de K}{\partial E}}}(E)\phi$ is provided by Lemma \[le:K\_C1\]. By assumptions , , we thus see that $h_\phi$ is in $C^1(I,L^2(G\times S))$.

By Theorem \[evoth\] there exists a unique solution $\phi\in C(I,W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$ of (\[ecsd7\]). Then $\psi(x,\omega,E):=
e^{C(E_m-E)}\phi(x,\omega,E_m-E)$ is the required solution of the problem , , for $g=0$.

B. Suppose that more generally $g\in C^2(I,T^2(\Gamma_-))$ and that the compatibility condition (\[cc\]) holds. By [@tervo14 Lemma 5.10] there exists a lift $Lg\in C^2(I,\tilde W^2(G\times S))$ for which $\gamma_-(Lg)=g$, and $(Lg)(\cdot,\cdot,E_m)=0$ (follows from (\[cc\])), and furthermore $\omega\cdot\nabla_x(Lg)=0$. Substituting into the problem (\[se1\]), (\[se2\]), (\[se3\]) the function $u:=\psi-Lg$ for $\psi$ we obtain the following problem for $u$, $$\begin{gathered}
-{{\frac{\partial (S_0u)}{\partial E}}}+\omega\cdot\nabla_x u+\Sigma u
- Ku = \tilde{f},\nonumber\\
u_{|\Gamma_-}=0,\nonumber\\
u(\cdot,\cdot,E_m)=0,\label{ecsd8}\end{gathered}$$ where $$\tilde{f}:=f-\Big(-{{\frac{\partial (S_0(Lg))}{\partial E}}}+\Sigma (Lg)-K(Lg)\Big),$$ and we have $\tilde{F}\in C^1(I,L^2(G\times S))$ under the assumption , , .

By Part A of the proof, the problem , , (for $g=0$) has a unique solution $u\in C(I,\tilde{W}^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$. As argued above, $\psi:=u+Lg$ is then then desired unique solution for the problem , , , for the given, arbitrary $g\in C^2(I,T^2(\Gamma_-))$.

We claim that the solution $\psi$ belongs to $W^2_1(G\times S\times I)$. Indeed, since $\psi\in C^1(I,L^2(G\times S))$, we have $\psi\in L^2(G\times S\times I)$ and ${{\frac{\partial \psi}{\partial E}}}\in L^2(G\times S\times I)$. On the other hand, since $\psi$ solves , these imply, together with the assumptions made, that $\omega\cdot\nabla_x\psi\in L^2(G\times S\times I)$, which confirms the claim.

Hence, under the additional assumptions (\[ass2b\]), the estimate (\[evoest\]) follows from Corollary \[csdaco1\] (part (iii)). This comple