& \lesssim \fundbootsmall,
\label{E:DIVERGENCEOFROUGHTORITANGENTVECTORFIELD}
\\
|\mytr_{\gtorusroughfirstfund} \deform{\argLrough{\muxmulevelsetvalue}}|
& \lesssim 1,
\label{E:POINTWISEBOUNDFORROUGHTOROIDALTRACEOFDEFORMATIONTENSOROFROUGHNULLVECTORFIELD}
\\
|\mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}|
& \lesssim 1.
\label{E:POINTWISEBOUNDFORROUGHTOROIDALTRACEOFDEFORMATIONTENSOROFRTRANS}
\end{align}

We refer to Sect. <ref> for some results that we will silently use throughout the analysis.

Proofs of (<ref>)–(<ref>):
To prove (<ref>),
we first use
Lemma <ref>,
Lemma <ref>
to deduce that:

\begin{align} \label{E:SCHEMATICEXPRESSIONFORRTRANSSMALL}
\Rtransnormsmallfactorarg{\muxmulevelsetvalue}
\frac{1}{(\Lunit \timefunctionarg{\muxmulevelsetvalue})^2}
(\gtorus^{-1})^{AB} (\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}) \geop{x^B} \timefunctionarg{\muxmulevelsetvalue}
\frac{1}{(\Lunit \timefunctionarg{\muxmulevelsetvalue})^2}
\smoothfunction(\controlvars)
\cdot
\cdot
\end{align}

where the expression to the right of the last equality is schematic.
From (<ref>),
and the bootstrap assumptions,
and Cor. <ref>,
we deduce that $|\Rtransnormsmallfactorarg{\muxmulevelsetvalue}| \lesssim \fundbootsmall^2$ as desired.
To prove that $|\tander \Rtransnormsmallfactorarg{\muxmulevelsetvalue}|, \, |\muX \Rtransnormsmallfactorarg{\muxmulevelsetvalue}| \lesssim \fundbootsmall^2$,
we differentiate the expression for
\Rtransnormsmallfactorarg{\muxmulevelsetvalue}
with elements $\tander \in \lbrace \Lunit, \Yvf{2}, \Yvf{3} \rbrace$ and $\muX$ and apply a similar argument,
where we use Lemma <ref>
to express the vectorfield derivatives in terms of geometric coordinate partial derivatives
when they fall on $\timefunctionarg{\muxmulevelsetvalue}$.

The estimates (<ref>)
follow from similar arguments based on the identity

Proof of (<ref>):
The estimate (<ref>) follows from the decomposition
$\Rtransarg{\muxmulevelsetvalue}^{\alpha}= \upmu X^{\alpha}+ \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu} \Lunit^{\alpha} - \upmu \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}^{\alpha}$
(see (<ref>)),
Lemma <ref>,
the bootstrap assumptions (in particular (<ref>)),
and (<ref>).

Proofs of (<ref>)–(<ref>):
These estimates follow from the decompositions
(<ref>) and
and the arguments given in the previous paragraphs.

Proof of (<ref>):
First, computing relative to
the coordinates $(x^2,x^3)$ on the rough tori $\twoargroughtori{\timefunctionarg{\muxmulevelsetvalue},u}{\muxmulevelsetvalue}$,
we deduce that:

\begin{align} \label{E:EXPANSIONOFROUGHDIVERGENCEOFROUGHTORITANGENTVECTORFIELD}
\begin{split}
\roughangdiv \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}
\roughgeop{x^A} \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}^A
\\
& \ \
\frac{1}{2} \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}^A
\left\lbrace
\roughgeop{x^B} \gtorusroughfirstfund\left(\roughgeop{x^A},\roughgeop{x^C}\right)
\roughgeop{x^C} \gtorusroughfirstfund\left(\roughgeop{x^A},\roughgeop{x^B}\right)
\roughgeop{x^A} \gtorusroughfirstfund\left(\roughgeop{x^B},\roughgeop{x^C}\right)
\right\rbrace.
\end{split}
\end{align}

Next, we use
(<ref>) to write
= (\gtorusroughinversefirstfund)(dx^A,dx^B)
\frac{\geop{x^B}\timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}$.
We then use (<ref>)–(<ref>),
and Lemma <ref>
to schematically express
\smoothfunction
\left(\controlvars,
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tanderY \timefunctionarg{\muxmulevelsetvalue}
\right)
\Yvf{2}
\smoothfunction
\left(\controlvars,
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tanderY \timefunctionarg{\muxmulevelsetvalue}
\right)
\Yvf{3}
\smoothfunction
\left(\controlvars,
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\right)
\cdot
(\tanderY \timefunctionarg{\muxmulevelsetvalue})
\Lunit
and we also use
and Lemma <ref>
to deduce the schematic identities
\smoothfunction
\left(\controlvars,
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tanderY \timefunctionarg{\muxmulevelsetvalue}
\right)
\smoothfunction
\left(\controlvars,
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}},
\tanderY \timefunctionarg{\muxmulevelsetvalue}
\right)
Also using (<ref>), we arrive at the following schematic identity:

\begin{align} \label{E:SCHEMATICIDENTITYFORROUGHDIVERGENCEOFROUGHTORITANGENTVECTORFIELD}
\roughangdiv \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}
\smoothfunction
\left(\tander^{\leq 1} \controlvars,
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}},
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tander^{\leq 2} \timefunctionarg{\muxmulevelsetvalue}
\right)
\cdot
\tanderY^{[1,2]} \timefunctionarg{\muxmulevelsetvalue}.
\end{align}

From (<ref>),
Lemma <ref>,
the estimates
for the rough time function, the bootstrap assumptions,
and Cor. <ref>,
we arrive at the desired estimate

Proof of (<ref>):
First, using (<ref>), we compute that relative to the coordinates $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$, we have
\mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
\gtorusroughinversefirstfund\left(dx^A,dx^B\right)
\Rtransarg{\muxmulevelsetvalue} \gtorusroughfirstfund \left(\roughgeop{x^A},\roughgeop{x^B}\right)
2 \roughgeop{x^A} \Rtransarg{\muxmulevelsetvalue}^A
Considering this identity,
arguing as in the proof of (<ref>),
and also using (<ref>) and (<ref>),
we deduce the following schematic identity:

\begin{align} \label{E:SCHEMATICIDENTITYFORROUGHTORUSTRACEOFDEFORMATIONTENSOROFRTRANS}
\mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
\smoothfunction
\left(\tander^{\leq 1} \badcontrolvars,
\comder \controlvars,
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}},
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tander^{\leq 2} \timefunctionarg{\muxmulevelsetvalue},
\muX \tander \timefunctionarg{\muxmulevelsetvalue}
\right).
\end{align}

From (<ref>),
Lemmas <ref> and <ref>,
the estimates
for the rough time function, and the bootstrap assumptions,
we arrive at the desired estimate (<ref>).

