in [@Zill6] and [@Zhang2015inviscid]. Here, additionally the degeneracy of the coefficients and the need for localized estimates (with associated boundary conditions) pose strong technical challenges.

-   Furthermore, even in the setting without boundary, due to the degeneracy of the coefficient functions, the derivative of the Biot-Savart law introduces several non-small commutators. We hence adapt these into a modified elliptic operator (c.f. Lemma \[lem:H1equation\]), for which in turn estimates have to be developed. This additional modification further necessitates a longer argument in Section \[sec:H2\_2\], where need to take these changes into account in a recursive argument.

Auxiliary functions and related equations {#sec:auxil-funct}
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We begin by studying the equations for $\dg F$. For this, we define the bounded coefficient functions (related to $\dc$ in ) $$\begin{aligned}
\dc^{(1)}= \sum_{j} \frac{\max_{I_{j}}|U'|}{U'(y)}\chi_{j}',\qquad \dc^{(2)}= \sum_{j} \frac{\max_{I_{j}}|U'|}{U'(y)}\chi_{j}''.\end{aligned}$$ By our choice of $I_{j}$ in terms of level sets of $U'$, $\dc^{(1)}, \dc^{(2)}$ can be estimates in terms of $\epsilon_{0}$ and $\|\chi_{j}'\|_{L^{\infty}}$ and $\|\chi_{j}''\|_{L^{\infty}}$, respectively. Taking into account the scaling of this norms in terms of the interval size $|I_{j}|$ it follows by conditions \[P\] that $$\begin{aligned}
  |\dc^{(1)}|+ |\dc^{(2)}| \leq 2 C \epsilon_{0} (1+1/\kappa)^{2} \leq \frac{\gamma_{0}k^{2}}{2},\end{aligned}$$ which we use in our elliptic estimates to control these contributions as 