\l \omega(0), A(0) \omega(0) \r
  \end{aligned}$$ for all $T\geq 0$. As the left-hand side is a bounded increasing function, we can take the limit as $T\to\infty$ and use Lemma \[lem:local\_elliptic\] to deduce that $$\begin{aligned}
    \int_{0}^\infty\int_{I} |U'(y)| |\nabla_k  \psi(t,y)|^2 \dd y \dd t \leq \l \omega(0), A(0) \omega(0) \r,
  \end{aligned}$$ and conclude that proof.

Boundary layers and H1 stability {#sec:higher}
================================

Building on the results of the previous section, we extend our result to also establish $H^{1}$ stability and prove the intermediate part of Theorem \[thm:L2\]. Due to the potentially degenerate behavior of (derivatives of) the coefficient functions $U$ and $B$, here we rely on the scattering formulation and the differential operator $\dg$ in to derive sufficient control of commutator terms. Here, principal challenges arise from several sources:

-   Taking derivatives of the evolution equation , we obtain commutator terms involving derivatives of the coefficient functions. As these coefficient functions are mildly degenerate, we need to establish localized estimates controlling derivatives of these functions.

-   While the stream function satisfies zero Dirichlet conditions, this does not hold for its derivative. Hence, a boundary layer forms that develops a logarithmic singularity near the boundary as time tends to infinity, which was studied 