Proof of (<ref>):
Considering definition (<ref>) and arguing as in the proof of
we find that:

\begin{align} \label{E:SCHEMATICIDENTITYFORROUGHTORUSTRACEOFDEFORMATIONTENSOROFROUGHNULLVECTORFIELD}
\mytr_{\gtorusroughfirstfund} \deform{\argLrough{\muxmulevelsetvalue}}
\smoothfunction
\left(\tander^{\leq 1} \controlvars,
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}},
\frac{1}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}},
\tander^{\leq 2} \timefunctionarg{\muxmulevelsetvalue}
\right).
\end{align}

From (<ref>),
Lemmas <ref> and <ref>,
the estimates
for the rough time function, and the bootstrap assumptions,
we arrive at the desired estimate (<ref>).

§.§ Pointwise estimates for the elliptic-hyperbolic integral identity error terms

Recall that to prove our top-order $L^2$ estimates for the specific vorticity and entropy gradient,
we will rely on the elliptic-hyperbolic integral identity (<ref>)
with $\tander^{\Ntop} \vortrenormalized$ and $\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$.
In the next proposition, we derive pointwise estimates for the error terms appearing in the identity.

Let $\varsigma \in (0,1]$.
Then the error terms appearing in the elliptic-hyperbolic integral identity (<ref>)
(with $\tander^{\Ntop} \vortrenormalized$ and $\tander^{\Ntop} \GradEnt$ in the role of $\SigmatTan$)
satisfy the following pointwise estimates
on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
where the implicit constants are independent of $\varsigma$.

Estimates for controlling spacetime error integrals.

\begin{align}
\begin{split} \label{E:ANTISYMMETRICVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Antisymmetric})}
[\pmb{\partial} \tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\right|
\lesssim
\left|
\tander^{\Ntop} \VortVort
\right|^2
\frac{1}{\upmu^2}
\left|
\tander^{\leq \Ntop-1} \VortVort
\right|^2
\frac{1}{\upmu^2}
\left|
\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)
\right|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^2}
\left|
\muX \tander^{[1,\Ntop]} \wavearray
\right|^2
\varepsilon^2
\left|
\tander^{\Ntop+1} \wavearray
\right|^2
\frac{\varepsilon^2}{\upmu^2}
\left|
\tander^{[2,\Ntop]} \badcontrolvars
\right|^2,
\end{split}
\\
\begin{split} \label{E:ANTISYMMETRICENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Antisymmetric})}[\pmb{\partial} \tander^{\Ntop}\GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
& \lesssim
\frac{1}{\upmu^2}
\left|
\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)
\right|^2
\\
\ \
\frac{\varepsilon^2}{\upmu^2}
\left|
\muX \tander^{[1,\Ntop]} \wavearray
\right|^2
\varepsilon^2
\left|
\tander^{\Ntop+1} \wavearray
\right|^2
\frac{\varepsilon^2}{\upmu^2}
\left|
\tander^{[2,\Ntop]} \badcontrolvars
\right|^2,
\end{split}
\end{align}

\begin{align}
\begin{split} \label{E:DIVVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Div})}[\pmb{\partial} \tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\right|
& \lesssim
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\\
\begin{split} \label{E:DIVENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Div})}[\pmb{\partial} \tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop}\GradEnt]
\right|
& \lesssim
\left|
\tander^{\Ntop} \DivGradEnt
\right|^2
\frac{1}{\upmu^2}
\left|
\tander^{\leq \Ntop-1} \DivGradEnt
\right|^2
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\end{align}

\begin{align}
\upmu
\left|
\mathfrak{J}_{(\pmb{\partial} \frac{1}{\upmu})}
[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}
\vortrenormalized]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} \vortrenormalized|^2,
\label{E:DERIVATIVEOFONEOVERMUWEIGHTVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\\
\upmu
\left|
\mathfrak{J}_{(\pmb{\partial} \frac{1}{\upmu})}[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop}
\GradEnt]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} \GradEnt|^2,
\label{E:DERIVATIVEOFONEOVERMUWEIGHTENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\end{align}

\begin{align}
\begin{split} \label{E:FIRSTABSORBTERMVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Absorb-1})}[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\right|
\lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\left(1 + \frac{1}{\varsigma} \right)
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\\
\begin{split} \label{E:FIRSTABSORBTERMENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Absorb-1})}[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
\lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\left|
\tander^{\Ntop} \DivGradEnt
\right|^2
\left(1 + \frac{1}{\varsigma} \right)
\frac{1}{\upmu^2}
\left|
\tander^{\leq \Ntop-1} \DivGradEnt
\right|^2
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\left(1 + \frac{1}{\varsigma} \right)
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\end{align}

\begin{align}
\left|
\mathfrak{J}_{(\textnormal{Absorb-2})}[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\Ntop} \vortrenormalized|^2,
\label{E:SECONDABSORBTERMVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\\
\left|
\mathfrak{J}_{(\textnormal{Absorb-2})}[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\Ntop} \GradEnt|^2,
\label{E:SECONDABSORBTERMENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\end{align}

\begin{align}
\begin{split} \label{E:QUADRATICINMATERIALDERIVATIVETERMVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Material})}[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\right|
\lesssim
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\\
\begin{split} \label{E:QUADRATICINMATERIALDERIVATIVETERMENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\left|
\mathfrak{J}_{(\textnormal{Material})}[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
\lesssim
\frac{1}{\upmu^2}
|\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^2}
|\muX \tander^{[1,\Ntop]} \wavearray|^2
\varepsilon^2 |\tander^{\Ntop+1} \wavearray|^2
\frac{\varepsilon^2}{\upmu^2}
|\tander^{[2,\Ntop]} \badcontrolvars|^2,
\end{split}
\end{align}

\begin{align}
\left|
\mathfrak{J}_{(\textnormal{Null Geometry})}[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop}\vortrenormalized]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\Ntop} \vortrenormalized|^2,
\label{E:NULLGEOMETRYVORTICITYELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\\
\left|
\mathfrak{J}_{(\textnormal{Null Geometry})}[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
& \lesssim
\varsigma
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial}\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\frac{1}{\varsigma}
\frac{1}{\upmu^2}
|\tander^{\Ntop} \GradEnt|^2.
\label{E:NULLGEOMETRYENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITYERORTERMPOINTWISE}
\end{align}

Estimates for controlling spatial error integrals.

