We apply the energy identity (<ref>) with multiplier $N$ and commutators  $N^i$, in view of (<ref>).

We obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 8}
\begin{aligned}
&		\left(\int_{\{\bar{t}=\tau_2\}\cap D_\delta(\tau_1,\tau_2)}+\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}^+_\delta\cap D_\delta(\tau_1,\tau_2)}\right)\mathbb{T}(g(\nabla\psi))(N,n)[N^i\psi]\\
&    +\int\int_{D_\delta(\tau_1,\tau_2)} 	 \frac{1}{2}\:^{(N)}\pi_{\mu\nu}(\mathring{g})\mathbb{T}^{\mu\nu}(\mathring{g})[N^i\psi]\\
&   = \quad \int_{\{\bar{t}=\tau_1\}\cap D_\delta(\tau_1,\tau_2)} \mathbb{T}(g(\nabla\psi))(N,n)[N^i\psi]\\
&	\quad-\int\int_{D_\delta(\tau_1,\tau_2)}\:^{(N)}\pi_{\textit{non-lin}}(N^i\psi)-\int\int_{D_\delta(\tau_1,\tau_2)}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[\Box_{g(\nabla\psi)},N\right]N^{i-1-l}\psi+N^i\left(a^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi\right)\right)
\end{aligned}
\end{equation}

for all $1\leq i\leq k$.

Therefore, for a sufficiently small $\delta$ we obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 9}
\begin{aligned}
&		\left(\int_{\{\bar{t}=\tau_2\}\cap D_\delta(\tau_1,\tau_2)}+\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}^+_\delta\cap D_\delta(\tau_1,\tau_2)}\right)\mathbb{T}(g(\nabla\psi))(N,n)[N^{i}\psi]\\
&    +\int\int_{D_\delta(\tau_1,\tau_2)\cap \left(\{r_+-q\leq r\leq r_++q\}\cup\{\bar{r}_+-q\leq r\leq\bar{r}_++q\}\right)} 	 \frac{1}{2}\:^{(N)}\pi_{\mu\nu}(\mathring{g})\mathbb{T}^{\mu\nu}(\mathring{g})[N^{i}\psi]\\
&   = \quad \int_{\{\bar{t}=\tau_1\}\cap D_\delta(\tau_1,\tau_2)} \mathbb{T}(g(\nabla\psi))(N,n)[N^{i}\psi]\\
&	\quad -\int\int_{D_\delta(\tau_1,\tau_2)\setminus \left(\{r_+-q\leq r\leq r_++q\}\cup\{\bar{r}_+-q\leq r\leq\bar{r}_++q\}\right)} 	 \frac{1}{2}\:^{(N)}\pi_{\mu\nu}(\mathring{g})\mathbb{T}^{\mu\nu}(g(\nabla\psi))[N^{i}\psi]\\
&	\quad-\int\int_{D_\delta(\tau_1,\tau_2)}\:^{(N)}\pi_{\textit{non-lin}}(N^i\psi)-\int\int_{D_\delta(\tau_1,\tau_2)}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[\Box_{g(\nabla\psi)},N\right]N^{i-1-l}\psi+N^i\left(a^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi\right)\right),
\end{aligned}
\end{equation}

for all $1\leq i\leq k$, where for $q$ see Lemma <ref>. Now, we estimate the second term on the right hand side of (<ref>), and obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 10}
\begin{aligned}
&		\left(\int_{\{\bar{t}=\tau_2\}\cap D_\delta(\tau_1,\tau_2)}+\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}^+_\delta\cap D_\delta(\tau_1,\tau_2)}\right)\mathbb{T}(g(\nabla\psi))(N,n)[N^{i}\psi]\\
&    +\int\int_{D_\delta(\tau_1,\tau_2)\cap \left(\{r_+-q\leq r\leq r_++q\}\cup\{\bar{r}_+-q\leq r\leq\bar{r}_++q\}\right)} 	 \frac{1}{2}\:^{(N)}\pi_{\mu\nu}(\mathring{g})\mathbb{T}^{\mu\nu}(\mathring{g})[N^{i}\psi]\\
&   \leq \quad \int_{\{\bar{t}=\tau_1\}\cap D_\delta(\tau_1,\tau_2)} \mathbb{T}(g(\nabla\psi))(N,n)[N^{i}\psi]\\
&	\quad +C\tau_{\textit{max}}\sup_{\tau\in[\tau_1,\tau_2]}\int_{\{\bar{t}=\tau\}\cap\{r_++q\leq r\leq\bar{r}_+-q\}}\sum_{1\leq i_1+i_2+i_3= i+1}\sum_\alpha\left(\partial_{\bar{t}}^{i_1}\partial_r^{i_2}\Omega_\alpha^{i_3}\psi\right)^2\\
&	\quad-\int\int_{D_\delta(\tau_1,\tau_2)}\:^{(N)}\pi_{\textit{non-lin}}(N^i\psi)-\int\int_{D_\delta(\tau_1,\tau_2)}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[\Box_{g(\nabla\psi)},N\right]N^{i-1-l}\psi+N^i\left(a^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi\right)\right),
\end{aligned}
\end{equation}

for all $1\leq i\leq k$, where the constant $C>0$ depends only on the black hole parameters.

