\| f \|_{W_{\textnormal{rough}}^{1,\infty}((\timefunction_0,\timefunctionboot) \times (- \rightu,\leftu) \times \mathbb{T}^2)}

In the next lemma, we continue our analysis of the change of variables map
$\CHOVgeotorough{\muxmulevelsetvalue}$. More precisely, we exhibit its properties on the closure
of $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
We also derive the quasi-convexity of the closure of
$\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$
and, as a consequence, prove a standard Sobolev embedding result.

The following results hold.

* $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$
is precompact in the topology of the geometric coordinates $(t,u,x^2,x^3)$.

* The change of variables map $\CHOVgeotorough{\muxmulevelsetvalue}(t,u,x^2,x^3) = (\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$
extends to a $C^{2,1}$ diffeomorphism on the closure of
$\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
which we denote by $\mbox{\upshape cl} \left(\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue} \right)$.
$\mbox{\upshape cl} \left(\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue} \right)
= \twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
$\CHOVgeotorough{\muxmulevelsetvalue}\left(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue} \right)
= [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$.

* The following estimates hold for the extended maps:

\begin{align} \label{E:CLOSEDVERSIONKEYJACOBIANDETERMINANTESTIMATECHOVGEOTOROUGH}
\frac{1}{2} \updelta_*
\leq
\geop{t} \timefunctionarg{\muxmulevelsetvalue}
\leq
\frac{3}{2} \updelta_*,
\mbox{on } \twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue},
\\
\frac{1}{2} \updelta_*
\leq
\Lunit \timefunctionarg{\muxmulevelsetvalue}
\leq
\frac{3}{2} \updelta_*,
\mbox{on } \twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}.
\label{E:CLOSEDVERSIONLUNITROUGHTTIMEFUNCTION}
\end{align}

\begin{align} \label{E:CLOSEDKEYJACOBIANDETERMINANTESTIMATECHOVROUGHTOGEO}
\roughgeop{\timefunction} t
& =
\mbox{\upshape det} \left(d_{\textnormal{rough}} \InverseCHOVgeotorough{\muxmulevelsetvalue} \right)
\approx 1,
\mbox{on } [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2,
\end{align}

\begin{align}
\left\|
\CHOVgeotorough{\muxmulevelsetvalue}
\right\|_{C_{\textnormal{geo}}^{2,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})}
& \leq C,
\label{E:CLOSEDVERSIONC21BOUNDFORCHOVROUGHTOGEO}
\\
\left\|
\CHOVgeotorough{\muxmulevelsetvalue}^{-1}
\right\|_{C_{\textnormal{rough}}^{2,1}([\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2)}
& \leq C,
\label{E:CLOSEDVERSIONC21BOUNDFORINVERSECHOVGEOTOROUGH}
\\
\left\|
\left(\geop{x^2} \timefunctionarg{\muxmulevelsetvalue},\geop{x^3} \timefunctionarg{\muxmulevelsetvalue} \right)
\right\|_{C_{\textnormal{geo}}^{1,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})}
& \leq C \varepsilon^{1/2}.
\label{E:SMALLC11ESTIMATESFORROUGHTIMEFUNCTION}
\end{align}

* (Quasi-convexity of
$\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$).
For every pair of points
$q_1,q_2 \in [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$,
we have:

\begin{align} \label{E:COMPARABLEDISTANCES}
\mbox{\upshape dist}_{\mbox{\upshape flat}}
\left(\CHOVgeotorough{\muxmulevelsetvalue}^{-1}(q_1),\CHOVgeotorough{\muxmulevelsetvalue}^{-1}(q_2) \right)
& \approx
\mbox{\upshape dist}_{\mbox{\upshape flat}}(q_1,q_2),
\end{align}

where $\mbox{\upshape dist}_{\mbox{\upshape flat}}(q_1,q_2)$ is the
standard Euclidean distance between $q_1$ and $q_2$ in the flat space
$\mathbb{R}_{\timefunction} \times \mathbb{R}_u \times \mathbb{T}^2$.

Moreover, $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$ is quasi-convex.[Here,
we are not just interested in a qualitative version of quasi-convexity, but rather in obtaining control over the constants “$C$.”
Similar remarks apply for the Sobolev embedding result (<ref>) and for other
quasi-convexity and Sobolev embedding results derived throughout the paper. ]
That is,
every pair of points
$p_1,p_2 \in \twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
are connected by a $C_{\textnormal{geo}}^1$ curve in $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
whose length with respect to the standard flat Euclidean metric on geometric coordinate space
$\mathbb{R}_t \times \mathbb{R}_u \times \mathbb{T}^2$
is $\leq C \mbox{\upshape dist}_{\mbox{\upshape flat}}(p_1,p_2)$.

* (Sobolev embedding).
There is a constant $C > 0$ that is independent of $\timefunctionboot$
such that the following Sobolev embedding result holds for scalar functions $f$ on
$\twoargMrough{(\timefunction_0,\timefunctionboot),(- \rightu,\leftu)}{\muxmulevelsetvalue}$:

\begin{align} \label{E:SOBOELVEMBEDDINGRELYINGONQUASICONVEXITY}
\| f \|_{C_{\textnormal{geo}}^{0,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})}
& \leq
\| f \|_{W_{\textnormal{geo}}^{1,\infty}(\twoargMrough{(\timefunction_0,\timefunctionboot),(- \rightu,\leftu)}{\muxmulevelsetvalue})}.
\end{align}

* ($\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$ is a graph).
For $\timefunction \in [\timefunction_0,\timefunctionboot]$,
there exists a function
[- \rightu,\leftu] \times \mathbb{T}^2 \rightarrow \mathbb{R}$,
depending on $\timefunction$ and $\muxmulevelsetvalue$,
such that:

\begin{align} \label{E:C21BOUNDFORCONSTANTTIMEFUNCTIONGRAPH}
\| \Cartesiantisafunctiononlevelsetsofroughtimefunctionarg{\timefunction}{\muxmulevelsetvalue} \|_{C^{2,1}([- \rightu,\leftu] \times \mathbb{T}^2)}
& \leq C
\end{align}

and such that relative to the geometric coordinates, we have:

\begin{align} \label{E:LEVELSETSOFTIMEFUNCTIONAREAGRAPH}
\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}
& =
\left\lbrace
\ | \
t = \Cartesiantisafunctiononlevelsetsofroughtimefunctionarg{\timefunction}{\muxmulevelsetvalue}(u,x^2,x^3),
\,
(u,x^2,x^3) \in [- \rightu,\leftu] \times \mathbb{T}^2
\right\rbrace.
\end{align}

In (<ref>) and just above it,
we showed that $\InverseCHOVgeotorough{\muxmulevelsetvalue}$
extends as a $C_{\textnormal{rough}}^{2,1}$ function to the compact, convex domain
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$
such that (<ref>) holds on
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$.
In particular, this yields (<ref>).
From these facts,
and the fact that $(t,u,x^2,x^3) = \InverseCHOVgeotorough{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)$,
we conclude that the map $\InverseCHOVgeotorough{\muxmulevelsetvalue}$
with domain $[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$
has a global inverse,
i.e., that $\CHOVgeotorough{\muxmulevelsetvalue}$ extends to the compact domain
$\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
as an invertible map such that $\timefunctionarg{\muxmulevelsetvalue}$ satisfies

We now prove (<ref>).
For $i=1,2$, we set
$q_i \eqdef (\timefunction_i,u_i,x_i^2,x_i^3)$
$p_i \eqdef \CHOVgeotorough{\muxmulevelsetvalue}^{-1}(q_i) \eqdef (t_i,u_i,x_i^2,x_i^3)$.
We define
$\Delta \timefunction \eqdef \timefunction_2 - \timefunction_1$,
$\Delta u \eqdef u_2 - u_1$,
$\Delta x^2 \eqdef x_2^2 - x_1^2$,
$\Delta x^3 \eqdef x_2^3 - x_1^3$,
\Delta q \eqdef (\Delta \timefunction,\Delta u,\Delta x^2,\Delta x^3) = q_2 - q_1
Similarly, we define $\Delta t \eqdef t_2 - t_1$ and
\Delta p
\eqdef (\Delta t,\Delta u,\Delta x^2,\Delta x^3) = p_2 - p_1
Without loss of generality, we assume
$\Delta \timefunction \geq 0$.
Then by
there is a constant $C > 1$ such that
\frac{1}{C} \Delta \timefunction
C(|\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}})
\leq
\Delta t
\leq C \Delta \timefunction + C(|\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}})
where for $j=2,3$,
$|\Delta x^j|_{\mathbb{T}}$ is the Euclidean distance between $x_2^j$ and $x_1^j$ in the torus.
with $|\Delta p|_{Taxi} \eqdef |\Delta t| + |\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}}$,
we see that there exists a (different) constant $C > 1$ such that:

\begin{align} \label{E:QUANTITATIVESTEPINPROOFOFCOMPARABLEDISTANCES}
\left|
\frac{1}{C} \Delta \timefunction
C\left(|\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}} \right)
\right|
|\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}}
\leq
|\Delta p|_{Taxi}
\leq
C\left(|\Delta \timefunction| + |\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}} \right).
\end{align}

From (<ref>),
it follows that
|\Delta p|_{Taxi}
\approx
|\Delta q|_{Taxi}
\eqdef
|\Delta \timefunction| + |\Delta u| + |\Delta x^2|_{\mathbb{T}} + |\Delta x^3|_{\mathbb{T}},
which implies (<ref>).

We now prove the quasi-convexity of
$\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
Let $p_1,p_2 \in \twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
and let $q_1,q_2 \in [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$ be the unique points
such that $p_i = \CHOVgeotorough{\muxmulevelsetvalue}^{-1}(q_i)$,
as above. Let $\ell$ be a straight line in
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$
whose flat length is equal to $\mbox{\upshape dist}_{\mbox{\upshape flat}}(q_1,q_2)$.
it follows that the image curve $\CHOVgeotorough{\muxmulevelsetvalue}^{-1}(\ell)$
has a flat length that is $\lesssim \mbox{\upshape dist}_{\mbox{\upshape flat}}(q_1,q_2)$,
which by (<ref>) is
$\approx \mbox{\upshape dist}_{\mbox{\upshape flat}}(p_1,p_2)$
as desired.

is a standard Sobolev embedding result (see, for example, <cit.>),
which relies on the
quantitative quasi-convexity of
$\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
proved in the previous paragraph.

follows from (<ref>)

(<ref>) follows from (<ref>)
and the fact $\timefunctionarg{\muxmulevelsetvalue}$ is an element of
$C_{\textnormal{geo}}^{2,1}\left(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue} \right)$,
as is shown by

The existence of the function
such that (<ref>)
holds, as well as the estimate (<ref>),
follow from the fact that $(t,u,x^2,x^3) = \InverseCHOVgeotorough{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)$
and the estimate (<ref>),
is the first component of $\InverseCHOVgeotorough{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)$.

§.§ Hölder-space extensions to the compact set $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$

With the help of the bootstrap assumptions and
Lemma <ref>, we now show that various solution variables extend to
the compact set $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
as functions with substantial Hölder regularity relative to the geometric coordinates.

The following quantities extend to the compact set
$\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
as elements of the following Hölder spaces, and their
corresponding spacetime Hölder norms on $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$
are bounded by $\leq C$:

* $\wavearray, \, \vortrenormalized^i, \, \GradEnt^i, \, \VortVort^i, \, \DivGradEnt
\in C_{\textnormal{geo}}^{3,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})$

* $\Upsilon \in C_{\textnormal{geo}}^{3,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})$

* $\Lunit^i, \, \upmu \in C_{\textnormal{geo}}^{2,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})$

* $\timefunctionarg{\muxmulevelsetvalue} \in C_{\textnormal{geo}}^{2,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue})$

The results for $\timefunctionarg{\muxmulevelsetvalue}$ were already proved as (<ref>).
For the remaining results, we give the proof only for $\wavearray$ since the
other solution variables can be handled using nearly identical arguments.
To proceed, we note that
Lemma <ref>,
Lemma <ref>,
and the bootstrap assumptions imply that
$\| \wavearray \|_{W_{\textnormal{geo}}^{4,\infty}(\twoargMrough{(\timefunction_0,\timefunctionboot),(- \rightu,\leftu)}{\muxmulevelsetvalue})}
\leq C
From this bound and (<ref>),
we conclude that
\| \wavearray \|_{C_{\textnormal{geo}}^{3,1}(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu)}{\muxmulevelsetvalue}]}
\leq C
as desired.

§.§ Properties of $\CHOVroughtomumuxmu{\muxmulevelsetvalue}$ and consequences

In this section,
we provide a detailed analysis of the change of variables map
$\CHOVroughtomumuxmu{\muxmulevelsetvalue} (\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3) = (\upmu, \muX \upmu,x^2,x^3)$ and its Jacobian matrix $\CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}$.

The map $\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3) = (\upmu,\muX \upmu,x^2,x^3)$
from Def. <ref>
extends to a $C_{\textnormal{rough}}^{1,1}$ map on $[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$

\begin{align} \label{E:MUXMUCHOVMAPC112BOUND}
\left\|
\CHOVroughtomumuxmu{\muxmulevelsetvalue}
\right\|_{C_{\textnormal{rough}}^{1,1}\left([\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2 \right)}
& \leq C.
\end{align}

Moreover, the Jacobian matrix
\frac{\partial (\upmu,\muX \upmu,x^2,x^3)}{\partial(\timefunction,u,x^2,x^3)}$
is invertible at every point $q \in [\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2$
and	satisfies:

\begin{align} \label{E:JACOBIANDETERMINANTRATIOBOUND}
\max_{q_1,q_2 \in [\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2}
\left|
\InverseCHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}(q_1) \CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}(q_2)
\mbox{\upshape ID}
\right|_{\mbox{\upshape}Euc}
& \leq \frac{1}{2},
\end{align}

where $|\cdot|_{\mbox{\upshape}Euc}$ is the standard Frobenius norm on matrices
(equal to the square root of the sum of the squares of the matrix entries)
and $\mbox{\upshape ID}$ denotes the $4 \times 4$ identity matrix.

Furthermore, $\CHOVroughtomumuxmu{\muxmulevelsetvalue}$
is a diffeomorphism from the compact, convex set
$[\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2$
onto its (compact) image
$\CHOVroughtomumuxmu{\muxmulevelsetvalue}\left([\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2 \right)$,
where $\CHOVroughtomumuxmu{\muxmulevelsetvalue}\left([\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2 \right)$
enjoys the following properties,
and we recall that $\upmuboot = -\timefunctionboot$ and $\mupositive = -\timefunction_0$:

* It contains
[\upmuboot,\mupositive] \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2

* It contains
(\upmuboot,\mupositive) \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2
in its interior.

* It
is quasi-convex in the following sense:
every pair of points
$r_1, r_2 \in \CHOVroughtomumuxmu{\muxmulevelsetvalue}\left([\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2 \right)$
are connected by a $C^1$ curve in
$\CHOVroughtomumuxmu{\muxmulevelsetvalue}\left([\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2 \right)$
whose length is $\lesssim \mbox{\upshape dist}_{\mbox{\upshape flat}}(r_1,r_2)$,
where $\mbox{\upshape dist}_{\mbox{\upshape flat}}(r_1,r_2)$ is the
standard Euclidean distance between $r_1$ and $r_2$ in the flat space
$\mathbb{R}_{\timefunction} \times \mathbb{R}_u \times \mathbb{T}^2$.

