and scattering results of [@Zill6] to the *mildly degenerate* setting, which in particular allows for $U'$ to converge to zero or infinity as $y$ approaches the boundary.

Mildly degenerate flows {#sec:mildly_degen}
-----------------------

We call the coefficient functions $U,B$ *mildly degenerate* with respect to a torus $\T_{L}$ if the following conditions hold.

1.  \[H1\] There exists a constant $\gamma_0\in(0,1)$ such that $$\begin{aligned}
    \label{eq:mildy1}
    k^{2}|U'(y)| - \frac12\sign(U'(y))U'''(y) \geq \gamma_0 k^{2}|U'(y)|, \qquad \forall y\in I, 