Now, in view of Lemma <ref> and the elliptic estimates of Lemma <ref>, there exist $\delta,\epsilon(\delta,C_{\text{b}})$, sufficiently small, such that the estimate (<ref>) implies

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 11.1}
\begin{aligned}
&		cE_j(\tau_2)+c\int_{\tau_1}^{\tau_2}d\tau E_j(\tau)\\
&	 +c\left(\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}_\delta^+\cap D_\delta(\tau_1,\tau_2)}\right)\sum_{1\leq i\leq j-1}\mathbb{T}(N,n)[N^i\psi]d\mathring{g}_{\mathcal{H}_\delta}\\
&   \leq C E_j(\tau_1) +C\tau_{\textit{max}}\sup_{\tau\in[\tau_1,\tau_2]}E_{j,q}(\tau)\\
&	\quad-\int\int_{D_\delta(\tau_1,\tau_2)}\sum_{1\leq i\leq j-1}\:^{(N)}\pi_{\textit{non-lin}}(N^i\psi)\\
&	\quad -\int\int_{D_\delta(\tau_1,\tau_2)}\sum_{1\leq i\leq j-1}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[\Box_{g(\nabla\psi)},N\right]N^{i-1-l}\psi+N^i\left(a^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi\right)\right)\\
&   = C E_j(\tau_1) +C\tau_{\textit{max}}\sup_{\tau\in[\tau_1,\tau_2]}E_{j,q}(\tau)\\
&	\quad-\int\int_{D_\delta(\tau_1,\tau_2)}\sum_{1\leq i\leq j-1}\:^{(N)}\pi_{\textit{non-lin}}(N^i\psi)\\
&	\quad -\int\int_{D_\delta(\tau_1,\tau_2)}\sum_{1\leq i\leq j-1}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[\Box_{\mathring{g}},N\right]N^{i-1-l}\psi\right)\\
&	\quad -\int\int_{D_\delta(\tau_1,\tau_2)}\sum_{1\leq i\leq j-1}N^{i+1}\psi\left(\sum_{l=0}^{i-1}N^l\left[h^{ab}\partial_a\partial_b +S^c\:_{ab}\partial_c,N\right]N^{i-1-l}\psi+N^i\left(a^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi\right)\right),
\end{aligned}
\end{equation}

for all $1\leq j\leq k+1$, for some constants $c(k,M,\Lambda,A_{[k+1]},B_{[k+1]}),C(k,M,\Lambda,A_{[k+1]},B_{[k+1]})$, where for $S^c\:_{ab}$ see (<ref>).

Similarly to (<ref>), we estimate the two last (non-linear) terms of (<ref>). Note that we estimate the third to last bulk term of (<ref>), in view on the linear $N$-commutation Lemma <ref>. Specifically, in view of the bootstrap (<ref>), and the definitions of the smooth tensors $a,\: h$, see Sections <ref>, <ref> respectively, there exists a $\delta>0$ sufficiently small, and an $\epsilon(\delta,C_{\text{b}})>0$ sufficiently small, such that for the two non-linear terms we use the Lemma <ref> and the coarea formula (<ref>), to obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 11.2}
\begin{aligned}
&		E_{k+1}(\tau_2)+\int_{\tau_1}^{\tau_2}d\tau E_{k+1}(\tau)+\left(\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}_\delta^+\cap D_\delta(\tau_1,\tau_2)}\right)\sum_{1\leq i\leq k}\mathbb{T}(N,n)[N^i\psi]d\mathring{g}_{\mathcal{H}_\delta}\\
&   \leq \quad C E_{k+1}(\tau_1) +C\tau_{\textit{max}}\sup_{\tau\in[\tau_1,\tau_2]}E_{j,q}(\tau)\\
\end{aligned}
\end{equation}

for $k\geq 7$, where $ 0<C=C(k,M,\Lambda,A_{[k+1]},B_{[k+1]})$. We appropriately distributed derivatives and used a Sobolev injection, see Lemma <ref>, in view of that $k= 7$ is the smallest integer such that

\begin{equation}
\floor{\frac{k+1}{2}}+2\leq k-1.
\end{equation}

We obtain

The combination of the $\partial_{\bar{t}}$ and $N$ multiplier estimates.

Now, we combine the $\partial_{\bar{t}}$-estimate, namely (<ref>), and the $N$-estimate, namely (<ref>).

We obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 11.3}
\begin{aligned}
&		E_{k+1}(\tau_2)+\int_{\tau_1}^{\tau_2}d\tau E_{k+1}(\tau)+\left(\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}_\delta^+\cap D_\delta(\tau_1,\tau_2)}\right)\sum_{1\leq i\leq k}\mathbb{T}(N,n)[N^i\psi]d\mathring{g}_{\mathcal{H}_\delta}\\
&   \leq \quad C E_{k+1}(\tau_1) +C\tau_{\textit{max}}\Bigg(C E_{k+1}(\tau_1)+C\delta E_{k+1}(\tau_2)\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +	C\sqrt{\delta}\left(\int_{\mathcal{H}^+_{\delta}\cap D_\delta(\tau_1,\tau_2)}+\int_{\bar{\mathcal{H}}_\delta^+\cap D_\delta(\tau_1,\tau_2)}\right)\sum_{1\leq i\leq k}\mathbb{T}(N,n)[N^i\psi]  d\mathring{g}_{\mathcal{H}_\delta}\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +\sqrt{\epsilon(\delta)} C \int_{\tau_1}^{\tau_2}d\tau E_{k+1}(\tau)\Bigg),
\end{aligned}
\end{equation}

for $k\geq 7$.

The integral inequality.
Therefore, for $\delta(\tau_{\textit{max}}),\epsilon(\tau_{\textit{max}},\delta)>0$ sufficiently small, then from inequality (<ref>) we obtain

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 11.4}
\begin{aligned}
&		E_{k+1}(\tau_2)+\int_{\tau_1}^{\tau_2}d\tau E_{k+1}(\tau)  \leq C E_{k+1}(\tau_1) +C\left(\tau_2-\tau_1\right) E_{k+1}(\tau_1),\\
\end{aligned}
\end{equation}

for $k\geq 7$, and for a constant $C(k,M,\Lambda,A_{[k+1]},B_{[k+1]})>0$. Note that (<ref>) is the quasilinear analogue of the integral inequality (<ref>).

Finishing the proof, improving the bootstrap (<ref>).

Note that the estimate (<ref>) holds for any $\tau_1^\prime\geq \tau_1$ in the place of $\tau_1$.

Therefore, arguing as in the linear case, see the integral inequality (<ref>), we obtain from (<ref>) that there exists a constant

\begin{equation}
\end{equation}

independent of $\tau_{\textit{max}}$, such that

\begin{equation}\label{eq: proof prop: well posedness quasilinear, eq 12}
\begin{aligned}
E_{k+1}[\psi](\tau^\prime)	&	\leq C_{\textit{wp}} E_{k+1}[\psi](\tau_1),\\
\end{aligned}
\end{equation}

for all $\tau^\prime\in [\tau_1,\tau_1+\tau_{\textit{max}}]$. Moreover, by using (<ref>) and the smallness $E_{k+1}[\psi](\tau_1)\leq \epsilon$, we also prove that

\begin{equation}
E_{k+2}[\psi](\tau^\prime)	\leq C_{\textit{wp}} E_{k+2}[\psi](\tau_1),
\end{equation}

by repeating the arguments of the above proof, and after redefining $C_{\textit{wp}}(k,M,\Lambda,A_{[k+1]},B_{[k+1]})>1$ appropriately, provided that $E_{k+2}[\psi](\tau_1)<\infty$.

Finally, for a sufficiently large $C_{\text{b}}(k,M,\Lambda,A_{[k+1]},B_{[k+1]})\gg C_{\textit{wp}}$ and a sufficiently small $\epsilon>0$, we improve the bootstrap assumption (<ref>) by Sobolev inequalities, see Lemma <ref>, on the left hand side of inequality (<ref>).

We conclude the energy estimates (<ref>), (<ref>) and therefore the Proposition.

Note also the following Remark

A posteriori, one can remove the dependence of $\delta$ on the $\tau_{\textit{step}}$ parameter. To do so, note that we dropped the terms on the hypersurfaces $\mathcal{H}^+_\delta,\bar{\mathcal{H}}^+_\delta$ from (<ref>). By using the $N$ redshift vector field in the spacetime regions

\begin{equation}
\{r_+-\tilde{\delta}\leq r\leq r_+-\delta\},\qquad  \{\bar{r}_++\delta\leq r\leq\bar{r}_++\tilde{\delta}\}
\end{equation}

for $\delta>0$ as in Theorem <ref>, and for a sufficiently small $\tilde{\delta}$ independent of $\tau_{\textit{step}}$, we can absorb the contributions at $\mathcal{H}^+_\delta,\bar{\mathcal{H}}^+_\delta$ by (<ref>), and conclude the Cauchy stability result of Proposition <ref> for a $\tilde{\delta}>0$ in the place of $\delta$.

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