Moreover, with $\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}$ denoting the inverse map
and $\InverseCHOVgeotorough{\muxmulevelsetvalue}$ denoting the inverse of
the change of variables map $\CHOVgeotorough{\muxmulevelsetvalue}$ defined in (<ref>),
the following holds for $\mulevelsetvalue \in [\upmuboot,\mupositive]$:

\begin{align} \label{E:PHIINVERSEIMAGEOFTORUSISTORUSCONTAINEDINROUGHTIMEFUNCTIONLEVELSET}
\InverseCHOVgeotorough{\muxmulevelsetvalue}
\circ
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}\left(\lbrace \mulevelsetvalue \rbrace \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2 \right)
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
\subset
\hypthreearg{-\mulevelsetvalue}{[-\frac{\interestingu}{2},\frac{\interestingu}{2}]}{\muxmulevelsetvalue},
\end{align}

\begin{align} \label{E:PHIINVERSEIMAGEOFTORUSCROSSMUINTERVALISTORUSCROSSINTERVALCONTAINEDININTERESTINGREGION}
\InverseCHOVgeotorough{\muxmulevelsetvalue}
\circ
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}
\left([\upmuboot,\mupositive] \times
\lbrace
\muxmulevelsetvalue
\rbrace
\times \mathbb{T}^2 \right)
\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}
\subset
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\frac{\interestingu}{2},\frac{\interestingu}{2}]}{\muxmulevelsetvalue}.
\end{align}

In addition, $\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}$ satisfies the following estimate:

\begin{align} \label{E:INVERSEOFMUXMUCHOVMAPC112BOUND}
\left\|
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue} \right
\|_{C^{1,1}\left(
\CHOVroughtomumuxmu{\muxmulevelsetvalue}\left([\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2 \right)
\right)}
& \leq C.
\end{align}

In addition, for each fixed
$(\timefunction,x^2,x^3) \in [\timefunction_0,\timefunctionboot] \times \mathbb{T}^2$,
the map
$u \rightarrow \muX \upmu(\timefunction,u,x^2,x^3)$
is strictly increasing on $[-\interestingu,\interestingu]$.

the map $(\timefunction,u,x^2,x^3) \rightarrow \left(\timefunction,\muX \upmu,x^2,x^3 \right)$
is a $C_{\textnormal{rough}}^{1,1}$ diffeomorphism from
$[\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2$ onto its image,
which contains
\times
\left[-\muxmulevelsetvalue - \frac{\secondtransversalderivativemulowerbound \interestingu}{16},
- \muxmulevelsetvalue + \frac{\secondtransversalderivativemulowerbound \interestingu}{16}\right] \times \mathbb{T}^2$.
In particular, by (<ref>),
the image set contains
$[\timefunction_0,\timefunctionboot] \times [-\muxmulevelsetvalue_0,0] \times \mathbb{T}^2$.

Finally, there
exists a family of functions
$\lbrace \Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue} \rbrace_{\mulevelsetvalue \in [\upmu_{Boot},\mupositive]}$
on $\mathbb{T}^2$
that, for $\mulevelsetvalue \in (\upmu_{Boot},\mupositive]$, are equal to the functions from
from Sect. <ref>,
such that
for each $\timefunction \in [\timefunction_0,\timefunctionboot]$,
we have:

\begin{align} \label{E:C11BOUNDFORUROUGHCOORDINATEGRAPHTORI}
\sup_{\timefunction \in [\timefunction_0,\timefunctionboot]}
\| \Eikonalisafunctiononmumuxtoriarg{-\timefunction}{-\muxmulevelsetvalue} \|_{C^{1,1}(\mathbb{T}^2)}
& \leq C,
\\
\CHOVgeotorough{\muxmulevelsetvalue}\left(\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}\right)
\left\lbrace
\left(\timefunction,\Eikonalisafunctiononmumuxtoriarg{-\timefunction}{-\muxmulevelsetvalue}(x^2,x^3),x^2,x^3 \right)
\ | \
(x^2,x^3) \in \mathbb{T}^2
\right\rbrace,
\label{E:LASTSLICETORIAREGRAPHSABOVEFLATTORIINGEOMETRICCOORDINATES}
\end{align}

where $\CHOVgeotorough{\muxmulevelsetvalue}$ is the change of variables map defined in (<ref>).
In particular,

\begin{align} \label{E:LASTSLICELEVELSETSTRUCTUREANDLOCATIONOFMIN}
\twoargmumuxtorus{-\timefunctionboot}{-\muxmulevelsetvalue}
\subset
\hypthreearg{\timefunctionboot}{[-\frac{3}{4}\interestingu,\frac{3}{4}\interestingu]}{\muxmulevelsetvalue}.
\end{align}

Proof of (<ref>):
Lemmas <ref> and <ref> yield (<ref>).

Proof of (<ref>):
We first use Lemma <ref>
(in particular, (<ref>)),
Lemma <ref>,
and (<ref>)
to deduce that on the compact, convex set
$\mathscr{D} \eqdef [\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2$,
the Jacobian matrix
\frac{\partial (\upmu,\muX \upmu,x^2,x^3)}{\partial(\timefunction,u,x^2,x^3)}$
$\| \CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue} \|_{C_{\textnormal{rough}}^{0,1}(\mathscr{D})} \leq C$

\begin{align}
\begin{split} \label{E:DETERMINANTOFJACOBIANFROMROUGHCOORDINATESTOMUMUXMUCOORDINATES}
\mbox{\upshape det} \CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}
& = \left(\roughgeop{\timefunction} \upmu \right) \roughgeop{u} \muX \upmu
\left(\roughgeop{u} \upmu \right) \roughgeop{\timefunction} \muX \upmu
\\
& =
\mbox{\upshape det} \left(d_{\textnormal{rough}} \InverseCHOVgeotorough{\muxmulevelsetvalue} \right)
\mbox{\upshape det} \frac{\partial (\upmu,\muX \upmu,x^2,x^3)}{\partial(t,u,x^2,x^3)}
\\
& =
\mbox{\upshape det} \left(d_{\textnormal{rough}} \InverseCHOVgeotorough{\muxmulevelsetvalue} \right)
\left\lbrace
\left(\geop{t} \upmu \right) \geop{u} \muX \upmu
\left(\geop{u} \upmu \right) \geop{t} \muX \upmu
\right\rbrace
\\
& <
- 1/C, \qquad (\mbox{on } \mathscr{D}).
\end{split}
\end{align}

From (<ref>),
it follows that $\CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}$ is invertible on $\mathscr{D}$.
Also using the definition of Lipschitz continuity, we deduce the pointwise bound
\left|
\CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
\CHOVJacobianroughtomumuxmu{\muxmulevelsetvalue}(\timefunction_0,u,x^2,x^3)
\right|_{Euc}
\leq C |\timefunction - \timefunction_0|
\leq
C \mupositive
From these bounds and the data assumption (<ref>),
we conclude (<ref>)
whenever $\mupositive$ is sufficiently small.

Next, using (<ref>),
the inverse function theorem
and (<ref>),
we deduce that $\CHOVroughtomumuxmu{\muxmulevelsetvalue}$ is a $C_{\textnormal{rough}}^{1,1}$ diffeomorphism from
$\mathscr{D}$ onto its image
such that $\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\mathscr{D})$ is quasi-convex.
We clarify that (<ref>), (<ref>), and the convexity of
$\mathscr{D}$ together guarantee the injectivity of $\CHOVroughtomumuxmu{\muxmulevelsetvalue}$ on $\mathscr{D}$
and the quasi-convexity of $\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\mathscr{D})$.

Next, we use
the fact that
is a diffeomorphism on $\mathscr{D}$,
and the facts that
$\upmu|_{\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}} = - \timefunction$
and $\Wtransarg{\muxmulevelsetvalue} \upmu|_{\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}} = 0$
(i.e., that $\muX \upmu|_{\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}} = - \muxmulevelsetvalue$)
to deduce that
\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\mathscr{D})
$[\upmuboot,\mupositive] \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2$
and that
(\upmuboot,\mupositive) \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2
is contained in the interior of
\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\mathscr{D})

We now use (<ref>),
(<ref>), and Lemma <ref>
to deduce that:

\begin{align} \label{E:SIMPLEESTIMATEFORROUGHUDERIVATIVEOFMUXMU}
\frac{\secondtransversalderivativemulowerbound}{4}
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\roughgeop{u} \muX \upmu
\leq \max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\roughgeop{u} \muX \upmu
\leq \frac{4}{\secondtransversalderivativemulowerbound}.
\end{align}

From (<ref>),
it follows that for each fixed
$(\timefunction,x^2,x^3) \in [\timefunction_0,\timefunctionboot] \times \mathbb{T}^2$,
the map
$u \rightarrow \muX \upmu(\timefunction,u,x^2,x^3)$
is strictly increasing on $[-\interestingu,\interestingu]$.
Hence the map $(\timefunction,u,x^2,x^3) \rightarrow \left(\timefunction,\muX \upmu,x^2,x^3 \right)$
is a $C^{1,1}$ diffeomorphism from
$[\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu] \times \mathbb{T}^2$ onto its image.
Since (<ref>)
implies that there is a $u_* \in [-\frac{3}{4}\interestingu,\frac{3}{4}\interestingu]$
such that the image of $(\timefunction,u_*,x^2,x^3)$ under the map is $(\timefunction,-\muxmulevelsetvalue,x^2,x^3)$,
we further deduce from (<ref>) that
the image of $[-\interestingu,\interestingu] \times \mathbb{T}^2$ under the map contains
$[\timefunction_0,\timefunctionboot] \times [-\muxmulevelsetvalue - \frac{\secondtransversalderivativemulowerbound \interestingu}{16},
- \muxmulevelsetvalue + \frac{\secondtransversalderivativemulowerbound \interestingu}{16}] \times \mathbb{T}^2$,
as is desired.

the properties of $\Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}$
and the estimates
follow from the form
of $\CHOVroughtomumuxmu{\muxmulevelsetvalue}$,
the fact that
is $C^{1,1}$ on
the compact, quasi-convex set
\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\mathscr{D})
and the bootstrap assumption (<ref>),
which implies that for $\mulevelsetvalue \in [\upmuboot,\mupositive]$
and $(x^2,x^3) \in \mathbb{T}^2$,
we have
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}(\mulevelsetvalue,-\muxmulevelsetvalue,x^2,x^3) \in
\hypthreearg{-\mulevelsetvalue}{[-\frac{3}{4}\interestingu,\frac{3}{4}\interestingu]}{\muxmulevelsetvalue}

Control over $\embeddatahypersurfacearg{\muxmulevelsetvalue}$.
The map
\left(\Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),\Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),x^2,x^3 \right)
\in
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
from (<ref>) extends to a $C^{1,1}$ embedding
from $[\upmuboot,\mupositive] \times \mathbb{T}^2$
onto its image, which is $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$.
In addition, there is a $C > 1$ such that the extended embedding satisfies:

\begin{align} \label{E:EMBEDDINGOFDATAHYPERSURFACEC11EXTENDEDC11BOUND}
\| \embeddatahypersurfacearg{\muxmulevelsetvalue} \|_{C^{1,1}([\upmuboot,\mupositive] \times \mathbb{T}^2)}
& \leq C
\end{align}

\begin{align} \label{E:CLOSEDIMPROVEMENTMUEQUALSMINUSKAPPAEMBEDDINGCARTESIANTIMEFUNCTIONNEGATIVEMUDERIVATIVE}
\min_{(\mulevelsetvalue,x^2,x^3) \in [\upmuboot,\mupositive] \times \mathbb{T}^2}
\frac{\partial}{\partial \mulevelsetvalue} \Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3)
\leq
\max_{(\mulevelsetvalue,x^2,x^3) \in [\upmuboot,\mupositive] \times \mathbb{T}^2}
\frac{\partial}{\partial \mulevelsetvalue} \Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3)
- \frac{1}{C}.
\end{align}

Furthermore, for $\mulevelsetvalue \in [\upmuboot,\mupositive]$,
we have:

\begin{align} \label{E:GRAPHDESCRIPTIONOFMUXMUTORUS}
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
& =
\left\lbrace
\left(\Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),\Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),x^2,x^3 \right)
\ | \
(x^2,x^3) \in \mathbb{T}^2
\right\rbrace
\end{align}

\begin{align}
\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}
\bigcup_{\mulevelsetvalue \in [\upmuboot,\mupositive]}
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}.
\label{E:CLOSEDIMPROVEMENTLEVELSETSTRUCTUREOFMUXEQUALSMINUSKAPPA}
\end{align}

Control over $\embeddingdatahypfortimefunctionarg{\muxmulevelsetvalue}$.
The map $\embeddingdatahypfortimefunctionarg{\muxmulevelsetvalue}(t,x^2,x^3)
\left(t,\scalarembeddingdatahypfortimefunctionarg{\muxmulevelsetvalue}(t,x^2,x^3),x^2,x^3 \right)$ from
Lemma <ref>
extends to a $C^{1,1}$ embedding
\eqdef
\lbrace
(t,x^2,x^3) \in \mathbb{R} \times \mathbb{T}^2
\ | \
\Cartesiantisafunctiononmumxtoriarg{\mupositive}{-\muxmulevelsetvalue}(x^2,x^3) \leq t \leq \Cartesiantisafunctiononmumxtoriarg{\upmuboot}{-\muxmulevelsetvalue}(x^2,x^3)
\rbrace
onto its image, which is $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$.
Moreover,  the extended embedding satisfies the following estimate:

\begin{align} \label{E:EXTENDEDEMBEDDINGOFXMUISMINUSKAPPADATAHYPERSURFACEC11BOUND}
\| \embeddingdatahypfortimefunctionarg{\muxmulevelsetvalue} \|_{C^{1,1}\left(\domainforembeddingdatahypfortimefunctiontwoarg{\muxmulevelsetvalue}{[\upmuboot,\mupositive]} \right)}
& \leq C.
\end{align}

$\embeddatahypersurfacearg{\muxmulevelsetvalue}$ is the composition of the maps
$(\mulevelsetvalue,x^2,x^3) \rightarrow \InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}(\mulevelsetvalue,\muxmulevelsetvalue,x^2,x^3)$
(see Def. <ref>)
and thus
all of the desired conclusions except for those concerning $\embeddingdatahypfortimefunctionarg{\muxmulevelsetvalue}$
follow from
Lemmas <ref>

To obtain the results for
we consider the map
\rightarrow
\left(\Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),x^2,x^3 \right)
on the domain $[\upmuboot,\mupositive] \times \mathbb{T}^2$.
and the inverse function theorem, we solve for the global inverse of the map.
Composing this inverse with $\embeddatahypersurfacearg{\muxmulevelsetvalue}$,
we obtain precisely the map $\embeddingdatahypfortimefunctionarg{\muxmulevelsetvalue}$.
The estimate (<ref>)
is a straightforward consequence of these arguments

§.§ Estimates for $\InverseCHOVJacobiangeotomumuxmu$ tied to the inverse function theorem

Recall that
$\CHOVgeotomumuxmu(t,u,x^2,x^3) = (\upmu,\muX \upmu,x^2,x^3)$
is the map defined in (<ref>).
In the next lemma, we prove estimates for its Jacobian matrix in the
region $\lbrace |u| \leq \interestingu \rbrace$.
Near the end of the paper, in Prop. <ref>,
we will use the estimates in our analysis of the invertibility properties
of $\CHOVgeotomumuxmu$, which ultimately will help us derive the structure of the
singular boundary.

The Jacobian matrix $\CHOVJacobiangeotomumuxmu$
defined in (<ref>)
is invertible on $\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}$
and satisfies the following bounds, where $C > 1$:

\begin{align}
- C
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\mbox{\upshape det} \CHOVJacobiangeotomumuxmu
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\mbox{\upshape det} \CHOVJacobiangeotomumuxmu
\leq
- \frac{1}{C},
\label{E:JACOBIANDETBOUNDCHOVGEOTOMUXMUCOORDS}
\end{align}

\begin{align}
\sup_{
\substack{p_1 \in \twoargmumuxtorus{-\timefunction_0}{0}
\\
p_2 \in \twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}
\left|
\CHOVJacobiangeotomumuxmu(p_1)
\InverseCHOVJacobiangeotomumuxmu(p_2)
\mbox{\upshape ID}
\right|_{\mbox{\upshape}Euc}
& \leq \frac{1}{2},
\label{E:GEOTOMUMUXCOORDINATESJACOBIANDETERMINANTRATIOINEVOLUTIONBOUND}
\end{align}

where $|\cdot|_{\mbox{\upshape}Euc}$ is the standard Frobenius norm on matrices
(equal to the square root of the sum of the squares of the matrix entries)
and $\mbox{\upshape ID}$ denotes the $4 \times 4$ identity matrix.