\begin{align}
\begin{split} \label{E:PRINCIPALSIGAMTILDEELLIPTICHYPERBOLICVORTICITYPOINTWISE}
\left|
\Currentboundaryerrorhavetocontrolprincipal[\tander^{\Ntop} \vortrenormalized,\pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\right|
\lesssim
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop} \VortVort
\right|^2
\frac{1}{\upmu^{3/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\leq \Ntop-1} \VortVort
\right|^2
\\
& \ \
\frac{1}{\upmu^{5/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)
\right|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^{3/2}}
\left|
\muX \tander^{[1,\Ntop]} \wavearray
\right|^2
\varepsilon^2
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop+1} \wavearray
\right|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^{3/2}}
\left|
\tander^{[2,\Ntop]} \badcontrolvars
\right|^2,
\end{split}
\\
\begin{split} \label{E:PRINCIPALSIGAMTILDEELLIPTICHYPERBOLICENTROPYGRADIENTPOINTWISE}
\left|
\Currentboundaryerrorhavetocontrolprincipal[\tander^{\Ntop} \GradEnt,\pmb{\partial} \tander^{\Ntop} \GradEnt]
\right|
\lesssim
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop} \DivGradEnt
\right|^2
\frac{1}{\upmu^{3/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\leq \Ntop-1} \DivGradEnt
\right|^2
\\
& \ \
\frac{1}{\upmu^{5/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\leq \Ntop} (\vortrenormalized,\GradEnt)
\right|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^{3/2}}
\left|
\muX \tander^{[1,\Ntop]} \wavearray
\right|^2
\varepsilon^2
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop+1} \wavearray
\right|^2
\\
& \ \
\frac{\varepsilon^2}{\upmu^{3/2}}
\left|
\tander^{[2,\Ntop]} \badcontrolvars
\right|^2,
\end{split}
\end{align}

\begin{align}
\left|
\Currentboundaryerrorhavetocontrollowerorder[\tander^{\Ntop} \vortrenormalized,\tander^{\Ntop} \vortrenormalized]
\right|
& \lesssim
\frac{1}{\upmu^{5/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop} \vortrenormalized
\right|^2,
\label{E:LOWERORDERSIGAMTILDEELLIPTICHYPERBOLICVORTICITYPOINTWISE}
\\
\left|
\Currentboundaryerrorhavetocontrollowerorder[\tander^{\Ntop} \GradEnt,\tander^{\Ntop} \GradEnt]
\right|
& \lesssim
\frac{1}{\upmu^{5/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\tander^{\Ntop} \GradEnt
\right|^2.
\label{E:LOWERORDERSIGAMTILDEELLIPTICHYPERBOLIENTROPYGRADIENTPOINTWISE}
\end{align}

Proofs of (<ref>)–(<ref>)
and (<ref>)–(<ref>):
First, using (<ref>)
and (<ref>),
we deduce that
$|\mathfrak{J}_{(\textnormal{Antisymmetric})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]|
\lesssim
|\mathrm{d} \SigmatTan_{\flat}|_{\hfour}^2
From this bound with
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$
the pointwise estimates
we conclude
the desired bounds

follow from similar arguments
based on (<ref>)
and the pointwise estimates (<ref>)–(<ref>).

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first note that since the elliptic hyperbolic current
\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
defined by (<ref>)
is tangent to $\nullhyparg{u}$,
we can use the identity (<ref>) to obtain the following
decomposition, where $\smoothtorusproject$ is the $\ell_{t,u}$-projection
tensorfield from Def. <ref>:
\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
- \frac{1}{2}
\uLunit_{\beta}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
\Lunit^{\alpha}
\smoothtorusproject_{\beta}^{\ \alpha}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
From this decomposition
and definition (<ref>),
we deduce the following pointwise bound, where
\smoothtorusproject \ehcurrent[\SigmatTan,\pmb{\partial} \SigmatTan]
is the $\ell_{t,u}$-projection of $\ehcurrent[\SigmatTan,\pmb{\partial} \SigmatTan]$:

\begin{align} \label{E:FIRSTPOINTWISEBOUNDFORCHARCURRENTDERIVATIVEOF1OVERMUWEIGHTERRORTERM}
\upmu
|\mathfrak{J}_{(\pmb{\partial} \frac{1}{\upmu})}[\SigmatTan,\pmb{\partial} \SigmatTan]|
\lesssim
\frac{1}{\upmu}
|\Lunit \upmu|
|\uLunit_{\beta} \ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]|+
|\angD \upmu|_{\gtorus}
|\smoothtorusproject \ehcurrent[\SigmatTan,\pmb{\partial} \SigmatTan]|_{\gtorus}.
\end{align}

Next, using (<ref>),
definition (<ref>),
Lemma <ref>,
the estimates of Prop. <ref>,
and the pointwise comparison estimates provided by Lemma <ref>,
we deduce the following pointwise bound:

\begin{align} \label{E:SECONDPOINTWISEBOUNDFORCHARCURRENTDERIVATIVEOF1OVERMUWEIGHTERRORTERM}
\upmu
|\mathfrak{J}_{(\pmb{\partial} \frac{1}{\upmu})}[\SigmatTan,\pmb{\partial} \SigmatTan]|
\lesssim
\frac{1}{\upmu}
|\pmb{\partial} \SigmatTan|_{\hfour}.
\end{align}

Using (<ref>) with
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$,
the bound $\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$
(see (<ref>)),
and Young's inequality (where we multiply and divide by powers of $\varsigma$),
we deduce the desired bounds

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use definition (<ref>),
Lemma <ref>,
the estimates of Prop. <ref>,
the bound $|\partial_{\alpha} \gfour_{\beta \gamma}| \lesssim |\pmb{\partial} \wavearray| \lesssim \frac{1}{\upmu}$
(which follows from Lemma <ref>
Lemma <ref>, and Prop. <ref>),
and the pointwise comparison estimates provided by Lemma <ref>
to deduce that:

\begin{align} \label{E:FIRSTPOINTWISEBOUNDFIRSTABSORBELLIPTICHYPERBOLICCURRENTERRORTERM}
\left|
\mathfrak{J}_{(\textnormal{Absorb-1})}[\SigmatTan,\pmb{\partial} \SigmatTan]
\right|
\lesssim
\frac{1}{\upmu}
|\pmb{\partial} \SigmatTan|_{\hfour}
|\partial_a \SigmatTan^a|
|\pmb{\partial} \SigmatTan|_{\hfour}.
\end{align}

Using (<ref>)
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$,
the bound $\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$
(see (<ref>)),
the pointwise estimates
and Young's inequality (where we multiply and divide by powers of $\upmu$ and $\varsigma$ as needed),
we deduce the desired bounds

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use definitions (<ref>)
and the same arguments used in the proof of (<ref>)
to deduce the pointwise bound
\left|
\mathfrak{J}_{(\textnormal{Absorb-2})}[\SigmatTan,\pmb{\partial} \SigmatTan]
\right|
\lesssim
\frac{1}{\upmu}
|\pmb{\partial} \SigmatTan|_{\hfour}
Then using the same arguments given just below
we conclude (<ref>)–(<ref>).

