k \in \frac{2\pi}{L}\mathbb{Z}\setminus\{0\}.\end{aligned}$$

2.  \[H2\]There exists $\eps_0 < \infty$ such that $$\begin{aligned}
    \label{eq:mildy2}
    |B(y)|+ |B'(y)| +|U''(y)|\leq \eps_0 |U'(y)|, \qquad \forall y\in I.\end{aligned}$$

3.  \[H3\] There exists a collection of smooth functions $\chi_j$ such that $$\begin{aligned}
    \sum_{j\in\J}\chi_j^2=1, \qquad \forall y\in I.\end{aligned}$$ and so that the support of each function $\chi_{j}$, $\spt \chi_j =: I_j$ is an interval with $$\begin{aligned}
    \label{eq:mildly31}
    \inf_{j\in \J}|I_j|=:\kappa >0,\end{aligned}$$ and $$\begin{aligned}
    \label{eq:mildly32}
    \sup_{j\in\J}\left\|\frac{\max_{y\in I_j} U'(y)}{U'(y)}\right\|_{L^\infty(I_j)}<\infty.\end{aligned}$$ In light of , this further implies that for some constant $C$ $$\begin{aligned}
    \label{eq:part1}
      \sup_{j\in\J} \|\chi_j\|_{W^{1,\infty}}< C (1+ \frac{1}{\kappa}), \qquad   \sup_{j\in\J} \|\chi_j\|_{W^{2,\infty}}< C^{2} (1+ \frac{1}{\kappa})^{2}.\end{aligned}$$

Unless otherwise stated, we will assume throughout the article that $U$ and $B$ are mildly degenerate. Furthermore, for some estimates we will require a perturbation condition:

1.  \[P\] Let $U,B$ be mildly degenerate and let $\epsilon_{0}, \kappa$ be as in the definition. Then we say that the flow is perturbative or satisfies a smallness condition if $ \epsilon_{0} \leq \frac{k}{2}$ and $ \epsilon_{0} C(1+1/\kappa)^{2} \leq
      \frac{\gamma_{0} k^{2}}{4}$ (c.f. Section \[sec:auxil-funct\]).

Let us briefly comment on these conditions:

-   The condition \[H1\] yields that the Laplacian is a strictly elliptic operator on $L^{2}(\T_{L} \times I, |U'(y)| \dd x \dd y)$ for any interval $I$ (c.f. Lemma \[lem:weighted\_elliptic\]). Combined 