Thanks to the initial data assumption
and the transversal convexity bootstrap assumption (<ref>),
the lemma can be proved using the same arguments we used to prove
(see especially the estimates in (<ref>)),
which in particular relied on the smallness of $\mupositive$.

§ CONTROL OF THE FLOW MAP OF  $\ARGLROUGH{\MUXMULEVELSETVALUE}$

In this section, we derive various estimates tied to the
flow map of the vectorfield $\argLrough{\muxmulevelsetvalue}$.
We then use these results to derive preliminary
estimates for solutions $f$ to transport equations of the form
$\argLrough{\muxmulevelsetvalue} f = F$.

§.§ Basic properties of the flow map of $\argLrough{\muxmulevelsetvalue}$

Let $\argLrough{\muxmulevelsetvalue}$ be the null vectorfield defined in (<ref>),
and let $\FlowmapLrougharg{\muxmulevelsetvalue}$ be the $\timefunction_0$-normalized flow map of
$\argLrough{\muxmulevelsetvalue}$ with respect to the rough adapted coordinates
$(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$, i.e., the solution to
the following initial value problem:

\begin{align} \label{E:FLOWMAPOFLROUGHINROUGHCOORDINATES}
\roughgeop{\timefunction}
\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
& =
\argLrough{\muxmulevelsetvalue} \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3),
\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_0,u,x^2,x^3)
& = (\timefunction_0,u,x^2,x^3).
\end{align}

Then for $A = 2,3$ there exist functions
: [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2
\rightarrow \mathbb{T}$
such that:

\begin{align} \label{E:FORMOFROUGHNULLGENERATORFLOWMAP}
\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
\left(
\timefunction,u,
\FlowmapLroughtwoarg{\muxmulevelsetvalue}{2}(\timefunction,u,x^2,x^3),
\FlowmapLroughtwoarg{\muxmulevelsetvalue}{3}(\timefunction,u,x^2,x^3)
\right).
\end{align}

Moreover, $\FlowmapLrougharg{\muxmulevelsetvalue}$
is a $C^{1,1}$ diffeomorphism
from $[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$
onto $[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$

\begin{align} \label{E:C11ROUGHBOUNDFORFLOWMAPOFWIDETILDEL}
\left\|
\FlowmapLrougharg{\muxmulevelsetvalue}
\mbox{\upshape I}
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunctionboot] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
& \lesssim \varepsilon^{1/2},
\left\|
\InverseFlowmapLrougharg{\muxmulevelsetvalue}
\mbox{\upshape I}
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunctionboot] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
& \lesssim \varepsilon^{1/2},
\end{align}

where $\mbox{\upshape I}(\timefunction,u,x^2,x^3) \eqdef (\timefunction,u,x^2,x^3)$
is the identity map and $\InverseFlowmapLrougharg{\muxmulevelsetvalue}$
is the inverse function of $\FlowmapLrougharg{\muxmulevelsetvalue}$.
In particular, for each fixed $(\timefunction,u) \in [\timefunction,\timefunctionboot] \times [- \rightu,\leftu]$,
the map
\mapsto
\left(\FlowmapLroughtwoarg{\muxmulevelsetvalue}{2}(\timefunction,u,x^2,x^3),\FlowmapLroughtwoarg{\muxmulevelsetvalue}{3}(\timefunction,u,x^2,x^3) \right)$
is a $C^{1,1}$ diffeomorphism from $\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}$ onto

(<ref>) is a trivial consequence of (<ref>)
the identities $\argLrough{\muxmulevelsetvalue} \timefunctionarg{\muxmulevelsetvalue} =  1$ and $\argLrough{\muxmulevelsetvalue} u = 0$.

To prove (<ref>),
we first note that
the functions $\left(\FlowmapLroughtwoarg{\muxmulevelsetvalue}{2} - x^2,\FlowmapLroughtwoarg{\muxmulevelsetvalue}{3} - x^3 \right)$
solve the transport system:

\begin{align} \label{E:LROUGHFLOWMAPODETORUSCOMPONENTS}
\roughgeop{\timefunction}
[\FlowmapLroughtwoarg{\muxmulevelsetvalue}{A} - x^A](\timefunction,u,x^2,x^3)
\twoargLrough{\muxmulevelsetvalue}{A}\left(\timefunction,u,\FlowmapLroughtwoarg{\muxmulevelsetvalue}{2},\FlowmapLroughtwoarg{\muxmulevelsetvalue}{3}\right),
\end{align}

with vanishing data at rough time $\timefunction_0$.
we use definition (<ref>),
Lemma <ref>,
the bootstrap assumptions,
and Lemma <ref>
to deduce that:

\begin{align} \label{E:L2ROUGHL3ROUGHC11BOUNDS}
\left\|
\left(\twoargLrough{\muxmulevelsetvalue}{2},\twoargLrough{\muxmulevelsetvalue}{3} \right)
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunctionboot] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
\lesssim \varepsilon^{1/2}.
\end{align}

Hence, commuting (<ref>)
up to one time with the rough adapted coordinate partial derivatives,
using (<ref>),
and integrating with respect to rough time,
we find that for $\timefunction \in [\timefunction_0,\timefunctionboot]$,
we have:

\begin{align}
\begin{split}\label{E:GRONWALLREADYC11ROUGHBOUNDFORFLOWMAPOFWIDETILDEL}
\max_{A=2,3}
\left\|
\FlowmapLroughtwoarg{\muxmulevelsetvalue}{A} - x^A
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunction] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
& \leq C \varepsilon^{1/2}
\\
& \ \
C  \varepsilon^{1/2}
\int_{\timefunction_0}^{\timefunction}
\max_{A=2,3}
\left\|
\FlowmapLroughtwoarg{\muxmulevelsetvalue}{A} - x^A
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunction'] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
\, \mathrm{d} \timefunction'.
\end{split}
\end{align}

From (<ref>) and Grönwall's inequality,
we find that
\max_{A=2,3}
\left\|
\FlowmapLroughtwoarg{\muxmulevelsetvalue}{A} - x^A
\right\|_{C_{\textnormal{rough}}^{1,1}([\timefunction_0,\timefunction] \times [- \rightu, \leftu] \times \mathbb{T}^2)}
\leq C \varepsilon^{1/2}
From this bound and (<ref>),
we conclude the first bound stated in (<ref>).
The second bound in (<ref>)
follows from differentiating the identity
\InverseFlowmapLrougharg{\muxmulevelsetvalue} \circ \FlowmapLrougharg{\muxmulevelsetvalue} = \mbox{\upshape I}
and using the first bound.

§.§ Estimate for $\mbox{\upshape det} \gtorusroughfirstfund$ and Minkowski's integral inequality

In the next lemma, we control the factor $\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_2,u,x^2,x^3)$
featured in the area form $\volroughtorus$ (see (<ref>))
on the rough tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$.
We then use this estimates
to prove a Minkowski's integral inequality-type estimate.

Recall that $\gtorusroughfirstfund$ is the first fundamental form of $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$
(see Def. <ref>) and that
$\FlowmapLrougharg{\muxmulevelsetvalue}$ is the
$\timefunction_0$-normalized flow map of $\argLrough{\muxmulevelsetvalue}$ from Lemma <ref>.
Then for every $\timefunction_1, \timefunction_2 \in [\timefunction_0,\timefunctionboot]$
and every $(u,x^2,x^3) \in [- \rightu,\leftu] \times \mathbb{T}^2$,
the following estimates hold,
where $\mbox{\upshape det} \gtorusroughfirstfund$ is evaluated relative to the rough adapted coordinates
via the formula (<ref>):

\begin{align} \label{E:VOLFORMESTIMATEROUGHTORI}
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_2,u,x^2,x^3)
& =
\left\lbrace
1 + \mathcal{O}(\auxbootsmall)
\right\rbrace
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_1,u,x^2,x^3)
1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha}),
\\
\mbox{\upshape det} \gtorusroughfirstfund \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_2,u,x^2,x^3)
\left\lbrace
1 + \mathcal{O}(\auxbootsmall)
\right\rbrace
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_1,u,x^2,x^3)
1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha}).
\label{E:NORMALIZEDROUGHNULLFLOWMAPDOESNOTSUBSTANTIALLYDISTORTVOLFORMESTIMATEROUGHTORI}
\end{align}

Moreover, for any scalar function $F$ on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
the function $\varphi(\timefunction,u,x^2,x^3)$
defined by:

\begin{align} \label{E:ANTIROUGHTIMEDERIVATIVE}
\varphi(\timefunction,u,x^2,x^3)
& \eqdef
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\, \mathrm{d} \timefunction'
\end{align}

satisfies the following estimate:

\begin{align} \label{E:MINKOWSKIFORINTEGRALS}
\| \varphi \|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
& \leq
\left\lbrace
1 + \mathcal{O}(\auxbootsmall)
\right\rbrace
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\| F \|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'.
\end{align}

First, using the expression of the rough metric components
in (<ref>)
and the identity
$\mbox{\upshape det} \gtorus = \frac{1}{\Speed^2 (X^1)^2}$
stated in (<ref>),
we compute that:

\begin{align} \label{E:IDENTITYNEEDEDFORVOLFORMESTIMATEROUGHTORI}
\mbox{\upshape det} \gtorusroughfirstfund
& =
\left(\frac{\Lunit \timefunctionarg{\muxmulevelsetvalue}}{\geop{t}\timefunction}\right)^2
\frac{1}{\Speed^2(X^1)^2}.
\end{align}

Using (<ref>)
our assumptions on the data from Sect. <ref>,
and recalling that
\Speed(\LogDensity = 0,\Ent = 0) = 1
$X^1 = - 1 + \Xsmall^1$,
we deduce that
$\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_0,u,x^2,x^3)
1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha})
Moreover, using (<ref>),
Lemma <ref>,
Lemma <ref>,
and the bootstrap assumptions,
we deduce that
\roughgeop{\timefunction} \mbox{\upshape det} \gtorusroughfirstfund
\mathcal{O}(\auxbootsmall)
which, in view of the mean value theorem,
yields that for any
$(\timefunction,u,x^2,x^3) \in [\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$,
we have
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction,u,x^2,x^3)
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_0,u,x^2,x^3)
\mathcal{O}(\auxbootsmall)
\left\lbrace
1 + \mathcal{O}(\auxbootsmall)
\right\rbrace
\mbox{\upshape det} \gtorusroughfirstfund(\timefunction_0,u,x^2,x^3)
In total, these estimates imply (<ref>).

To prove (<ref>),
we first note that definition (<ref>) and the estimates cited above yield the following estimate
relative to the rough adapted coordinates:
\roughgeop{\timefunction}
\left\lbrace
\gtorusroughfirstfund \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
\right\rbrace
[\argLrough{\muxmulevelsetvalue} \mbox{\upshape det} \gtorusroughfirstfund] \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
\mathcal{O}(\auxbootsmall)
Using this estimate,
arguing as in the previous paragraph, and using the initial condition
$\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_0,u,x^2,x^3) = (\timefunction_0,u,x^2,x^3)$,
we deduce the following estimate
for every
$(\timefunction,u,x^2,x^3) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu] \times \mathbb{T}^2$:
\mbox{\upshape det}  \gtorusroughfirstfund \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
\left\lbrace
1 + \mathcal{O}(\auxbootsmall)
\right\rbrace
\mbox{\upshape det}  \gtorusroughfirstfund(\timefunction_0,u,x^2,x^3)
Combining this estimate with (<ref>),
we conclude (<ref>).

The inequality (<ref>)
follows from the following estimates relative to the rough adapted coordinates,
which rely on (<ref>)
and Minkowski's inequality for integrals:

\begin{align}
\begin{split}\label{E:PROOFOFMINKOWSKIFORINTEGRALS}
& \| \varphi \|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2
\\
\int_{u' = - \rightu}^u
\int_{\mathbb{T}^2}
\left(
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\, \mathrm{d} \timefunction'
\right)^2
\sqrt{\mbox{\upshape det} \gtorusroughfirstfund(\timefunction,u',x^2,x^3)}
\, \mathrm{d} x^2 \mathrm{d} x^3
\mathrm{d} u'
\\
& \leq
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\int_{u' = - \rightu}^u
\int_{(x^2,x^3) \in \T^2}
\left(
\int_{\timefunction' = \timefunction_0}^{\timefunction}
[\mbox{\upshape det} \gtorusroughfirstfund(\timefunction',u',x^2,x^3)]^{1/4}
\, \mathrm{d} \timefunction'
\right)^2
\, \mathrm{d}x^2 \mathrm{d}x^3
\mathrm{d} u'
\\
& \leq
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\left\lbrace
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\left[
\int_{u' = - \rightu}^u
\int_{(x^2,x^3) \in \T^2}
\left(
\right)^2
\,
\sqrt{\mbox{\upshape det} \gtorusroughfirstfund(\timefunction',u',x^2,x^3)}
\, \mathrm{d}x^2 \mathrm{d} x^3 \mathrm{d} u'
\right]^{1/2}
\mathrm{d} \timefunction'
\right\rbrace^2
\\
& =
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\left\lbrace
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\| F \|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'
\right\rbrace^2.
\end{split}
\end{align}

§.§ Estimates for solutions to $\argLrough{\muxmulevelsetvalue} f = F$

The following lemma provides the simple transport equation estimates
that we use to control solutions to $\argLrough{\muxmulevelsetvalue} f = F$.