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use definition (<ref>),
and the pointwise comparison estimates provided by Lemma <ref>
to deduce the pointwise bound
\left|
\mathfrak{J}_{(\textnormal{Material})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\right|
\lesssim
|\Transport \SigmatTan|_{\hfour}^2
\lesssim
\sum_{a=1,2,3}
|\Transport \SigmatTan^a|^2
From this bound with
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$
and the pointwise estimates (<ref>),
we conclude

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use definitions
Lemma <ref>,
Lemma <ref>,
and the estimates of Prop. <ref>
to deduce the pointwise bound
\left|
\mathfrak{J}_{(\textnormal{Null Geometry})}[\SigmatTan,\pmb{\partial} \SigmatTan]
\right|
\lesssim
\frac{1}{\upmu}
|\pmb{\partial} \SigmatTan|_{\hfour}
Then using the same arguments given just below
we conclude (<ref>)–(<ref>).

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use
definitions (<ref>)
and (<ref>),
Lemma <ref>,
the estimates of Prop. <ref>,
and the pointwise comparison estimates provided by Lemma <ref>
to deduce the following pointwise estimate, where $\phi$ is the cut-off function from Def. <ref>:

\begin{align} \label{E:FIRSTPOINTWISEPROOFSTEPPRINCIPALSIGAMTILDEELLIPTICHYPERBOLICVORTICITYPOINTWISE}
\left|
\Currentboundaryerrorhavetocontrolprincipal[\SigmatTan,\pmb{\partial} \SigmatTan]
\right|
\lesssim
\frac{1}{\upmu}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left\lbrace
|\Transport \SigmatTan|_g
|\mathrm{d} \SigmatTan_{\flat}|_{\hfour}
|\partial_a \SigmatTan^a|
\right\rbrace.
\end{align}

Using (<ref>) with
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$,
the pointwise estimates
with $\Ntop$ in the role of $N$,
the pointwise comparison estimates
the estimate
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right| \lesssim 1$
(which follows from Prop. <ref> and (<ref>))
and Young's inequality (where we multiply and divide by powers of $\upmu$ and $\varsigma$ as needed),
we conclude
that the desired estimates
hold for any $\varsigma \in (0,1]$.

Proofs of (<ref>)–(<ref>):
Let $\SigmatTan$ be any $\Sigma_t$-tangent vectorfield.
We first use definition (<ref>),
Lemma <ref>,
Cor. <ref>,
Lemma <ref>,
Lemma <ref>,
Prop. <ref>,
Lemma <ref>,
and the pointwise comparison estimates provided by Lemma <ref>
to deduce the following pointwise bound, where $\phi$ is the cut-off function from Def. <ref>:
\left|
\Currentboundaryerrorhavetocontrollowerorder[\SigmatTan,\SigmatTan]
\right|
\lesssim
\frac{1}{\upmu^{5/2}}
\left\lbrace
\upmu
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right\rbrace
\left|
\SigmatTan
\right|_g^2
Using this bound with
$\tander^{\Ntop} \vortrenormalized$
$\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$
and the comparison estimate (<ref>),
we conclude the desired bounds

§ STATEMENT OF THE A PRIORI $L^2$ ESTIMATES,
DATA ESTIMATES FOR THE $L^2$-CONTROLLING QUANTITIES, AND
BOOTSTRAP ASSUMPTIONS FOR THE WAVE VARIABLE ENERGIES

In Sect. <ref>, we state all of the a priori energy estimates
for the fluid variables and the acoustic geometry.
In particular, we state the main estimates for the $L^2$-controlling quantities from
Sect. <ref>.
The proofs take considerable effort and form the focus of the paper
through Sect. <ref>.
In Sect. <ref>,
we show that along the data hypersurfaces
$\hypthreearg{\timefunction_0}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$
the $L^2$-controlling quantities are bounded by $\lesssim \initialsmall^2$,
i.e., the $L^2$-controlling quantities have small data.
Finally, in Sect. <ref>,
we state bootstrap assumptions for the $\totalcontrolwave_N$,
i.e., for the $L^2$-controlling quantities of the wave variables.

§.§ Statement of the a priori $L^2$ estimates

§.§.§ Statement of the a priori $L^2$ estimates for the wave variables

In the following proposition, we state our main a priori energy estimates
for the wave variables. Its proof is located in Sect. <ref>.

Let $\totalcontrolwave_{N}(\timefunction,u)$
be the $L^2$-controlling quantity for the wave variables
$\wavearray$, as defined in (<ref>).
Under the data assumptions of Sect. <ref>,
the parameter size-assumptions of Sect. <ref>,
and the bootstrap assumptions of Sect. <ref>,
there exists a constant $C > 0$
such that the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$:

\begin{align}
\totalcontrolwave_{\Ntop - K}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-15.6 + 2K}
\mbox{if } 0 \leq K \leq 7,
\label{E:MAINWAVEENERGYESTIMATESBLOWUP}
\\
\totalcontrolwave_{N}(\timefunction,u)
& \leq C \initialsmall^2
&& \mbox{if } 1 \leq N \leq \Ntop - 8.
\label{E:MAINWAVEENERGYESTIMATESREGULAR}
\end{align}

§.§.§ Statement of the a priori $L^2$ estimates for the transport variables

The following proposition provides an analog of
Prop. <ref> for the transport variables.
Its proof is located in Sects. <ref>
and <ref>.

be the $L^2$-controlling quantities for
$\vortrenormalized$ and $\GradEnt$,
as defined in (<ref>)–(<ref>),
and let $\hypersurfacecontrolVortVort_{N}(\timefunction,u)$
be the $L^2$-controlling quantities for
the modified fluid variables $\VortVort$ and $\DivGradEnt$,
as defined in (<ref>)–(<ref>).
Under the data assumptions of Sect. <ref>,
the parameter size-assumptions of Sect. <ref>,
and the bootstrap assumptions of Sect. <ref>,
there exists a constant $C > 0$
such that the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$:

\begin{align}
\hypersurfacecontrolVort_{\Ntop - K}(\timefunction,u),
\,
\hypersurfacecontrolGradEnt_{\Ntop - K}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-14.6 + 2K }, & & \text{for } 0 \leq K \leq 7,
\label{E:MAINL2ESTIMATESVORTANDENTROPYGRADIENTBLOWUP}
\\
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u),
\,
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-17.1},
\label{E:MAINTOPORDERENERGYESTIMATESMODIFIEDFLUIDVARIABLESBLOWUP}
\\
\hypersurfacecontrolVortVort_{\Ntop - 1 -K}(\timefunction,u),
\,
\hypersurfacecontrolDivGradEnt_{\Ntop - 1 - K}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-14.6 + 2K},  & & \text{for } 0 \leq K \leq 7.
\label{E:MAINL2BELOWTOPORDERESTIMATESMODIFIEDFLUIDBLOWUP}
\end{align}