Let $f$ be a function of the rough adapted coordinates on
$[\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu] \times \mathbb{T}^2$,
let $\FlowmapLrougharg{\muxmulevelsetvalue}$ be the $\timefunction_0$-normalized flow map of $\argLrough{\muxmulevelsetvalue}$ from
Lemma <ref>.
Then relative to the rough adapted coordinates, the following identity holds for any
$\timefunction_0 \le \timefunction_1 \le \timefunction_2 \leq \timefunctionboot \leq 0$
and any $(u,x^2,x^3) \in [- \rightu,\leftu] \times \mathbb{T}^2$:

\begin{align} \label{E:TRANSPORTIDENTITYALONGLROUGHINTEGRALCURVES}
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_2,u,x^2,x^3)
& =
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_1,u,x^2,x^3)
\int_{\timefunction' = \timefunction_1}^{\timefunction_2} (\argLrough{\muxmulevelsetvalue} f)
\circ
\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction',u,x^2,x^3)
\, \mathrm{d} \timefunction'.
\end{align}

Moreover, relative to the rough adapted coordinates,
we have the following estimate for the critically important factor $G_{\Lunit \Lunit}^0$:

\begin{align} \label{E:CRUCIALGLL0TRANSPORTESTIMATE}
\left|
G_{\Lunit \Lunit}^0 \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_2,u,x^2,x^3)
G_{\Lunit \Lunit}^0 \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_1,u,x^2,x^3)
\right|
& \lesssim \auxbootsmall |\timefunction_2 - \timefunction_1|.
\end{align}

In addition, we have the following $L^{\infty}$ and $L^2$ estimates:

\begin{align}
\| f \|_{L^{\infty}\left(\twoargroughtori{\timefunction_2,u}{\muxmulevelsetvalue}\right)}
& \leq
\| f\|_{L^{\infty}\left(\twoargroughtori{\timefunction_1,u}{\muxmulevelsetvalue}\right)}
\mupositive \sup_{\timefunction' \in [\timefunction_1,\timefunction_2]}
\| \argLrough{\muxmulevelsetvalue} f \|_{L^{\infty}\left(\twoargroughtori{\timefunction',u}{\muxmulevelsetvalue}\right)},
\label{E:LINFINITYTRANSPORTROUGHLFESTIMATE}
\\
\esssup_{(x^2,x^3) \in \mathbb{T}^2}
\left|
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_2,u,x^2,x^3)
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_1,u,x^2,x^3)
\right|
& \leq
\mupositive \sup_{\timefunction' \in [\timefunction_1,\timefunction_2]}
\| \argLrough{\muxmulevelsetvalue} f \|_{L^{\infty}\left(\twoargroughtori{\timefunction',u}{\muxmulevelsetvalue}\right)},
\label{E:MOREPRECISELINFINITYTRANSPORTROUGHLFESTIMATE}
\end{align}

\begin{align} 	\label{E:L2ONROUGHCONSTANTTIMEHYPERSURFACESTRANSPORTROUGHLFESTIMATE}
\| f \|_{L^2 \left(\hypthreearg{\timefunction_2}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
& \leq
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\| f \|_{L^2\left(\hypthreearg{\timefunction_1}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\int_{\timefunction' = \timefunction_1}^{\timefunction_2}
\| \argLrough{\muxmulevelsetvalue} f \|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'.
\end{align}

Furthermore, if $f$ is any function of the rough adapted coordinates on
$[\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu] \times \mathbb{T}^2$,
then for every $(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$,
the following estimate holds:

\begin{align} \label{E:DISTORTINGWITHROUGHFLOWMAPDOESNOTCHANGEL2NORMSMUCH}
\| f \circ \FlowmapLrougharg{\muxmulevelsetvalue} \|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\| f \|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}.
\end{align}

Finally, let $F$ be a scalar function of the rough adapted coordinates
on $[\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu] \times \mathbb{T}^2$,
and let $\varphi$ be the function of the rough adapted coordinates defined by:

\begin{align} \label{E:AGAINANTIROUGHTIMEDERIVATIVE}
\varphi(\timefunction,u,x^2,x^3)
& \eqdef
\int_{\timefunction' = \timefunction_0}^{\timefunction}
F \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction',u,x^2,x^3)
\, \mathrm{d} \timefunction'.
\end{align}

Then for every $(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$,
the following estimate holds:

\begin{align} \label{E:L2ONROUGHCONSTANTTIMEHYPERSURFACEOFROUGHTIMEINTEGRALWITHFLOWMAPFACTORSBOUND}
\| \varphi \|_{L^2 \left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\leq
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\| F \|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'.
\end{align}

follows from
and the fundamental theorem of calculus.

To prove (<ref>),
we first note that
and the bootstrap assumptions imply that
$\left|\argLrough{\muxmulevelsetvalue} G_{\Lunit \Lunit}^0 \right|
\lesssim \auxbootsmall
Hence, using this bound and
with $G_{\Lunit \Lunit}^0$ in the role of $f$,
we conclude (<ref>).

We now prove (<ref>).
First, from (<ref>),
we deduce:

\begin{align} \label{E:TRANSPORTINEEQUALITYALONGLROUGHINTEGRALCURVES}
\left|
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_2,u,x^2,x^3)
\right|
\leq
\left|
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction_1,u,x^2,x^3)
\right|
\int_{\timefunction' = \timefunction_1}^{\timefunction_2}
\left|
(\argLrough{\muxmulevelsetvalue} f)
\circ
\FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction',u,x^2,x^3)
\right|
\, \mathrm{d} \timefunction'.
\end{align}

We now take the essential supremum norm of both sides of (<ref>)
over $(x^2,x^3) \in \mathbb{T}^2$ and use Lemma <ref>
to deduce
\| f \|_{L^{\infty}\left(\twoargroughtori{\timefunction_2,u}{\muxmulevelsetvalue}\right)}
\leq
\| f \|_{L^{\infty}\left(\twoargroughtori{\timefunction_1,u}{\muxmulevelsetvalue}\right)}
\int_{\timefunction' = \timefunction_1}^{\timefunction_2}
\| \argLrough{\muxmulevelsetvalue} f \|_{L^{\infty}\left(\twoargroughtori{\timefunction',u}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'
From this bound and the fact that
|\timefunction_2 - \timefunction_1| \leq |\timefunction_0| = \mupositive
we conclude (<ref>).
The estimate (<ref>)
can be proved via a similar argument, and we omit the details.

To prove (<ref>),
we first take the
$\| \cdot \|_{L^2\left(\hypthreearg{\timefunction_2}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}$
norm of both sides of (<ref>).
We then use Lemma <ref>
(in particular (<ref>)),
and the standard formula for changing variables in an integral
(these estimates allow us in particular to replace all terms
f \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)
under the $L^2$ norms with
up to $1 + \mathcal{O}(\auxbootsmall)$ multiplicative factors)
to deduce:

\begin{align} 	\label{E:FIRSTSTEPL2TRANSPORTROUGHLFESTIMATE}
\| f \|_{L^2\left(\hypthreearg{\timefunction_2}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
& \leq
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\| f(\timefunction_1,\cdot) \|_{L^2\left(\hypthreearg{\timefunction_2}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\int_{\timefunction' = \timefunction_1}^{\timefunction_2}
\left\|
\argLrough{\muxmulevelsetvalue} f
\right\|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction'.
\end{align}

We then use (<ref>)
to deduce that the first term on RHS (<ref>)

\begin{align} \label{E:BOUNDFORFIRSTTERMONRHSOFFIRSTSTEPL2TRANSPORTROUGHLFESTIMATE}
\| f(\timefunction_1,\cdot) \|_{L^2\left(\hypthreearg{\timefunction_2}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\left\lbrace
\mathcal{O}(\auxbootsmall)
\right\rbrace
\| f(\timefunction_1,\cdot) \|_{L^2\left(\hypthreearg{\timefunction_1}{[- \rightu,u]}{\muxmulevelsetvalue}\right)},
\end{align}

which in total yields (<ref>).

The estimate (<ref>)
follows from (<ref>),
and the standard formula for changing variables in an integral.

Finally, (<ref>)
follows from

§ $L^{\INFTY}$ ESTIMATES AND IMPROVEMENT OF THE AUXILIARY BOOTSTRAP ASSUMPTIONS

In this section, we derive $L^{\infty}$ estimates for the fluid variables and eikonal function quantities
that in particular yield improvements of the auxiliary bootstrap assumptions stated in Sect. <ref>.

Under the parameter-size and initial data assumptions
of Sects. <ref>
and <ref>
and the bootstrap assumptions of
Sects. <ref> and <ref>,
the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$
(where we recall that $\wavearray$ and $\wavearraypartial$ are defined in Def. <ref>,
and that in Sect. <ref>,
we introduced notation for strings of commutation vectorfields).

$L^{\infty}$ estimates for small quantities.

\begin{align}
\left\| \RRiemann \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq \mathring{\upalpha}
C \fundbootsmall,
\label{E:LINFINITYIMPROVEMENTAUXR+}
\\
\left\| \wavearraypartial \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq C \fundbootsmall,
\label{E:IMPROVEAUXWAVEARRAYPARTIALLINFINITY}
\\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{\leq \Ntop-11;1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,\Ntop-11];1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\label{E:LINFINITYIMPROVEMENTAUXWAVEARRAY}
&		\\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,6];2} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,6];2} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
& \notag	\\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,5];3} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,5];3} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
& \notag	\\
\left\| \argLrough{\muxmulevelsetvalue} \muX \muX \muX \muX \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq
C \fundbootsmall,
\notag
\\
\left\|
\comder^{\leq \Ntop-11;1} (\vortrenormalized,\GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\label{E:LINFINITYIMPROVEMENTAUXVORTGRADENT}
\\
\left\|
\comder^{\leq 6;2} (\vortrenormalized,\GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
&		\notag \\
\left\|
\comder^{\leq 5;3} (\vortrenormalized,\GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\notag \\
\left\|
\muX \muX \muX \muX (\vortrenormalized,\GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq C \fundbootsmall,
\notag \\
\left\|
\comder^{\leq \Ntop-12;1} (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\label{E:LINFINITYIMPROVEMENTAUXMODIFIEDFLUID}
\\
\left\|
\comder^{\leq 6;2} (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
& \notag
\\
\left\|
\comder^{\leq 5;3} (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
& \notag
\\
\left\|
\muX \muX \muX \muX (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq C \fundbootsmall,
\notag
\\
\left\| \argLrough{\muxmulevelsetvalue} \tander^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \tander_*^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\label{E:LINFINITYIMPROVEMENTAUXMUSMALL}
\\
\left\| \argLrough{\muxmulevelsetvalue} \comdersmall^{[1,5];1}  \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comderdoublesmall^{[1,5];1} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
&	\notag
\\
\left\| \argLrough{\muxmulevelsetvalue} \comdersmall^{[1,4];2} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comderdoublesmall^{[1,4];2} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
& \leq C \fundbootsmall,
\notag \\
\left\| \Lsmall^1 \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq \mathring{\upalpha}
C \fundbootsmall,
\label{E:LINFINITYIMPROVEMENTAUXL1SMALL}
\\
\left\| \Lsmall^A \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\label{E:LINFINITYIMPROVEMENTAUXLASMALL}
\leq C \fundbootsmall
\\
\left\| \argLrough{\muxmulevelsetvalue} \tander^{\leq \Ntop-11} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \tander^{[1,\Ntop-11]} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\label{E:LINFINITYIMPROVEMENTAUXMIXECTANGENTIALTRANSVERSALDERIVATIVESLISMALL}
\\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\notag
\\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,5];2}\Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,5];2} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\notag \\
\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,4];3}\Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})},
\,
\left\| \comdersmall^{[1,4];3} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq C \fundbootsmall.
\notag
\end{align}

$L^{\infty}$ estimates tied to pure transversal derivatives.

\begin{align}
\left
\| \muX^M \RRiemann
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\left\|
\muX^M \RRiemann
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall,
1 \leq M \leq 4,
\label{E:LINFINITYIMPROVEMENTAUXTRANSVERSALPDERIVATIVESRRIEMANNLARGE}
\\
\left
\| \muX^M \wavearraypartial
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
C \fundbootsmall,
1 \leq M \leq 4,
\label{E:LINFINITYIMPROVEMENTAUXTRANSVERSALPDERIVATIVESPARTIALWAVEARRAYSMALL}
\\
\left\|
\muX^M \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\left\|
\muX^M
\left\lbrace
\Speed^{-1}
\right\rbrace
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
\frac{1}{2}
\mathring{\updelta}_*^{-1}
\left\|
\muX^M
\left\lbrace
\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
\muX \RRiemann
\right\rbrace
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall,
\label{E:LINFINITYIMPROVEMENTAUXMUANDTRANSVERSALDERIVATIVES}
0 \leq M \leq 3,
\\
\left\|
\argLrough{\muxmulevelsetvalue} \muX^M \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\frac{1}{\updelta_*}
\left\|
\muX^M
\left\lbrace
\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
\muX \RRiemann
\right\rbrace
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall,
\label{E:LINFINITYIMPROVEMENTAUXNULLDERIVATIVEMUANDTRANSVERSALDERIVATIVES}
0 \leq M \leq 3,
\\
\left\|
\muX^M \Lsmall^i
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\label{E:LINFINITYESTIMATESFORMUXDERIVATIVESOFLI}
1 \leq M \leq 3,
\end{align}

where $\argLrough{\muxmulevelsetvalue}$ is defined in (<ref>).

We refer to Sect. <ref> for various results that we will silently use throughout the analysis.
We will also silently use the assumptions on the initial data stated in
Sect. <ref>,
the parameter-relations (<ref>),
and the estimate
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\approx 1
implied by (<ref>).

Proof of (<ref>):
We use (<ref>) with $\timefunction_1 \eqdef \timefunction_0$ and $\timefunction_2 \eqdef \timefunction$,
the bootstrap assumptions (<ref>),
the data estimate (<ref>),
and (<ref>)
to conclude that:

\begin{align} \label{E:PROOFSTEPSIMPROVEAUXRRIEMANN}
\| \RRiemann \|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\| \RRiemann \|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
\mupositive \sup_{\timefunction' \in [\timefunction_0,\timefunction]}
\| \argLrough{\muxmulevelsetvalue} \RRiemann \|_{L^{\infty}\left(\twoargroughtori{\timefunction',u}{\muxmulevelsetvalue}\right)}
\leq
\mathring{\upalpha} + C \fundbootsmall
\end{align}

as desired.

Proof of (<ref>)–(<ref>):
We first use Lemma <ref>
to write the transport equation (<ref>) for $\Lunit^i$ schematically as:

\begin{align} \label{E:LUNITITRANSPORTSCHEMATIC}
\Lunit \Lsmall^i
& = \smoothfunction(\controlvars)
\cdot
\tander \wavearray.
\end{align}

From (<ref>) and the bootstrap assumptions,
we find that
$\|\argLrough{\muxmulevelsetvalue} \Lunit^i \|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
= \mathcal{O}(\fundbootsmall)$. From this bound,
the initial data assumptions, and the same arguments we used to prove (<ref>),
we conclude (<ref>)–(<ref>).

Proof of (<ref>) and
(<ref>) for $M = 1$:
Fix $\Psi \in \wavearray = (\RRiemann,\LRiemann,v^2,v^3,\Ent)$.
Multiplying (<ref>)
by $\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$ and using the bootstrap assumptions, we see that:

\begin{align} \label{E:LXPSISMALL}
\left\| \argLrough{\muxmulevelsetvalue} \muX \Psi \right \|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
& \leq
C \fundbootsmall.
\end{align}

From (<ref>) and the same arguments we used to prove (<ref>),
we conclude
(<ref>) and
for $M = 1$.

Proof of (<ref>)
for $\left\| \argLrough{\muxmulevelsetvalue} \comder^{\leq \Ntop-11;1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$
and $\left\| \comdersmall^{[1,\Ntop-11];1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$:
We first commute (<ref>)
with $\tander^{[1,\Ntop-12]}$ and use the commutator estimate
and the bootstrap assumptions
to deduce that
\left\| \Lunit \tander^{[1,\Ntop -12]} \muX \Psi \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim \fundbootsmall
and that the same bound holds with $\argLrough{\muxmulevelsetvalue}$ in place of $\Lunit$.
In particular, this yields (<ref>)
for the first term on the LHS.
Also using the same arguments we used to prove (<ref>)
(including our assumptions on the data),
we find that
\left\|
\tander^{[1,\Ntop -12]} \muX \Psi \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim \fundbootsmall
From this bound, the commutator estimate (<ref>),
and the bootstrap assumptions, we deduce that
$\left\| \comdersmall^{[1,\Ntop-11];1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim
\left\| \tander^{[1,\Ntop -12]} \muX \Psi \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\left\| \tander^{[1,\Ntop -12]} \Psi \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim
\fundbootsmall
which yields (<ref>)
for the second term on the LHS.

Proof of (<ref>) for
$\left\| \argLrough{\muxmulevelsetvalue} \tander^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$
and $\left\| \tander_*^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$:
We begin by using Lemmas <ref> and
to write the transport equation (<ref>) for $\upmu$ as:

\begin{align} \label{E:MUTRANSPORTSCHEMATICVERSION}
\Lunit \upmu
& =
- \frac{1}{2} \Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
\muX \RRiemann
\smoothfunction(\controlvars) \cdot \muX \wavearraypartial
\smoothfunction(\badcontrolvars) \cdot \tander \wavearray
\smoothfunction(\badcontrolvars) \cdot \GradEnt,
\end{align}

where the first product on RHS (<ref>) is written exactly (for use later on)
and the remaining ones schematically.
Commuting (<ref>) with $\tander^N$ for $1 \leq N \leq \Ntop -12$
and using (<ref>),
the bootstrap assumptions,
and the already
proved bound
\left\| \tander^{[1,\Ntop -12]} \muX \Psi \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim
\fundbootsmall
we find that
\left\| \Lunit \tander^{[1,\Ntop-12]} \upmu \right \|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim \fundbootsmall
and that the same bound holds with $\argLrough{\muxmulevelsetvalue}$ in place of $\Lunit$.
In particular, this yields (<ref>)
for the first term on the LHS.
In the case that $\tander^N = \tandersmall^N$,
we can combine this bound with the initial data assumptions
and the same arguments we used to prove (<ref>)
in order to conclude
$\left\| \tandersmall^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
which yields
for the second term on the LHS.

Proof of (<ref>) and
(<ref>) in the case $M=0$:
The arguments given in the previous paragraph in particular yield that
\left\| \argLrough{\muxmulevelsetvalue} \Lunit \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
From this bound and the same arguments we used to prove (<ref>),
we find that
\left\|
\Lunit \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\left\|
\Lunit \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall
From this bound, (<ref>),
the already proven bound
\left
\| \muX \wavearraypartial
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim
\fundbootsmall
and the bootstrap assumptions,
we further deduce that
\left\|
\Lunit \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\frac{1}{2}
\left\|
\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
\muX \RRiemann
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall
From this bound and (<ref>),
we find that
\left\|
\argLrough{\muxmulevelsetvalue} \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\frac{2}{\updelta_*}
\left\|
\Lunit \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall
which yields (<ref>)
in the case $M=0$. From this bound
and the same arguments we used to prove (<ref>),
we find that
\left\|
\upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\leq
\left\|
\upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
\frac{2}{\updelta_*}
\mupositive
\left\|
\Lunit \upmu
\right\|_{L^{\infty}(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}
C \fundbootsmall
which (assuming $\mupositive \leq 1$)
yields (<ref>) in the case $M=0$.

Proof of (<ref>)
for $\left\| \argLrough{\muxmulevelsetvalue} \tander^{\leq \Ntop-11} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$
$\left\| \tander^{[1,\Ntop-11]} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$:
We commute (<ref>) with $\tander^N$ for $1 \leq N \leq \Ntop - 11$ to obtain:

\begin{align} \label{E:COMMUTEDWITHTANGENTIALLUNITITRANSPORTSCHEMATIC}
\Lunit \tander^N \Lunit^i
& =
[\Lunit, \tander^N] \Lunit^i
\tander^N
\left\lbrace
\smoothfunction(\controlvars)
\cdot
\tander \wavearray
\right\rbrace.
\end{align}

From (<ref>),
the commutator estimate (<ref>),
and the bootstrap assumptions,
we find that
$\left\| \argLrough{\muxmulevelsetvalue} \tander^{\leq \Ntop-11} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
and that the same bound holds with $\argLrough{\muxmulevelsetvalue}$ in place of $\Lunit$.
The remainder of the proof relies on the same arguments used above.

Proof of (<ref>) in the case $M=1$
and (<ref>)
for $\left\| \comdersmall^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$:
The desired estimate (<ref>) in the case $M=1$
follows from the identity (<ref>)
and the bootstrap assumptions.

Next, we differentiate (<ref>)
with $\tander^N$ for $1 \leq N \leq \Ntop - 13$
and use the bootstrap assumptions and the already
proven bounds
$\left\| \comdersmall^{[1,\Ntop-11];1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim
\fundbootsmall
\left\| \tandersmall^{[1,\Ntop-12]} \upmu \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
$\left\|\tander^{[1,\Ntop-11]} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
to deduce
$\left\| \tander^{[1,\Ntop-13]} \muX \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
Also using the commutator estimate (<ref>),
we conclude that
$\left\| \comdersmall^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
as desired.

Proof of (<ref>)
$\left\| \argLrough{\muxmulevelsetvalue} \comder^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$:
We assume that $1 \leq N \leq \Ntop -12$ and that $\comder^{N;1}$
contains a factor of $\muX$.
We commute (<ref>) with $\comder^{N;1}$ to obtain:

\begin{align} \label{E:COMMUTEDWITHONETRANSVERSALANDTRANGENTIALLUNITITRANSPORTSCHEMATIC}
\Lunit \comder^{N;1} \Lunit^i
& =
[\Lunit,\comder^{N;1}] \Lunit^i
\comder^{N;1}
\left\lbrace
\smoothfunction(\controlvars)
\cdot
\tander \wavearray
\right\rbrace.
\end{align}

Using the commutator estimate (<ref>)
and the already proven bounds

\left\|\tander^{[1,\Ntop-11]} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall,
\qquad
\left\| \comdersmall^{[1,\Ntop-12];1} \Lsmall^i \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall,

we find that the first term on RHS (<ref>)
is bounded in magnitude by $\lesssim \varepsilon$. From the bootstrap assumptions
and the already proven bound
$\left\| \comdersmall^{[1,\Ntop-11];1} \wavearray \right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim
\fundbootsmall
we deduce that the second term on RHS (<ref>)
obeys the same bound. It follows that
\left\| \Lunit \comder^{[1,\Ntop - 12];1} \Lunit^i \right\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
\lesssim \fundbootsmall
and that the same bound holds with $\argLrough{\muxmulevelsetvalue}$ in place of $\Lunit$, as desired.

Proof of (<ref>) for
\comder^{\leq \Ntop-11;1} (\vortrenormalized,\GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}$
\left\|
\comder^{\leq \Ntop-12;1} (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
We apply $\tander^{\leq \Ntop - 11}$ to
and use the bootstrap assumptions (<ref>)
to deduce that
\left\|
\tander^{\leq \Ntop - 11 }\muX(\vortrenormalized, \GradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
\lesssim \fundbootsmall
From this bound,
the commutator estimate (<ref>),
and the bootstrap assumptions,
we conclude (<ref>) for the first term on the LHS.

The estimate (<ref>)
\left\|
\comder^{\leq \Ntop-12;1} (\VortVort,\DivGradEnt)
\right\|_{L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}
follows from a similar argument based on

Outline of the remainder of the proof:
The remaining estimates in the proposition can be derived
by commuting the equations with one additional $\muX$
derivative followed by elements of $\Tanset$,
using the above arguments and the estimates already proved,
and then repeating the process, adding one additional $\muX$ derivative each time.
The estimates need to be derived in the same order
as above. The commutator terms can be handled with the help of
Prop. <ref>.
The bootstrap assumptions guarantee that all terms that need to be controlled have
sufficient $L^{\infty}(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})$-regularity.
We omit the tedious, but straightforward details.

The following corollary is an immediate consequence of the fact that we have improved the auxiliary bootstrap assumptions.

All prior inequalities whose RHS feature an explicit factor of $\mathring{\upalpha}^{1/2}, \auxbootsmall$ remain true with $\mathring{\upalpha}^{1/2},\, \auxbootsmall$ respectively replaced by $C \mathring{\upalpha},\, C \fundbootsmall$.

§ SHARP CONTROL OF $\UPMU$ AND THE PROPERTIES OF $\UPSILON$

In this section, we derive sharp control of $\upmu$ and its derivatives, as well as strict improvements
of the bootstrap assumptions from Sects. <ref>–<ref>.
Because our $L^2$-type energies feature $\upmu$ weights,
these sharp estimates will play a fundamental role in our proof of the energy estimates.
We also derive homeomorphism and diffeomorphism properties
of the change of variables map from geometric coordinate to Cartesian coordinates,
which are crucial for understanding the structure of the singular boundary.

§.§ Sharp control of $\upmu$ and its derivatives

The following estimates hold.

Minima of $\upmu$ occur precisely along $\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}$.
For each fixed $\timefunction \in [\timefunction_0,\timefunctionboot] = [-\mupositive,-\upmuboot]$,
we have:

\begin{align} \label{E:MINVALUEOFMUONFOLIATION}
\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\upmu
& =
- \timefunction,
\end{align}

and the minimum value of $- \timefunction$
in (<ref>) is achieved by $\upmu$ precisely on the $\upmu$-adapted torus
\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}
$ defined in (<ref>).
In particular,
\upmuboot
\inf_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}} \upmu$,
and the infimum is not achieved.

$\upmu$ is large when $|u| \geq \interestingu$.
The following lower bound holds, where $\boringregionmupositive > 0$ is the constant appearing in

\begin{align} \label{E:MUISLARGEINBORINGREGION}
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}
\backslash
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \upmu
& \geq \frac{\boringregionmupositive}{2}.
\end{align}

Location of $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$
and $\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}$.
denoting the sets defined in (<ref>)–(<ref>),
we have:

\begin{align} \label{E:MUXMUKAPPALEVELSETLOCATION}
\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}
\subset
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{\muxmulevelsetvalue},
\\
\text{for each $ \timefunction \in [\timefunction_0,\timefunctionboot]$,} \qquad
\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue},
\subset
\hypthreearg{\timefunction}{[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{\muxmulevelsetvalue}.
\label{E:IMPROVEDLEVELSETSTRUCTUREANDLOCATIONOFMIN}
\end{align}

Moreover, with $\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}$
denoting the inverse function of the function
$\CHOVroughtomumuxmu{\muxmulevelsetvalue}$ defined in (<ref>),
we have:

\begin{align} \label{E:INTERMSOFPHICHOVMAPIMPROVEDLEVELSETSTRUCTUREANDLOCATIONOFMIN}
\text{For each $\mulevelsetvalue \in [\mupositive,\upmuboot]$,} \qquad
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}
\left(\lbrace \mulevelsetvalue \rbrace \times \lbrace - \muxmulevelsetvalue \rbrace \times \mathbb{T}^2 \right)
\subset
\lbrace - \mulevelsetvalue \rbrace \times \left[-\frac{\interestingu}{2},\frac{\interestingu}{2}\right] \times \mathbb{T}^2.
\end{align}

Transversal convexity of $\upmu$ and its consequences.
The following estimates hold,
where $\Wtransarg{\muxmulevelsetvalue}$ is the vectorfield from Def. <ref>:

\begin{align}
\begin{split} \label{E:MUTRANSVERSALCONVEXITY}
\frac{\secondtransversalderivativemulowerbound}{2}
& \leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\left\lbrace
\Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu,
\,
\Wtransarg{\muxmulevelsetvalue} \muX \upmu,
\,
\muX \muX \upmu,
\,
\muX \muX \upmu - \frac{(\muX \upmu) \Lunit \muX \upmu}{\Lunit \upmu},
\,
\geop{u} \muX \upmu,
\,
\geop{u} \muX \upmu - \frac{(\geop{u} \upmu) \geop{t} \muX \upmu}{\geop{t} \upmu},
\,
\roughgeop{u} \muX \upmu
\right\rbrace
\\
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\left\lbrace
\Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu,
\,
\Wtransarg{\muxmulevelsetvalue} \muX \upmu,
\,
\muX \muX \upmu,
\,
\muX \muX \upmu - \frac{(\muX \upmu) \Lunit \muX \upmu}{\Lunit \upmu},
\,
\geop{u} \muX \upmu,
\,
\geop{u} \muX \upmu - \frac{(\geop{u} \upmu) \geop{t} \muX \upmu}{\geop{t} \upmu},
\,
\roughgeop{u} \muX \upmu
\right\rbrace
\leq
\frac{2}{\secondtransversalderivativemulowerbound},
\end{split}
\end{align}

\begin{align}
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}
\backslash
\mathcal{M}^{(\muxmulevelsetvalue)}_{[\timefunction_0,\timefunctionboot],[-\frac{\interestingu}{2},\frac{\interestingu}{2}]}}
|\muX \upmu + \muxmulevelsetvalue|
& \geq \frac{\secondtransversalderivativemulowerbound \interestingu}{8}.
\label{E:REGIONWHEREMUXMUKAPPALEVELSETISNOTLOCATED}
\end{align}

Moreover, the following pointwise estimates hold on
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}
$, where $\Rtransarg{\muxmulevelsetvalue}$ is the vectorfield defined in (<ref>):

\begin{align} \label{E:WMANDRTRANSMUUBOUNDEDBYSQRTMU}
|\Wtransarg{\muxmulevelsetvalue} \upmu|,
\,
|\Rtransarg{\muxmulevelsetvalue} \upmu|
& \leq
C \sqrt{\upmu}.
\end{align}

Rough control of null and almost null derivatives of $\upmu$ in the interesting region.
The following estimates hold,
where $\argLrough{\muxmulevelsetvalue}$ is the vectorfield defined in (<ref>):

\begin{align} \label{E:BOUNDSONLMUINTERESTINGREGION}
\frac{9}{8}
\mathring{\updelta}_*
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \Lunit \upmu
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \Lunit \upmu
\leq
\frac{7}{8}
\mathring{\updelta}_*,
\\
\frac{9}{8}
\mathring{\updelta}_*
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \geop{t} \upmu
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \geop{t} \upmu
\leq
\frac{7}{8}
\mathring{\updelta}_*,
\label{E:BOUNDSONGEOMETRICTDERIVATIVEMUINTERESTINGREGION}
\\
\frac{3}{2}
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \argLrough{\muxmulevelsetvalue} \upmu
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}} \argLrough{\muxmulevelsetvalue} \upmu
\leq
- \frac{2}{3}.
\label{E:SIZEBOUNDSONWIDETILDELMUINTERESTINGREGION}
\end{align}

Sharp control of $\Lunit \timefunctionarg{\muxmulevelsetvalue}$.
The following estimates hold:

\begin{align} \label{E:ROUGHTIMEFUNCTIONLDERIVATIVEBOUNDS}
\frac{7}{9}
\mathring{\updelta}_*
\leq
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\Lunit \timefunctionarg{\muxmulevelsetvalue}
\leq
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\Lunit \timefunctionarg{\muxmulevelsetvalue}
\leq
\frac{9}{7} \mathring{\updelta}_*.
\end{align}

Estimates for $\upmu$ along the flow map of $\FlowmapLrougharg{\muxmulevelsetvalue}$.
Let $\FlowmapLrougharg{\muxmulevelsetvalue}$ denote the $\timefunction_0$-normalized
flow map of $\argLrough{\muxmulevelsetvalue}$ from Lemma <ref>.
Then the following estimates hold:

\begin{align} \label{E:MUMUSTDECREASEININTERESTINGREGION}
\sup_{\substack{|u| \leq \interestingu
\\
\timefunction_0 \leq \timefunction \leq \timefunction + \Delta \timefunction \leq \timefunctionboot
\\
(x^2,x^3) \in \mathbb{T}^2}}
\frac{\upmu \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction + \Delta,u,x^2,x^3)}
{\upmu \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)}
& \leq 1,
\\
\sup_{\substack{u \in [- \rightu,\leftu] \backslash [-\interestingu,\interestingu]
\\
\timefunction_0 \leq \timefunction \leq \timefunction + \Delta \timefunction \leq \timefunctionboot
\\
(x^2,x^3) \in \mathbb{T}^2}}
\frac{\upmu \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction + \Delta,u,x^2,x^3)}
{\upmu \circ \FlowmapLrougharg{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3)}
& \leq C.
\label{E:MURATIOBOUNDEDINBORINGREGION}
\end{align}

A lower bound tied to the blowup in the interesting region.
The following lower bounds holds:

\begin{align} \label{E:LOWERBOUNDONMAGNITUDEOFXRPLUS}
\min_{\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}}
\upmu |X \RRiemann|
& \geq
\frac{\mathring{\updelta}_*}{|\bar{\Speed}_{;\LogDensity} + 1|},
\end{align}

where $\bar{\Speed}_{;\LogDensity} \eqdef \Speed_{;\LogDensity}(\LogDensity = 0,\Ent=0)$
is $\Speed_{;\LogDensity}$ evaluated at the trivial solution,
$\bar{\Speed}_{;\LogDensity} + 1$ is a non-zero constant
by assumption, and the vectorfield $X$ has Euclidean length satisfying
$\sqrt{\sum_{a=1}^3 (X^a)^2} = 1 + \mathcal{O}(\mathring{\upalpha})$.