Moreover, for $ 0 \leq N_1 \leq \Ntop - 8$  and $0 \leq N_2 \leq \Ntop - 9$, we have:

\begin{align}
\hypersurfacecontrolVort_{N_1}(\timefunction,u),
\,
\hypersurfacecontrolGradEnt_{N_1}(\timefunction,u)
& \leq C \initialsmall^2,
\label{E:MAINL2BELOWTOPORDERESTIMATESVORTANDENTROPYGRADIENTREGULAR}
\\
\hypersurfacecontrolVortVort_{N_2}(\timefunction,u),
\,
\hypersurfacecontrolDivGradEnt_{N_2}(\timefunction,u) & \leq C \initialsmall^2.
\label{E:MAINL2BELOWTOPORDERESTIMATESMODIFIEDFLUIDVARIABLESREGULAR}
\end{align}

In addition to the $L^2$ estimates of Prop. <ref>,
which are estimates for the transport variables on constant-rough-time
hypersurfaces and null hypersurfaces,
we also derive $L^2$ estimates for the transport variables
on the rough tori $\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}$.
We need these estimates because rough tori integrals
are featured in the main elliptic-hyperbolic integral identity
(see Prop. <ref>)
that we use to control various top-order spacetime $L^2$ norms
of the $\vortrenormalized$ and $\GradEnt$.
The rough tori estimates of interest are provided by the next proposition.
Its proof is located in Sects. <ref>
and <ref>.

be the tori-$L^2$-controlling quantities for
and $\DivGradEnt$
as defined in (<ref>)–(<ref>)
Under the data assumptions of Sect. <ref>,
the parameter size-assumptions of Sect. <ref>,
and the bootstrap assumptions of Sect. <ref>,
there exists a constant $C > 0$
such that the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$:

\begin{align}
\toricontrolVort_{\Ntop}(\timefunction,u),
\,
\toricontrolGradEnt_{\Ntop}
\leq C \initialsmall^2 |\timefunction|^{-17.1},
\label{E:MAINTORIL2VORTGRADENTTOPORDERBLOWUP}
\\
\toricontrolVort_{\Ntop - 1 - K}(\timefunction,u),
\,
\toricontrolGradEnt_{\Ntop - 1 - K}
& \leq C \initialsmall^2 |\timefunction|^{-14.6  + 2 K}, & & \text{for } 0 \leq K \leq 7,
\label{E:MAINTORIL2VORTGRADENTBELOWTOPORDERBLOWUP}
\\
\toricontrolVortVort_{\Ntop-1}(\timefunction,u),
\,
\toricontrolDivGradEnt_{\Ntop-1}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-17.1},
\label{E:MAINTORIL2MODIFIEDFLUIDVARIABLESTOPORDERBLOWUP}
\\
\toricontrolVortVort_{\Ntop-2-K}(\timefunction,u),
\,
\toricontrolDivGradEnt_{\Ntop-2-K}(\timefunction,u)
& \leq C \initialsmall^2 |\timefunction|^{-14.6 + 2K},  & & \text{for } 0 \leq K \leq 7,
\label{E:MAINTORIL2MODIFIEDFLUIDVARIABLESBELOWTOPORDERBLOWUP}
\end{align}

\begin{align}
\toricontrolVort_N(\timefunction,u),
\,
\toricontrolGradEnt_N(\timefunction,u)
& \leq C \initialsmall^2,
& & \text{for } 0 \leq N \leq \Ntop - 9,
\label{E:MAINTORIL2VORTGRADENTBELOWTOPORDERREGULAR}
\\
\toricontrolVortVort_N(\timefunction,u),
\,
\toricontrolDivGradEnt_N(\timefunction,u)
& \leq C \initialsmall^2,
& & \text{for } 0 \leq N \leq \Ntop - 10.
\label{E:MAINTORIL2MODIFIEDFLUIDVARIABLESBELOWTOPORDERREGULAR}
\end{align}

§.§.§ Statement of the a priori $L^2$ estimates for the acoustic geometry

The following proposition provides our main a priori energy estimates
for the acoustic geometry. Its proof is located in

Under the data assumptions of Sect. <ref>,
the parameter size-assumptions of Sect. <ref>,
and the bootstrap assumptions of Sect. <ref>,
there exists a constant $C > 0$
such that the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$:

\begin{align}
\left\|
\tander^{\Ntop} \mytr_{\gtorus} \upchi,
\,
\angLie_{\tander}^{\Ntop} \upchi
\right\|_{L^2\left( \hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}\right)}
& \leq C \initialsmall |\timefunction|^{-8.8},
\label{E:MAINL2CHITOPORDERBLOWUP} \\
\left\|
\tander^{N - 1} \mytr_{\gtorus} \upchi,
\,
\angLie_{\tander}^{N - 1} \upchi,
\,
\tandersmall^N\upmu,
\,
\tander^N \Lsmall^i
\right\|_{L^2\left( \hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}\right)}
& \leq C \initialsmall |\timefunction|^{- 7.3 + \Ntop - N}, \qquad \Ntop - 7 \leq N \leq \Ntop,
\label{E:MAINACOUSTGEOMETRYBELOWTOPORDERBLOWUP}
\\
\left\|
\tander^{\leq \Ntop - 9}  \mytr_{\gtorus} \upchi,
\,
\angLie_{\tander}^{\Ntop - 9} \upchi,
\,
\tandersmall^{[1,\Ntop - 8]}\upmu,
\,
\tander^{\leq \Ntop - 8} \Lsmall^i
\right\|_{L^2\left( \hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}\right)}
& \leq C \initialsmall.
\label{E:MAINACOUSTGEOMETRYBELOWTOPORDERREGULAR}
\end{align}

§.§ Data estimates for the $L^2$-controlling quantities

In this section, we prove initial data estimates for
the $L^2$-controlling quantities
$\totalcontrolwave_{[1,\Ntop]}$, $\hypersurfacecontrolVort_{\leq \Ntop }$, etc.
We will use these data estimates in our proofs of
Props. <ref>–<ref>.

The following estimates hold for
$\timefunction \in [\timefunction_0,\timefunctionboot)$ and $u \in [-\rightu,\leftu]$:

\begin{align} \label{E:WAVEL2CONTROLLINGINITIALLYSMALL}
\totalcontrolwave_{[1,\Ntop]}(\timefunction,-\rightu)
& \leq
C \initialsmall^2,
\totalcontrolwave_{[1,\Ntop]}(\timefunction_0,u)
& \leq C \initialsmall^2,
\end{align}

\begin{align}
\hypersurfacecontrolVort_{\leq \Ntop }(\timefunction_0,u) & \leq  C \initialsmall^2,
& \hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,- \rightu) & \leq  C \initialsmall^2,
\label{E:VORTICITYL2CONTROLLINGINITIALLYSMALL}
\\
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction_0,u) & \leq  C \initialsmall^2,
& \hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,- \rightu) & \leq  C \initialsmall^2,
\label{E:GRADENTL2CONTROLLINGINITIALLYSMALL}
\end{align}