Especially sharp control in a small neighborhood.
Recall that $\flowmapWtransargtwoarg{\muxmulevelsetvalue}{\Delta u}$ is the flow map of $\Wtransarg{\muxmulevelsetvalue}$
(see Lemma <ref>).
There exists a constant $\Delta U$ with $0 < \Delta U < \frac{\interestingu}{2}$,
a neighborhood (in the geometric coordinate topology of $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$)
$\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}$ of
of the form:

\begin{align} \label{E:SMALLNEIGHBORHOOD}
\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{(-\Delta U,\Delta U)}
\left(\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)
\eqdef
\bigcup_{\Delta u' \in (-\Delta U,\Delta U)}
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{\Delta u'}
\left( \datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)
\end{align}

such that:

\begin{align} \label{E:SMALLNEIGHBORHOODCONTAINEDININTERESTINGREGION}
\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}
\subset
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue},
\end{align}

and a constant $\awayfromsmallneightborhoodnmupositive > 0$ defined by:

\begin{align} \label{E:MU2DEF}
\awayfromsmallneightborhoodnmupositive
\eqdef \min \left\lbrace \frac{\secondtransversalderivativemulowerbound}{4} (\Delta U)^2, \frac{\boringregionmupositive}{2} \right\rbrace
\end{align}

(where $\boringregionmupositive$ is as in (<ref>))
such that the following estimates hold:

\begin{align} \label{E:WIDETILDELMUISALMOSTMINUSONEINSMALLNEIGHBORHOOD}
- 1.01
\leq
\min_{\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}} \argLrough{\muxmulevelsetvalue} \upmu
\leq
\max_{\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}} \argLrough{\muxmulevelsetvalue} \upmu
\leq
\\
\awayfromsmallneightborhoodnmupositive
\leq
\min_{
\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}
\backslash
\smallneighborhoodofcreasetwoarg{[\timefunction_0,\timefunctionboot]}{\muxmulevelsetvalue}} \upmu.
\label{E:EASYREGIONLOWERBOUNDFORMU}
\end{align}

Throughout this proof, we sometimes silently use the continuous extension properties shown in Lemma <ref>.
In particular, these properties allow us to extend various results that
were proved prior to the lemma on the half-open rough time interval $[\timefunction_0,\timefunctionboot)$
to the closed rough time interval

Proof of (<ref>):
To prove (<ref>) for $\Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu$,
we use first use (<ref>),
Lemmas <ref> and <ref>,
and the estimate
$\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$ (see (<ref>))
to deduce that for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot] \times [-\interestingu,\interestingu]$,
we have
\argLrough{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu
\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)} \leq C$.
From this bound,
and the data assumption
we deduce that in the region $\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}$,
we have
\secondtransversalderivativemulowerbound
C \mupositive
\leq
\Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu
\leq
\frac{1}{\secondtransversalderivativemulowerbound}
C \mupositive
which, for $\mupositive$ sufficiently small,
implies (<ref>)
for $\Wtransarg{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu$
(recall that, as we highlighted in Sect. <ref>, the constants $C$
can be chosen to be independent of $\mupositive$).
The remaining estimates in (<ref>)
follow from a nearly identical argument,
where we use the identity
when deriving the estimates involving $\roughgeop{u} \muX \upmu$.

Proofs of
and (<ref>):
The arguments we used in the proof of (<ref>)
also yield that $|\argLrough{\muxmulevelsetvalue} \Wtransarg{\muxmulevelsetvalue} \upmu| \leq C$.
Hence, noting that $\Wtransarg{\muxmulevelsetvalue} \upmu = \muX \upmu + \muxmulevelsetvalue$,
and using (<ref>)
and the data assumption (<ref>),
we find that
in the spacetime region $\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}
\backslash
\mathcal{M}^{(\muxmulevelsetvalue)}_{[\timefunction_0,\timefunctionboot],[-\frac{\interestingu}{2},\frac{\interestingu}{2}]}$,
we have
$|\muX \upmu + \muxmulevelsetvalue|
\geq
\frac{\secondtransversalderivativemulowerbound \interestingu}{4} - C \mupositive
which, for $\mupositive$ sufficiently small, implies (<ref>).

The estimates
(<ref>) and (<ref>)
follow from similar arguments
based on the data assumptions
(<ref>) and (<ref>)
and the estimates
$|\argLrough{\muxmulevelsetvalue} \upmu| \leq C$
$|\argLrough{\muxmulevelsetvalue} \Lunit \upmu | \leq C \fundbootsmall$,
which follow from (<ref>)
and the estimates of Prop. <ref>.
We also conclude (<ref>)
by combining these arguments with
the relation $\geop{t} \upmu = \Lunit \upmu + \mathcal{O}(\fundbootsmall)$,
which follows from the identity $\geop{t} \upmu = \Lunit \upmu - \Lunit^A \geop{x^A} \upmu$,
Lemma <ref>,
Lemma <ref>,
and the estimates of Prop. <ref>.

Proof of (<ref>):
This estimate follows from the proof of (<ref>),
except we now use the
bound (<ref>)
in place of the
bootstrap assumption (<ref>) used in the proof of

Proof of (<ref>) and (<ref>):
(<ref>) follows from definition (<ref>)
and the estimates (<ref>) and (<ref>).
(<ref>) then follows from (<ref>),
which implies that $\upmu$ decreases along the integral curves of $\argLrough{\muxmulevelsetvalue}$ when $|u| \leq \interestingu$.

Proof of (<ref>):
We use (<ref>),
Lemma <ref>,
the estimates of Prop. <ref>,
to deduce that in $\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}$,
we have the estimate
\frac{1}{2}
\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
|\muX \RRiemann|
\geq
\frac{7}{8} \mathring{\updelta}_*
\mathcal{O}(\fundbootsmall)
\geq
\frac{3}{4} \mathring{\updelta}_*$,
and, also Taylor expanding
$\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)$
around the background solution $\wavearray = 0$
and using
the estimate
\Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
|\muX \RRiemann|
\left\lbrace
1 + \mathcal{O}(\mathring{\upalpha})
\right\rbrace
|\bar{\Speed}_{;\LogDensity} + 1|
|\muX \RRiemann|
Combining these two estimates, we conclude (<ref>).

The fact that $\sqrt{\sum_{a=1}^3 (X^a)^2} = 1 + \mathcal{O}(\mathring{\upalpha})$
follows from
and the estimates of Prop. <ref>.

Proof of (<ref>) and (<ref>):
Since $|\muX \upmu + \muxmulevelsetvalue|  = 0$ along $\breve{\mathbb{X}}^{(-\muxmulevelsetvalue)}$,
(<ref>) follows directly from (<ref>).

(<ref>) then follows from
(<ref>) since
\subset \datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$.

Proof of (<ref>) and the fact
that $\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} \upmu$
occurs in
\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}
Since the construction of the rough time function $\timefunctionarg{\muxmulevelsetvalue}$ is such that
\timefunctionarg{\muxmulevelsetvalue}|_{\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}}$,
it follows from
and (<ref>)
that for each fixed $\timefunction \in [\timefunction_0,\timefunctionboot] = [-\mupositive,-\upmuboot]$,
$\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} \upmu$ is achieved only in
the subset $\hypthreearg{\timefunction}{[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{\muxmulevelsetvalue}$,
which is interior to $\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
Hence, since $\Wtransarg{\muxmulevelsetvalue}$ is tangent to
$\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
it must be that $\Wtransarg{\muxmulevelsetvalue} \upmu = 0$ at the minima.
Considering also the definition of $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$
and the identity
$\Wtransarg{\muxmulevelsetvalue} \upmu = \muX \upmu + \muxmulevelsetvalue$,
we see that the minima of $\upmu$ in $\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$ must belong to
\cap
\hypthreearg{\timefunction}{[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{\muxmulevelsetvalue}$.
Hence, we conclude that
$\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} \upmu = - \timefunction$, which is
Moreover, also using
Lemma <ref>,
we conclude
that the torus
\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}
\InverseCHOVroughtomumuxmu{\muxmulevelsetvalue}(\lbrace -\timefunction \rbrace \times \lbrace -\muxmulevelsetvalue \rbrace \times \mathbb{T}^2)
is exactly the set of points
\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}
where $\upmu$ achieves its minimum value of $-\timefunction$.

Proof of (<ref>):
First, we use
Lemma <ref>,
Lemma <ref>,
Lemma <ref>,
and the estimates of Lemma <ref>,
Prop. <ref>,
and Cor. <ref>
to deduce that
$|\Wtransarg{\muxmulevelsetvalue} \upmu| \leq C$
$\Rtransarg{\muxmulevelsetvalue} \upmu
\Wtrans \upmu + \upmu \mathcal{O}(\fundbootsmall)
= \Wtrans \upmu + \mathcal{O}(\fundbootsmall)$,
where we clarify that we will use the first equality in the next paragraph.
From these bounds
and (<ref>),
we see that the bounds stated in (<ref>)
hold in $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}
\backslash
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}$.

It remains for us to prove the desired bounds in
To this end, we assume that
$\timefunction \in [\timefunction_0,\timefunctionboot]$
$q \in \hypthreearg{\timefunction}{[- \interestingu,\interestingu]}{\muxmulevelsetvalue}$.
By Lemma <ref>,
there is a unique integral curve of $\Wtransarg{\muxmulevelsetvalue}$
that joins $q$ to a point $q_0 \in \twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}$
on the primal torus $\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}$
(along which $\upmu \equiv - \timefunction$),
which, in view of the already proven result (<ref>),
is contained in $\hypthreearg{\timefunction}{[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{\muxmulevelsetvalue}$.
Let $\iota = \iota(u')$ denote this integral curve, parameterized by the eikonal function,
and let $u_0$ and $u$ respectively denote the eikonal function values corresponding to $q_0$ and $q$.
In particular, $\iota(u) = q$, $\iota(u_0) = q_0$,
$\upmu \circ \iota(u_0) = - \timefunction$,
and by (<ref>), $\Wtrans \upmu \circ \iota(u_0) = 0$.
Using (<ref>) and the mean value theorem,

we see that $|\Wtrans \upmu \circ \iota(u)| \leq \frac{2}{\secondtransversalderivativemulowerbound} |u - u_0|$
and $\upmu \circ \iota(u) \geq - \timefunction + \frac{\secondtransversalderivativemulowerbound}{4} (u- u_0)^2
\geq \frac{\secondtransversalderivativemulowerbound}{4} (u- u_0)^2$.
Combining the above results, we find that at $q$, we have
$|\Wtrans \upmu| \leq \frac{2}{\secondtransversalderivativemulowerbound}
\sqrt{\frac{4}{\secondtransversalderivativemulowerbound}} \sqrt{\upmu}
\leq \frac{4}{\secondtransversalderivativemulowerbound^{3/2}} \sqrt{\upmu}$,
which yields (<ref>) for the first term on the LHS.
From this estimate,
the bound
$\Rtransarg{\muxmulevelsetvalue} \upmu = \Wtrans \upmu + \upmu \mathcal{O}(\fundbootsmall)$ noted in the previous paragraph,
and the estimate $\upmu \leq C$ implied by (<ref>),
we conclude the desired estimate (<ref>)
for the second term on the LHS.

Proof of (<ref>):
Recalling that $\flowmapWtransargtwoarg{\muxmulevelsetvalue}{\Delta u}$ is the flow map of $\Wtransarg{\muxmulevelsetvalue}$,
for real numbers $\Delta u$ small and positive, we consider the set
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{(-\Delta u,\Delta u)}
\left(\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)
\eqdef
\bigcup_{\Delta u' \in (-\Delta u,\Delta u)}
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{\Delta u'}
\left( \datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)
which by Lemma <ref>
and (<ref>)
is a neighborhood of $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}$
in $\twoargMrough{[\timefunction_0,\timefunctionboot],[-\interestingu,\interestingu]}{\muxmulevelsetvalue}$.
Next, using (<ref>),
Lemma <ref>,
Lemma <ref>,
and the estimates of Lemma <ref>
Prop. <ref>,
we deduce that
|\Wtransarg{\muxmulevelsetvalue} \argLrough{\muxmulevelsetvalue} \upmu|
\leq C
Moreover, since Lemma <ref>
and definition (<ref>) imply that
\argLrough{\muxmulevelsetvalue} \upmu|_{\breve{\mathbb{X}}^{(-\muxmulevelsetvalue)}}
= -1
we can use the mean value theorem to deduce that in
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{(-\Delta u,\Delta u)}
\left( \datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)
the following estimates hold:
- 1 - C \Delta u
\leq
\argLrough{\muxmulevelsetvalue} \upmu \circ \iota(u)
\leq -1
+ C \Delta u$.
Choosing and fixing a value of $\Delta u$, which we denote by
$\Delta U$, to be sufficiently small such that $C \Delta U < .01$,
we arrive at (<ref>)
\eqdef
\flowmapWtransargtwoarg{\muxmulevelsetvalue}{(-\Delta U,\Delta U)}
\left( \datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]} \right)

Proof of (<ref>) and (<ref>):
We recall that
\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot]}
\bigcup_{\mulevelsetvalue \in [-\timefunctionboot,\mupositive]}
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
and that for $\timefunction \in [\timefunction_0,\timefunctionboot]$,
we have
\upmu|_{\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue}}
= -\timefunction
Moreover, the arguments we used in the proof of (<ref>)
imply that for $|\Delta U| \leq \frac{\interestingu}{2}$,
we have
\frac{\secondtransversalderivativemulowerbound}{4} (\Delta U)^2
\leq
\min_{\flowmapWtransargtwoarg{\muxmulevelsetvalue}{(-\Delta U,\Delta U)}\left(\twoargmumuxtorus{-\timefunction}{-\muxmulevelsetvalue} \right)}
\upmu
Hence, in view of (<ref>)
and definition (<ref>),
we see that
holds with $\Delta U$ defined to be the small constant fixed in the proof of
(<ref>) then follows as a simple
consequence of (<ref>),
and the bound
\max_{\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\upmu
\lesssim
implied by the third item in
Lemma <ref>.

§.§ Homeomorphism and diffeomorphism properties of $\Upsilon$

Our main goal in this section is to reveal the homeomorphism and diffeomorphism properties of
the change of variables map
$\Upsilon(t,u,x^2,x^3) = (t,x^1,x^2,x^3)$.
We start with the following monotonicity lemma, which plays an important role in
controlling $\Upsilon$.

The following identity holds, where $\phi = \phi(u)$ is the cut-off from Def. <ref>:

\begin{align} \label{E:ROUGHUDERIVATIVEOFCATERSIANX1}
\begin{split}
\roughgeop{u} x^1
& =
\upmu
\left\lbrace
\frac{X^A X^A}{X^1}
\frac{X^A (\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}) \Lunit^B X^B}{(\geop{t} \timefunctionarg{\muxmulevelsetvalue}) X^1}
\frac{X^A \geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
\Lunit^1
\right\rbrace
\\
& \ \
\phi
\frac{\muxmulevelsetvalue}{\Lunit \upmu}
\left\lbrace
\Lunit^1
\frac{\Lunit^A X^A}{X^1}
\frac{\Lunit^A (\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}) \Lunit^B X^B}{(\geop{t} \timefunctionarg{\muxmulevelsetvalue}) X^1}
\frac{\Lunit^A \geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
\Lunit^1
\right\rbrace.
\end{split}
\end{align}

Moreover, the following estimate holds on
$\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$:

\begin{align}  \label{E:X1MONOTONOCITYESTIMATE}
\roughgeop{u} x^1
& = - \upmu
\left\lbrace
1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha})
\right\rbrace
\phi
\frac{\muxmulevelsetvalue}{\Lunit \upmu}
\left\lbrace
1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha})
\right\rbrace.
\end{align}

for every fixed
$(\timefunction,x^2,x^3) \in [\timefunction_0,\timefunctionboot] \times \mathbb{T}^2$,
the map $u \rightarrow x^1(\timefunction,u,x^2,x^3)$ is strictly decreasing
on $[- \rightu,\leftu]$.

follows from

(<ref>) then follows from (<ref>),
Lemma <ref>,
the estimates of Lemma <ref>
and Prop. <ref>,
and Cor. <ref>,
and (<ref>),
which in particular imply that
$X^1 = -1 + \Xsmall^1 = - 1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha})$,
$\Lunit^1 = 1 + \Lsmall^1 = 1 + \mathcal{O}_{\mydiam}(\mathring{\upalpha})$,
$X^A, \, \Lunit^A = \mathcal{O}(\varepsilon) = \mathcal{O}_{\mydiam}(\mathring{\upalpha})$.

To prove the monotonicity of $x^1$, we note that
(<ref>) and Prop. <ref>
imply that $\roughgeop{u} x^1 < 0$,
except in the case $\timefunctionboot = \muxmulevelsetvalue = 0$,
where $\roughgeop{u} x^1$ vanishes precisely along the torus
which is contained
in $\hypthreearg{0}{[-\frac{1}{2}\interestingu,\frac{1}{2}\interestingu]}{0}$.
Moreover, (<ref>)
implies that
$\roughgeop{u} \muX \upmu|_{\twoargmumuxtorus{0}{0}} > 0
Thus, since (<ref>) implies that
(i.e., the image of the crease in rough adapted coordinate space
$\mathbb{R}_{\timefunction} \times \mathbb{R}_u \times \mathbb{T}^2$)
is a graph over $\mathbb{T}^2$,
we see that every integral curve of
$\roughgeop{u}$ in $\hypthreearg{0}{[- \rightu,\leftu]}{0}$
intersects $\twoargmumuxtorus{0}{0}$ in precisely one point.
In total, we have shown that for every fixed
$(\timefunction,x^2,x^3) \in [\timefunction_0,\timefunctionboot] \times \mathbb{T}^2$,
the map $u \rightarrow x^1(\timefunction,u,x^2,x^3)$ on the domain $[- \rightu,\leftu]$
has a negative derivative, except at possibly a single point
in $[-\frac{\interestingu}{2},\frac{\interestingu}{2}]$.
From this fact, we conclude that the map is strictly decreasing
as desired.

The change of variables map $\Upsilon(t,u,x^2,x^3) = (t,x^1,x^2,x^3)$
is injective on the compact set
$\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$
and satisfies:

\begin{align} \label{E:C31BOUNDFORCHOVFROMGEOTOCARTESIAN}
\| \Upsilon \|_{C_{\textnormal{geo}}^{3,1}(\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)})}
\leq C.
\end{align}

In particular, $\Upsilon$ is a homeomorphism from $\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$
onto its image.

Moreover, with $d_{\textnormal{geo}} \Upsilon$ denoting the Jacobian matrix of $\Upsilon$,
we have:

\begin{align} \label{E:GEOTOCARTESIANJACOBIANDETERMINANTESTIMATE}
\mbox{\upshape det} d_{\textnormal{geo}} \Upsilon
& =
\upmu \frac{\Speed^2}{X^1}
\approx
\upmu.
\end{align}

In addition, if $\timefunctionboot < 0$, then $\Upsilon$ is a diffeomorphism from
$\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$ onto its image.

Finally, let $\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}$ be the $\upmu$-adapted torus defined in
(<ref>), and let $\Upsilon\left(\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue} \right)$
be the image in Cartesian coordinate space of
$\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}$ under $\Upsilon$.
Then for $\mulevelsetvalue \in [\upmuboot,\mupositive] = [-\timefunctionboot,-\timefunction_0]$,
$\Upsilon\left(\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue} \right)$ is $C^{1,1}$
embedded submanifold of Cartesian
space that is diffeomorphic to $\mathbb{T}^2$. More precisely,
and $\Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}$ denoting the functions on $\mathbb{T}^2$
from (<ref>), the map $\embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$ defined by:

\begin{align} \label{E:MUADAPATEDTORIINCARTESIANSPACE}
\embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}(x^2,x^3)
\eqdef
\Upsilon
\circ
\left(\Cartesiantisafunctiononmumxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),
\Eikonalisafunctiononmumuxtoriarg{\mulevelsetvalue}{-\muxmulevelsetvalue}(x^2,x^3),
\right)
\end{align}

is a $C^{1,1}$ embedding, i.e., a $C^{1,1}$
diffeomorphism from $\mathbb{T}^2$ onto
$\Upsilon\left(\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue} \right)$.

We already proved the bound (<ref>) in Lemma <ref>.

Next, we use
Lemma <ref>,
Prop. <ref>
to compute that
\frac{\partial \Upsilon(t,u,x^2,x^3)}{\partial (t,u,x^2,x^3)}
= \begin{pmatrix}
1 & 0 & 0 & 0
\\
\Lunit^1 + * & \upmu \frac{\Speed^2}{X^1} & * & *
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{pmatrix}
where here and in the rest of the proof,
“$*$” denotes any quantity that is pointwise bounded in magnitude by $\mathcal{O}(\mathring{\upalpha})$.
Thus, $\mbox{\upshape det} \frac{\partial \Upsilon(t,u,x^2,x^3)}{\partial (t,u,x^2,x^3)} = \upmu \frac{\Speed^2}{X^1}$,
and therefore, using (<ref>),
Lemma <ref>,
Prop. <ref>,
we compute that
$\mbox{\upshape det} \frac{\partial \Upsilon(t,u,x^2,x^3)}{\partial (t,u,x^2,x^3)}
- \left\lbrace 1 + \mathcal{O}(\mathring{\upalpha}) \right\rbrace \upmu$,
which yields (<ref>).

If $\timefunctionboot < 0$, then by Prop. <ref>,
$\upmu$ is uniformly positive on $\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$,
and from (<ref>) and the inverse function
theorem, we see that
$\Upsilon$ is a local diffeomorphism on $\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$.
Thus, to complete the proof, we need to show that $\Upsilon$ is injective on
$\Mrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}^{(\muxmulevelsetvalue)}$,
even if $\timefunctionboot = 0$. We will achieve this by
proving the injectivity of the map
$(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3) \rightarrow (\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3)$
on the domain
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$,
and then the injectivity of the map
$(\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3) \rightarrow (t,x^1,x^2,x^3)$;
since the composition of two injective functions is injective, this
would finish the proof.

To proceed, we first note that the injectivity of the map
$(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3) \rightarrow (\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3)$
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \mathbb{T}^2$
follows from the monotonicity of the map $u \rightarrow x^1(\timefunction,u,x^2,x^3)$
guaranteed by Lemma <ref>.
For use below, we also note that by
Lemma <ref>,
and the estimates of Prop. <ref>,
we have:

\begin{align} \label{E:ROUGHTIMEDERIVATIVEOFCARTESIANX1ISAPPROXIMATELYUNITY}
\roughgeop{\timefunction} x^1
\approx \geop{t} x^1
& =
\Lunit x^1
\Lunit^A \geop{x^A} x^1
= 1
\Lsmall
\Lunit^A \geop{x^A} x^1
\approx 1.
\end{align}

Now for each fixed $(x^2,x^3) \in \mathbb{T}^2$,
let ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$ denote the image of the set
$[\timefunction_0,\timefunctionboot] \times [- \rightu,\leftu] \times \lbrace (x^2,x^3) \rbrace$
under the map $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3) \rightarrow (\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3)$.
The arguments given above, including the monotonicity guaranteed by (<ref>),
imply that for each fixed $(u,x^2,x^3) \in [- \rightu,\leftu] \times \mathbb{T}^2$,
the map $\timefunction \rightarrow x^1(\timefunction,u,x^2,x^3)$
is strictly increasing on $[\timefunction_0,\timefunctionboot]$, and that
for each fixed $(\timefunction,x^2,x^3) \in[\timefunction_0,\timefunctionboot] \times \mathbb{T}^2$,
the map
$u \rightarrow x^1(\timefunction,u,x^2,x^3)$ is strictly decreasing on $[- \rightu,\leftu]$.
It follows that there exist scalar functions $\timefunction \rightarrow {^{(\muxmulevelsetvalue)}a}_{x^2,x^3}(\timefunction)$
$\timefunction \rightarrow {^{(\muxmulevelsetvalue)}b}_{x^2,x^3}(\timefunction)$
on $[\timefunction_0,\timefunctionboot]$
such that
\lbrace (\timefunction,x^1,x^2,x^3) \ | \
\timefunction \in [\timefunction_0,\timefunctionboot],
\,
{^{(\muxmulevelsetvalue)}a}_{x^2,x^3}(\timefunction) \leq x^1 \leq {^{(\muxmulevelsetvalue)}b}_{x^2,x^3}(\timefunction) \rbrace
where ${^{(\muxmulevelsetvalue)}a}_{x^2,x^3}(\cdot)$ and ${^{(\muxmulevelsetvalue)}b}_{x^2,x^3}(\cdot)$ are $C^1$ functions of $\timefunction$
such that ${^{(\muxmulevelsetvalue)}a}_{x^2,x^3}(\timefunction) < {^{(\muxmulevelsetvalue)}b}_{x^2,x^3}(\timefunction)$ and
\frac{d}{d \timefunction} {^{(\muxmulevelsetvalue)}a}_{x^2,x^3},
\,
\frac{d}{d \timefunction} {^{(\muxmulevelsetvalue)}b}_{x^2,x^3}
\approx 1

To complete the proof, it remains for us to show that the map
$(\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3) \rightarrow (t,x^1,x^2,x^3)$
is injective. Let ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$ be the set from the previous paragraph.
It suffices to show that for each fixed $(x^2,x^3) \in \mathbb{T}^2$,
any two distinct points in ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$ with the same $x^1$ coordinate
must be mapped to distinct points under the map
$(\timefunctionarg{\muxmulevelsetvalue},x^1,x^2,x^3) \rightarrow (t,x^1,x^2,x^3)$.
The structure of ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$ revealed in the previous paragraph
shows that for any two distinct points in ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$ with the same $x^1$ coordinate,
the straight line segment joining them
(along which $x^1,x^2,x^3$ are constant and $\newtimefunction$ varies)
is contained in ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$
(this can be thought of as the vertical convexity of ${^{(\muxmulevelsetvalue)}\mathcal{I}}_{x^2,x^3}$).
Hence, to complete the proof, it suffices for us to show that the partial derivative of $t$
with respect to $\timefunctionarg{\muxmulevelsetvalue}$ in the coordinate system
is positive, except possibly when $\timefunctionarg{\muxmulevelsetvalue} = 0$.
To proceed, we note that the partial derivative of interest is equal to
$\frac{1}{\partial_t \timefunctionarg{\muxmulevelsetvalue}}$,
where $\partial_t$ is the Cartesian partial derivative.
We now use
Lemma <ref>,
Lemma <ref>,
and the estimates of
Lemma <ref>,
Prop. <ref>,
and (<ref>)
to compute that
$\partial_t = \Lunit + (1 + *) X + * \Yvf{2} + * \Yvf{3}$
and that
$\frac{1}{\partial_t \timefunctionarg{\muxmulevelsetvalue}}
\approx
\frac{1}{1 + \frac{\muxmulevelsetvalue \phi}{\upmu}}
\frac{\upmu}{\upmu + \muxmulevelsetvalue \phi}
$, where $\phi$ is the cut-off function from Def. <ref>.
We now recall that by (<ref>),
$\upmu$ can vanish only when $\timefunctionarg{\muxmulevelsetvalue} = 0$.
follows that
\frac{1}{\partial_t \timefunctionarg{\muxmulevelsetvalue}} > 0
except possibly when $\timefunctionarg{\muxmulevelsetvalue} = 0$,
which is the desired result.

Finally, since the map $\embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$ from (<ref>)
is the composition of the injective $C^{3,1}$ map $\Upsilon$ with the $C^{1,1}$ embedding
of $\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}$
given by (<ref>),
it follows that $\embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$ is a $C^{1,1}$ injection from $\mathbb{T}^2$ into Cartesian coordinate space.
Moreover, its differential $d_{(x^2,x^3)} \embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$ is a $4 \times 2$ matrix
whose lower $2 \times 2$ block is the identity
\begin{pmatrix}
1 & 0
\\
0 & 1
\end{pmatrix}
That is, $d_{(x^2,x^3)} \embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$
is full rank, and therefore $\embeddingofmuadapatedtorinCartesianspace{\muxmulevelsetvalue}$ is a $C^{1,1}$ embedding.
This concludes the proof of the proposition.

§.§ Control of the size of $t$ and $x^1$

In the next lemma,
we derive improvements of the bootstrap assumptions of Sect. <ref>.

The following estimates hold for $\timefunction \in [\timefunction_0,\timefunctionboot]$:

\begin{align} \label{E:SIZEOFCARTESIANT}
\frac{1}{3 \mathring{\updelta}_*}
\leq
\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} t
\leq
\sup_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} t
\leq
\frac{3}{\mathring{\updelta}_*},
\\
- \leftu
\frac{1}{3 \mathring{\updelta}_*}
\leq
\min_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} x^1
\leq
\sup_{\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}} x^1
\leq
\rightu
\frac{3}{\mathring{\updelta}_*}.
\label{E:SIZEOFCARTESIANX1}
\end{align}

We first prove (<ref>).
we have $\argLrough{\muxmulevelsetvalue} t = \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit t
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\approx 1
Hence, recalling that
$\argLrough{\muxmulevelsetvalue} u  = 0$
$\argLrough{\muxmulevelsetvalue} \timefunction = 1$,
we can integrate along the integral curves of
$\argLrough{\muxmulevelsetvalue}$ starting from any point in
$\hypthreearg{\timefunction_0}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$
and use the data assumption
and the identity $|\timefunction_0| = \mupositive$
to deduce that in $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
we have
\frac{1}{2 \mathring{\updelta}_*}
\frac{1}{C} \mupositive
\leq
\leq
\frac{2}{\mathring{\updelta}_*}
C \mupositive
which, for $\mupositive$ sufficiently small,
implies (<ref>).

To prove (<ref>),
we first use
and (<ref>)
to deduce that
we have $\argLrough{\muxmulevelsetvalue} x^1 = \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit x^1
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (1 + \Lsmall^1)
\approx 1
$. We now argue
as in the proof of (<ref>)
using the data assumption (<ref>),
thereby arriving at (<ref>).

§ MODIFIED QUANTITIES FOR CONTROLLING THE ACOUSTIC GEOMETRY

In this section, we construct “modified” versions of the eikonal function quantity $\mytr_{\gtorus} \upchi$,
and we derive the transport equations that they satisfy.
There are two kinds of modified quantities: “partially modified” and “fully modified.”
The partially modified quantities, when combined with integration by parts, will allow us to avoid
uncontrollable error integrals in the wave equation energy identities.
The fully modified quantities,
when combined with elliptic estimates on the tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$,
will allow us to control the top-order
$\Yvf{A}$-derivatives of $\mytr_{\gtorus} \upchi$ without losing derivatives.

§.§ Decompositions of $\Ricfour_{\Lunit \Lunit}$

In the next lemma, we provide two key decompositions of $\Ricfour_{\Lunit \Lunit}$, where $\Ricfour$ is the Ricci curvature
of the acoustical metric $\gfour$. The decompositions are a crucial ingredient
in our derivation of the transport equations satisfied by the modified quantities;
see the proofs of Props. <ref> and <ref>.