\begin{align}
\hypersurfacecontrolVortVort_{\leq \Ntop}(\timefunction_0,u) & \leq  C \initialsmall^2,
& \hypersurfacecontrolVortVort_{\leq \Ntop}(\timefunction,- \rightu) & \leq  C \initialsmall^2,
\label{E:MODIVIEDVORTL2CONTROLLINGINITIALLYSMALL}
\\
\hypersurfacecontrolDivGradEnt_{\leq \Ntop}(\timefunction_0,u)& \leq  C \initialsmall^2,
& \hypersurfacecontrolDivGradEnt_{\leq \Ntop}(\timefunction,- \rightu) & \leq  C \initialsmall^2,
\label{E:MODIFIEDDIVGRADENTL2CONTROLLINGINITIALLYSMALL}
\end{align}

\begin{align}
\toricontrolVort_{\leq \Ntop }(\timefunction_0,u) & \leq  C \initialsmall^2,
\label{E:VORTICITYTORIL2CONTROLLINGINITIALLYSMALL}
\\
\toricontrolGradEnt_{\leq \Ntop}(\timefunction_0,u)
& \leq C \initialsmall^2,
\label{E:GRADENTTORIL2CONTROLLINGINITIALLYSMALL}
\end{align}

\begin{align}
\toricontrolVortVort_{\leq \Ntop-1}(\timefunction_0,u) & \leq  C \initialsmall^2,
\label{E:MODIFIEDVORTICITYTORIL2CONTROLLINGINITIALLYSMALL}
\\
\toricontrolDivGradEnt_{\leq \Ntop-1}(\timefunction_0,u)
& \leq C \initialsmall^2.
\label{E:DIVGRADENTTORIL2CONTROLLINGINITIALLYSMALL}
\end{align}

The estimates stated in (<ref>) are straightforward consequences of
the data assumptions
and (<ref>),
and (<ref>),
the identities
Lemma <ref>,
the bootstrap assumptions,
and the estimates

The estimates (<ref>)–(<ref>)
follow from applying similar reasoning
based on definitions
the data assumptions
and (<ref>).

The estimates
follow from applying similar reasoning
based on definitions
the data assumptions
and (<ref>).

The estimates
follow from
definitions (<ref>)–(<ref>)
and the data assumptions

The estimates
follow from
definitions (<ref>)–(<ref>)
and the data assumptions

§.§ Bootstrap assumptions for the $\totalcontrolwave_N$

In proving Props. <ref>–<ref>,
we find it convenient to make bootstrap assumptions for the $L^2$-controlling quantities
for the wave variables.
Specifically, with
(see definition (<ref>))
denoting the $L^2$-controlling quantity for the wave variables $\wavearray$,
we assume that the following bootstrap assumptions hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$,
where $\fundbootsmall$ is the bootstrap parameter from Sect. <ref>:

\begin{align}
\totalcontrolwave_{\Ntop - K}(\timefunction,u)
\leq \fundbootsmall |\timefunction|^{-15.6 + 2K},
\qquad 0 \leq K \leq 7
\label{E:MAINWAVEENERGYBOOTSTRAPBLOWUP}
\\
\totalcontrolwave_{[1,\Ntop - 8]}(\timefunction,u)
\leq \fundbootsmall.
\label{E:MAINWAVEENERGYBOOTSTRAPREGULAR}
\end{align}

Note that when $\initialsmall$ is sufficiently small,
the estimates of Prop. <ref>
yield strict improvements over the bootstrap assumptions

§ PRELIMINARY BELOW-TOP-ORDER $L^2$ ESTIMATES FOR THE ACOUSTIC GEOMETRY AND A DERIVATIVE-LOSING ESTIMATE

In this short section,
we derive preliminary $L^2$ estimates for the below-top-order derivatives of the
eikonal function quantities $\upmu$, $\Lunit^i$, $\upchi$, and $\mytr_{\gtorus}\upchi$.
We state the bounds in terms of the wave $L^2$-controlling quantities $\hypersurfacecontrolwave_{[1,N]}(\timefunction,u)$
and the initial data size parameter $\initialsmall$.
We also derive related $L^2$ estimates for top-order derivatives of $\upchi$ and $\mytr_{\gtorus} \upchi$ in the case that one
$\Lunit$-differentiation is involved. We provide the main estimates in Lemma <ref>.
The estimates are rather straightforward
consequences of the transport inequalities provided by
Prop. <ref>
Lemma <ref>.
In Cor. <ref>,
we derive $L^2$ estimates for $\wavearray$ that do not involve
any singular powers of $|\timefunction|^{-1}$.

Most of the $L^2$ estimates we derive in this section lose one derivative.
In Prop. <ref>, we will derive complementary
$L^2$ estimates for the top-order derivatives of $\upchi$ and $\mytr_{\gtorus}\upchi$.
Those estimates are much harder to prove because we cannot afford to lose
any derivatives, which forces us to rely on the modified quantities from Sect. <ref>
and elliptic estimates for the top-order derivatives of $\upchi$.

§.§ Preliminary below-top-order $L^2$ estimates for the eikonal function quantities

Let $N \leq \Ntop$. Then the following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$:

\begin{align}
\begin{pmatrix}
\left\| \Lunit \tandersmall^{[1,N]} \upmu \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \Lunit \tander^{\leq N} \Lsmall^i \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \Lunit \tander^{\leq N-1} \mytr_{\gtorus} \upchi \right \|_{L^2\left(\hypthreearg{\timefunction}{[\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \angLie_{\Lunit} \angLie_{ \tander}^{\leq N-1} \upchi \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)} \\
\left\| \Lunit \comder^{\leq N;1} \Lsmall^i  \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \Lunit \comder^{\leq N-1;1} \mytr_{\gtorus}\upchi  \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \angLie_{\Lunit} \angLie_{\comder}^{\leq N-1;1} \upchi \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\end{pmatrix}
& \lesssim \initialsmall
\frac{\hypersurfacecontrolwave_{[1,N]}^{1/2} (\timefunction,u)}{|\timefunction|^{1/2}},
\label{E:PRELIMINARYEIKONALWITHL}
\\
\begin{pmatrix}
\left\|  \tandersmall^{[1,N]} \upmu \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \tander^{[1,N]} \Lsmall^i\right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \tander^{\leq N-1} \mytr_{\gtorus} \upchi \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\|  \angLie_{\tander}^{\leq N-1} \upchi \right \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \comdersmall^{[1,N];1} \Lsmall^i \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \comder^{\leq N-1;1} \mytr_{\gtorus}\upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \angLie_{\comder}^{\leq N-1;1} \upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\end{pmatrix}
& \lesssim
\initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{\hypersurfacecontrolwave_{[1,N]}^{1/2} (\timefunction',u)}{|\timefunction'|^{1/2}}
\, \mathrm{d} \timefunction'.
\label{E:PRELIMINARYEIKONALWITHOUTL}
\end{align}