Assume that the entries of $\wavearray = (\RRiemann,\LRiemann, v^2,v^3,\Ent)$ solve the geometric wave equations
Then the following identity holds, where $\Ricfour$ is the Ricci curvature of $\gfour$:

\begin{align}
\begin{split} \label{E:RICCICONTRACTEDLANDLFORFULLYMODIFIED}
\upmu \Ricfour_{\Lunit \Lunit}
& =
\Lunit
\left\lbrace
\vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray
\frac{1}{2} \upmu \mytr_{\gtorus} \angG \diamond \Lunit \wavearray
\frac{1}{2} \upmu \vec{G}_{\Lunit \Lunit} \diamond \Lunit \wavearray
\upmu \angG_{\Lunit}^{\#} \diamond \cdot \angrmD \wavearray
\right\rbrace
\\
& \ \
\mathfrak{A},
\end{split}
\end{align}

where $\mathfrak{A}$ has the following schematic structure:

\begin{align} \label{E:AINHOMRIC}
\mathfrak{A}
& =
\upmu
\smoothfunction(\wavearray)
\cdot
\smoothfunction(\badcontrolvars,\comder \wavearray)
\cdot
\tander \wavearray
\smoothfunction(\badcontrolvars,\vortrenormalized,\GradEnt,\comder \wavearray)
\cdot
\end{align}

Moreover, without the assumption that the geometric wave equations
are satisfied, the following identity holds:

\begin{align}
\begin{split} \label{E:RICCICONTRACTEDLANDLFORPARTIALLYMODIFIED}
\Ricfour_{\Lunit \Lunit}
\frac{\Lunit \upmu}{\upmu} \mytr_{\gtorus} \upchi
\Lunit
\left\lbrace
-\frac{1}{2} \mytr_{\gtorus} \angG \diamond \Lunit \wavearray
\angG_{\Lunit}^{\#} \diamond \cdot \angrmD \wavearray
\right\rbrace
\\
& \ \
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \angLap \wavearray
\mathfrak{B},
\end{split}
\end{align}

where $\mathfrak{B}$ has the following schematic structure:

\begin{align} \label{E:BINHOMRIC}
\mathfrak{B}
& =
\smoothfunction(\controlvars)(\tander \wavearray) \cdot \tander \controlvars.
\end{align}

In <cit.>,
in the case of two spatial dimensions,
analogs of
(<ref>) and (<ref>)
were derived.
Analogous identities were also derived in <cit.>
in the case of quasilinear wave equations in three space dimensions.
The proof of (<ref>)
given in <cit.> is based
on first writing $\Ricfour_{\Lunit \Lunit}$ relative to the Cartesian coordinates,
then writing all derivatives of $\wavearray$ in terms of derivatives with respect to
elements of $\lbrace \Lunit, X \rbrace$ and $\ell_{t,u}$-tangent differentiations,
and then finally expressing (with the help of Lemma <ref>)
all of the principal terms (i.e., the terms that depend on the second derivatives of $\wavearray$)
as a perfect $\Lunit$ derivative up to lower-order terms,
except for the term
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \angLap \wavearray
on the last line of RHS (<ref>).
The detailed proof given in <cit.> goes through nearly verbatim,
except we have used Lemma <ref>
to simplify our schematic presentation of the term $\mathfrak{B}$.

Similarly, (<ref>) can be proved using the same
arguments given in <cit.>, but the new feature of
the present work is the structure of the terms on RHS (<ref>).
To see how these terms arise, we explain how the proof of
(<ref>) is connected to the identity (<ref>).
To pass from (<ref>) to (<ref>),
one uses the identity (<ref>),
the wave equations (<ref>)–(<ref>),
Lemma <ref>, and
Lemma <ref>
to express the product of $\upmu$
and the term
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \angLap \wavearray
on RHS (<ref>) as follows:

\begin{align}
\begin{split} \label{E:DETAILEDINHOM}
\frac{1}{2}
\upmu \vec{G}_{\Lunit \Lunit} \diamond \angLap \wavearray
& =
\frac{1}{2}
\Lunit
\left\lbrace
\vec{G}_{\Lunit \Lunit}
\diamond
(\upmu \Lunit\wavearray + 2 \muX \wavearray)
\right\rbrace
\frac{1}{2} \mytr_{\gtorus} \upchi
\vec{G}_{\Lunit \Lunit}
\diamond
\muX \wavearray
\\
& \ \
\smoothfunction(\controlvars) \cdot \text{Inhom}
\smoothfunction(\badcontrolvars,\comder \wavearray)
\cdot
\tander \wavearray.
\end{split}
\end{align}

In (<ref>), “Inhom” denotes the inhomogeneous terms
$\upmu \times \mbox{RHS~\eqref{E:VELOCITYWAVEEQUATION}--\eqref{E:ENTROPYWAVEEQUATION}}$.
In a detailed proof (see <cit.>), one finds that the term
$-\frac{1}{2} \mytr_{\gtorus}\upchi \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray$
on RHS (<ref>) is canceled
(and hence does not appear in (<ref>)–(<ref>))
by part of the first product on RHS (<ref>),
where one uses (<ref>)
to substitute for the factor $\Lunit \upmu$ on RHS (<ref>).
Next, decomposing $\upmu \times \mbox{RHS~\eqref{E:VELOCITYWAVEEQUATION}--\eqref{E:ENTROPYWAVEEQUATION}}$
using (<ref>),
we find that the term
$\smoothfunction(\controlvars) \cdot \text{Inhom}$
on RHS (<ref>) can be schematically expressed as follows:
\smoothfunction(\controlvars) \cdot \text{Inhom}
\upmu
\smoothfunction(\wavearray)
\cdot
\smoothfunction(\badcontrolvars,\comder \wavearray)
\cdot
\tander \wavearray
\smoothfunction(\badcontrolvars,\vortrenormalized,\GradEnt,\comder \wavearray)
\cdot
Placing these terms on RHS (<ref>), and also incorporating
the last term on RHS (<ref>) into RHS (<ref>),
we arrive at (<ref>)–(<ref>).

§.§ Definition of the modified quantities

We are now ready to define the modified quantities.
The definitions are motivated by the structure of the terms in Lemma <ref>;
this will become clear in the proofs of Props. <ref>
and <ref>.

Let $N = \Ntop$,
and let $\tander^N \in \mathfrak{P}^{(N)}$, where $\mathfrak{P}^{(N)}$
is the set of order $N$ $\nullhyparg{u}$-tangential commutator operators from
Sect. <ref>.
We define the fully modified quantity $\fullymodquant{\tander^N}$
as follows:

\begin{align}
\fullymodquant{\tander^N}
\eqdef
\upmu \tander^N \mytr_{\gtorus} \upchi
\tander^N \mathfrak{X},
\label{E:FULLYMODIFIEDQUANTITY}
\\
\mathfrak{X}
\eqdef
\vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray
\frac{1}{2} \upmu \mytr_{\gtorus} \angG \diamond \Lunit \wavearray
\frac{1}{2} \upmu \vec{G}_{\Lunit \Lunit} \diamond \Lunit \wavearray
\upmu \angG_{\Lunit}^{\#} \diamond \cdot \angrmD \wavearray.
\label{E:MODIFIEDQUANTITYINHOM}
\end{align}

Moreover, with $N = \Ntop - 1$ and $\tander^N \in \mathfrak{P}^{(N)}$,
we define the partially modified quantity $\partialmodquant{\tander^N}$
as follows:

\begin{align}
\partialmodquant{\tander^N}
\eqdef
\tander^N \mytr_{\gtorus} \upchi
\partialmodquantinhom{\tander^N},
\label{E:PARTIALMODIFIEDQUANTITY}
\\
\partialmodquantinhom{\tander^N}
& \eqdef
\frac{1}{2} \mytr_{\gtorus} \angG \diamond \Lunit \tander^N \wavearray
\angG_{\Lunit}^{\#} \diamond \cdot \angrmD \tander^N \wavearray.
\label{E:PARTIALMODIFIEDQUANTITYINHOM}
\end{align}

Finally, we define the following
“$0^{\text{th}}$-order” version of (<ref>):

\begin{align} \label{E:PARTIALMODIFIEDQUANTITYINHOMZEROORDER}
\widetilde{\mathfrak{X}}
& \eqdef
\frac{1}{2} \mytr_{\gtorus} \angG \diamond \Lunit \wavearray
\angG_{\Lunit}^{\#} \diamond \cdot \angrmD \wavearray.
\end{align}

§.§ Transport equations for the modified quantities

In this section, we derive transport equations for the fully modified quantities
from Def. <ref>.
We start with the following lemma, which
provides the transport equation satisfied by $\mytr_{\gtorus}\upchi$.
This transport equation is an analog of
the well-known Raychaudhuri equation in General Relativity <cit.>.

$\mytr_{\gtorus} \upchi$ obeys the following transport equation:

\begin{align}  \label{E:RAYCHAUDHURITRANSPORTCHI}
\upmu \Lunit \mytr_{\gtorus} \upchi
& =
(\Lunit \upmu)
\mytr_{\gtorus} \upchi
\upmu \Ricfour_{\Lunit \Lunit}
\upmu |\upchi|_{\gtorus}^2.
\end{align}

The same proof of <cit.> holds in the current setting.

Assume that $\wavearray = (\RRiemann,\LRiemann,v^2,v^3,\Ent)$
solve the geometric wave equations (<ref>)–(<ref>).
Let $N = \Ntop$,
and let $\tander^N \in \mathfrak{P}^{(N)}$, where $\mathfrak{P}^{(N)}$
is the set of order $N$ $\nullhyparg{u}$-tangential commutator operators from
Sect. <ref>.
Let $\fullymodquant{\tander^N}$ be the fully modified quantity defined in (<ref>),
let $\mathfrak{X}$ be as defined in (<ref>),
and let $\mathfrak{A}$ be the term on RHS (<ref>).
Then $\fullymodquant{\tander^N}$ obeys the following transport equation,
where $\tander^N$ is the same differential operator every time it appears in (<ref>):

\begin{align}
\begin{split} \label{E:TRANSPORTEQUATIONFORFULLYMODIFIEDQUANTITY}
\Lunit \fullymodquant{\tander^N}
\left(
2 \frac{\Lunit \upmu}{\upmu}
\right)
\fullymodquant{\tander^N}
& =
\left(2 \frac{\Lunit \upmu}{\upmu}\right) \tander^N \mathfrak{X}
\upmu [\Lunit,\tander^N] \mytr_{\gtorus} \upchi
\\
\ \
[\Lunit,\tander^N] \mathfrak{X}
[\upmu,\tander^N] \Lunit \mytr_{\gtorus}\upchi
[\tander^N,\Lunit \upmu] \mytr_{\gtorus}\upchi
\\
& \ \
\tander^N \left(\upmu |\upchi|_{\gtorus}^2 \right)
\tander^N \mathfrak{A}.
\end{split}
\end{align}

First, we use (<ref>) to substitute for the product
$\upmu \Ricfour_{\Lunit \Lunit}$ in (<ref>).
We then differentiate the resulting equation with $\tander^N$ and carry out tedious but straightforward
commutations. Also taking into account definition (<ref>),
we conclude (<ref>).
We refer to the proof of <cit.> for more details.

Let $N = \Ntop$,
and let $\tander^{N-1} \in \mathfrak{P}^{(N-1)}$, where $\mathfrak{P}^{(N-1)}$
is the set of order $N-1$ $\nullhyparg{u}$-tangential commutator operators from
Sect. <ref>.
Let $\partialmodquant{\tander^{N-1}}$ be the corresponding partially modified quantity defined in (<ref>),
let $\partialmodquantinhom{\tander^{N-1}}$ be the term defined in (<ref>),
let $\widetilde{\mathfrak{X}}$ be the term defined in (<ref>),
and let $\mathfrak{B}$ be the term on RHS (<ref>).
Then $\partialmodquant{\tander^{N-1}}$ obeys the following transport equation,
where $\tander^{N-1}$ is the same differential operator every time it appears in

\begin{align} \label{E:TRANSPORTEQUATIONFORPARTIALMODIFIEDQUANTITY}
\Lunit \partialmodquant{\tander^{N-1}}
& =
\frac{1}{2} \vec{G}_{\Lunit \Lunit}\diamond \angLap \tander^{N-1} \wavearray
\end{align}

\begin{align}
\begin{split} \label{E:COMMUTEDPARTIALMODIFIEDQUANTITYINHOM}
\eqdef
\tander^{N-1} \mathfrak{B}
\tander^{N-1} \left(|\upchi|_{\gtorus}^2 \right)
\\
& \ \
\frac{1}{2} [\tander^{N-1},\vec{G}_{\Lunit \Lunit}]
\diamond
\angLap \wavearray
\frac{1}{2} \vec{G}_{\Lunit \Lunit}
\diamond
\\
& \ \
[\Lunit,\tander^{N-1}] \mytr_{\gtorus}\upchi
[\Lunit,\tander^{N-1}] \widetilde{\mathfrak{X}}
\Lunit
\left\lbrace
\partialmodquantinhom{\tander^{N-1}}
\tander^{N-1} \widetilde{\mathfrak{X}}
\right\rbrace.
\end{split}
\end{align}

Substituting the identity (<ref>) into
(<ref>), dividing the resulting equation by $\upmu$,
and appealing to the definition
of $\widetilde{\mathfrak{X}}$,
we deduce that:

\begin{align} \label{E:TRANSPORTEQUATIONFORPARTIALMODIFIEDQUANTITY0THORDER}
\Lunit
\left(\mytr_{\gtorus} \upchi
\widetilde{\mathfrak{X}}
\right)
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \angLap \wavearray
\mathfrak{B}.
\end{align}

The transport equation (<ref>) then follows
from differentiating (<ref>) with $\tander^{N-1}$,
carrying out straightforward commutations,
and accounting for definitions

§ BASIC INGREDIENTS IN THE $L^2$ ANALYSIS

In this section, we establish some preliminary ingredients that we will use when we derive energy estimates.
In Sect. <ref>, we will derive one more crucial ingredient:
elliptic-hyperbolic integral identities that we use to control the top-order derivatives of the specific vorticity and entropy gradient.
In Sect. <ref>, we derive some differential and integral identities involving the rough tori
In Sect. <ref>, we establish some basic Sobolev embedding estimates
and fundamental theorem of calculus-type estimates on the rough tori.
In Sect. <ref>,
we use the identities from Sect. <ref>
to prove various integration by parts identities
that will play a key role in our energy estimates.
Next, in Sect. <ref>,
we use the vectorfield multiplier method to construct
the building block $L^2$-based energies and null-fluxes that we will use to control the wave variables $\wavearray$.
We also construct a companion set of building block energies and null-fluxes that we will use to control the transport variables
$\vortrenormalized$, $\GradEnt$, $\VortVort$, and $\DivGradEnt$.
Furthermore, in Prop. <ref>,
we establish the fundamental energy-null-flux
integral identities that we exploit in our $L^2$ analysis.
In Sect. <ref>, we use the building block energies and null fluxes to
define the quantities that we will use to control the solution in $L^2$.
Of particular interest are the spacetime integrals
defined in
(<ref>) and (<ref>)
respectively. These spacetime integrals
appear on the left-hand side of our energy identity (<ref>),
and they are fundamental for controlling error integrals that involve the quantities $\angrmD \tander^N \Psi$.
Finally, in Sect. <ref>,
we exhibit the key coerciveness properties of our $L^2$-controlling quantities
with respect to the $L^2$ norms from Sect. <ref>.

§.§ Differential and integral identities involving $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$

The following lemma, though standard, plays an important role in our proof of the
energy–null-flux identities for the wave variables
(see Prop. <ref>).
Moreover, the identity (<ref>) plays a crucial role in
our proof that one of the rough tori error integrals
(specifically, the second term on LHS (<ref>))
in our elliptic-hyperbolic identities for the vorticity and entropy