We fix $u \in [-\rightu,\leftu]$ and define the following
functions for $\timefunction \in [\timefunction_0,\timefunctionboot)$:

\begin{align}
\eqdef
\sum_{i = 1}^3
\left\| \tander^{[1,N]} \Lsmall^i  \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\left\| \tander^{\leq N-1} \mytr_{\gtorus}\upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\left\| \angLie_{\tander}^{\leq N-1}\upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)},
\label{E:QNDEFPROOFOFPRELIMINARYEIKONALL2}
\\
\begin{split} \label{E:PNDEFPROOFOFPRELIMINARYEIKONALL2}
& \eqdef
\sum_{i = 1}^3 \left\| \comdersmall^{[1,N];1} \Lsmall^i \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\left\| \comder^{\leq N-1;1} \mytr_{\gtorus}\upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\\
\left\| \angLie_{\comder}^{\leq N-1;1}\upchi \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\left\| \tandersmall^{[1,N]}\upmu \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}.
\end{split}
\end{align}

Next, we record the following pointwise estimate:
$\left|\Lunit\left(|\angLie_{\tander}^{N-1}\upchi|_{\gtorus} \right) \right|
\lesssim
|\angLie_{\Lunit} \angLie_{\tander}^{N-1} \upchi|_{\gtorus}
\fundbootsmall |\angLie_{\tander}^{N-1} \upchi|_{\gtorus} $,
which follows from the Leibniz rule for $\ell_{t,u}$-projected Lie derivatives,
the identity $\angLie_{\Lunit} \gtorus^{-1} = - 2 \upchi^{\# \#}$,
and the estimate
$|\upchi|_{\gtorus} \lesssim \fundbootsmall$,
which follows from (<ref>),
Lemma <ref>,
the bootstrap assumptions,
and Prop. <ref>.
(<ref>)–(<ref>) by
$\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}$
and using (<ref>),
Cor. <ref>,
as well as the estimates
and the pointwise estimate recorded above,
we deduce:

\begin{align} \label{E:QNGRONWALLREADYPROOFOFPRELIMINARYEIKONALL2}
\lesssim
\initialsmall
\fundbootsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\, \mathrm{d} \timefunction'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}}
\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)
\, \mathrm{d} \timefunction'.
\end{align}

The $L^2$ assumptions on the data stated in Sect. <ref> imply that
$q_N(\timefunction_0) \lesssim \initialsmall$. Inserting this estimate into RHS (<ref>)
and applying Grönwall's inequality, we find that
\lesssim
\initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}}
\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)
\, \mathrm{d} \timefunction'
which yields
(<ref>) for
$\tander^{[1,N]}\Lsmall^i$, $\tander^{\leq N-1} \mytr_{\gtorus}\upchi$, and $\angLie_{\tander}^{\leq N-1}\upchi$.

We now prove (<ref>) for
$\comdersmall^{[1,N];1} \Lsmall^i$,
$\comder^{\leq N-1;1} \mytr_{\gtorus} \upchi$,
$\angLie_{\comder}^{\leq N-1;1} \upchi$,
and $\tandersmall^{[1,N]}\upmu$.
We begin by examining the first term on RHS (<ref>).
Using the commutator estimates
the bootstrap assumptions,
and the estimates of Prop. <ref>,
we deduce:

\begin{align} \label{E:PRELIMINARYEIKONALWITHOUTLINTERMEDIATE}
|\comdersmall^{[1,N+1];1} \wavearray|
\lesssim
|\muX \tander^{[1,N]} \wavearray|
|\tander^{[1,N+1]} \wavearray|
\fundbootsmall
\fundbootsmall |\tandersmall^{[1,N]} \badcontrolvars|.
\end{align}

Arguing as in the proof of (<ref>),
but using (<ref>) and (<ref>)–(<ref>)
in place of (<ref>)–(<ref>),
and using (<ref>),
and (<ref>),
we find that:

\begin{align} \label{E:PRELIMINARYEIKONALWITHOUTLINTERMEDIATE2}
& \lesssim
\initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}  p_N(\timefunction') \, \mathrm{d} \timefunction'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}} \hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u) \,
\mathrm{d} \timefunction'.
\end{align}

As above,
the $L^2$ assumptions on the data stated in Sect. <ref> imply that
$p(\timefunction_0) \lesssim \initialsmall$,
and we can use Grönwall's inequality to deduce
\lesssim
\initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}}
\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)
\, \mathrm{d} \timefunction'
thereby concluding
(<ref>) for
$\comdersmall^{[1,N];1} \Lsmall^i$,
$\comder^{\leq N-1;1} \mytr_{\gtorus} \upchi$,
$\angLie_{\comder}^{\leq N-1;1} \upchi$,
and $\tandersmall^{[1,N]}\upmu$.

We now prove (<ref>).
Taking the norm $\| \cdot \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}$ of inequalities
and arguing as above using the already proven estimate (<ref>),
we obtain the desired result. We clarify that this argument generates the time integrals
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}} \hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)
\, \mathrm{d} \timefunction'$,
which we bound by
$\lesssim \hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction,u)
\lesssim \mbox{RHS~\eqref{E:PRELIMINARYEIKONALWITHL}}
by exploiting the monotonicity of $\hypersurfacecontrolwave_{[1,N]}(\timefunction,u)$ with respect to
its arguments.

§.§ $L^2$ estimates for $\wavearray$ that lose one derivative

Let $1 \leq N \leq \Ntop$. The following estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$:

\begin{align} \label{E:NONDEGENERATEBUTDERIVATIVELOSINGL2ESTIMATEPSI}
\left\| \comdersmall^{N;1} \wavearray \right\|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}
\lesssim \initialsmall
\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction,u).
\end{align}

Using (<ref>) with $N+1$ replaced by $N$, we deduce the pointwise estimate

\begin{align} \label{E:MAINPOINTWISESEMINORMNEEDEDTOPROVENONDEGENERATEBUTDERIVATIVELOSINGL2ESTIMATEPSI}
\left|\comdersmall^{N;1} \wavearray \right|
\lesssim
\left|\muX \tander^{[1,N-1]} \wavearray \right|
\left|\tander^{[1,N]} \wavearray \right|
\fundbootsmall
\left|\comdersmall^{[1,N-1];1}\controlvars \right|
\fundbootsmall
\left|\tandersmall^{[1,N-1]} \badcontrolvars \right|,
\end{align}

where when $N = 1$, we must have $\comdersmall^{1;1}  = \tander$ on
LHS (<ref>)
and then only the second term on RHS (<ref>) is present.
Taking the norm
$\| \cdot \|_{L^2\left(\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}\right)}$ of
and using
and the estimate
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}} \hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)
\, \mathrm{d} \timefunction'
\lesssim \hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction,u)$,
which follows from the fact that
$\hypersurfacecontrolwave_{[1,N]}(\timefunction,u)$ is increasing in its arguments,
we conclude (<ref>).

§ BELOW-TOP-ORDER HYPERBOLIC $L^2$ ESTIMATES FOR THE SPECIFIC VORTICITY AND ENTROPY GRADIENT

In this short section,
we prove the below-top-order $L^2$ estimates for the specific vorticity and entropy gradient.
Specifically, we prove
We derive a preliminary energy integral inequality
in Sect. <ref>,
and we prove the final estimates in
Sects. <ref>–<ref>.
These estimates are relatively straightforward consequences of
the transport energy identity (<ref>)
and various pointwise estimates we have already derived,
including the ones provided by Prop. <ref>.
In Sect. <ref>,
we will prove the top-order estimate (<ref>)
and the related estimates (<ref>) and (<ref>).
The proofs of these estimates are much more difficult because they rely on
the intricate elliptic-hyperbolic integral identity (<ref>).

In Sect. <ref>, we will use the estimates for
$\VortVort$, and $\DivGradEnt$
that we derive in this section in our proof of the wave a priori estimates,
which we stated as Prop. <ref>.
Hence, we highlight that for the logic of the paper, it is important that
the estimates we derive in this section do not rely on the wave estimates of Prop. <ref>;
our proofs of
instead rely on the bootstrap assumptions
for the wave energies, which are weaker than the estimates that we derive in Prop. <ref>.

§.§ Integral inequalities for the below-top-order vorticity- and entropy gradient-controlling quantities

We begin with the following preliminary lemma,
which provides integral inequalities
for the below-top-order vorticity- and entropy gradient-controlling quantities.

Let $0 \leq N \leq \Ntop$.
Then the following integral inequalities hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$:

\begin{align}   \label{E:BELOWTOPORDERENERGYINTEGRALINEQUALITIESSPECIFICVORTICITYANDENTROPYGRADIENT}
\hypersurfacecontrolVort_{\leq N}(\timefunction,u)
\hypersurfacecontrolGradEnt_{\leq N}(\timefunction,u)
\lesssim
\initialsmall^2
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVort_{\leq N}(\timefunction,u')
\hypersurfacecontrolGradEnt_{\leq N}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\hypersurfacecontrolwave_{[1,N]}(\timefunction',u)
\, \mathrm{d} \timefunction'.
\end{align}

Moreover, the following estimates hold for $0 \leq N \leq \Ntop - 1$:

\begin{align}
\begin{split} \label{E:BELOWTOPORDERENERGYINTEGRALINEQUALITIESFORMODIFIEDFLUID}
\hypersurfacecontrolVortVort_{\leq N}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\leq N}(\timefunction,u)
& \lesssim
\initialsmall^2
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\leq N}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\leq N}(\timefunction,u)
\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVort_{\leq N+1}(\timefunction,u')
\hypersurfacecontrolGradEnt_{\leq N+1}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\hypersurfacecontrolwave_{[1,N]}(\timefunction',u)
\, \mathrm{d} \timefunction'
\varepsilon^2
\int_{u' = - \rightu}^u
\hypersurfacecontrolwave_{[1,N]}(\timefunction,u')
\, \mathrm{d} u'
\fundbootsmall^2 \spacetimeintegralcontrolwave_{[1,N]}(\timefunction,u).
\end{split}
\end{align}

We first prove (<ref>).
For $0 \leq N \leq \Ntop$, we consider the transport energy identity (<ref>)
with $(\tander^N \vortrenormalized,\tander^N \GradEnt)$ in the role of $f$.
We use the bootstrap assumptions and Prop. <ref> to deduce the bound
$|\Lunit \upmu + \upmu \mytr_{\gtorus} \angk| \lesssim 1$ for the integrand factors in the last
integral on RHS (<ref>),
and we use (<ref>) to deduce that
the integrand factors
$\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}$
in (<ref>)
verify $\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$.
We also use
Lemma <ref>
to bound the data-dependent terms
$\mathbb{E}[f]_{(\textnormal{Transport})}(\timefunction_0,u) + \mathbb{F}[f]_{(\textnormal{Transport})}(\timefunction,-\rightu)$
on RHS (<ref>) by $\lesssim \initialsmall^2$.
Considering also Def. <ref>, Lemma <ref>,
and using Young's inequality,
we deduce:

\begin{align}  \label{E:FIRSTSTEPPROOFBELOWTOPORDERENERGYINTEGRALINEQUALITIESSPECIFICVORTICITYANDENTROPYGRADIENT}
\begin{split}
\hypersurfacecontrolVort_{\leq N}(\timefunction,u)
\hypersurfacecontrolGradEnt_{\leq N}(\timefunction,u)
\lesssim
\initialsmall^2
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVort_{\leq N}(\timefunction,u')
\hypersurfacecontrolGradEnt_{\leq N}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
|\upmu \Transport \tander^{\leq N} (\vortrenormalized,\GradEnt)|^2
\, \volMRoughCoordinates.
\end{split}
\end{align}

Next, we use the pointwise estimate (<ref>)
to bound the integrand factors of $|\upmu \Transport \tander^{\leq N} (\vortrenormalized,\GradEnt)|$
on RHS (<ref>).
Again appealing to Def. <ref> and Lemma <ref>,
and also using (<ref>),
and (<ref>),
we conclude (<ref>),
but with the additional double integral
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\left\lbrace
\int_{\timefunction'' = \timefunction_0}^{\timefunction'}
\frac{\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction'',u)}{|\timefunction''|^{1/2}}
\, \mathrm{d} \timefunction''
\right\rbrace^2
\, \mathrm{d} \timefunction'
on the RHS arising from the estimate (<ref>),
which we use to handle the terms $\fundbootsmall |\tander^{[2,N]} \badcontrolvars|$
on RHS (<ref>).
By using that $\hypersurfacecontrolwave_{[1,N]}(\timefunction,u)$ is increasing in its arguments,
we can bound this double integral by
\lesssim
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\hypersurfacecontrolwave_{[1,N]}(\timefunction',u)
\, \mathrm{d} \timefunction'
which in turn is bounded by RHS (<ref>) as desired.
We have therefore proved (<ref>).

The estimate (<ref>)
can be proved using similar arguments
based on the pointwise estimates
for $0 \leq N \leq \Ntop-1$.
However, we need two additional ingredients, which we now point out:
to bound the spacetime integrals
\varepsilon^2
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
|\tander^{N+1} \wavearray|^2
\, \volMRoughCoordinates
generated by the terms $\varepsilon |\tander^{N+1} \wavearray|$
RHSs (<ref>)–(<ref>),
we also need to use
the $L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})$-control
guaranteed by
which leads to the presence of
the term
\varepsilon^2
\int_{u' = - \rightu}^u
\hypersurfacecontrolwave_{[1,N]}(\timefunction,u')
\, \mathrm{d} u'
on RHS (<ref>);
and ii) the spacetime integral coerciveness estimate
which leads the presence of the term
$\fundbootsmall^2 \spacetimeintegralcontrolwave_{[1,N]}(\timefunction,u)$
on RHS (<ref>).

§.§ Proof of the estimates (<ref>) and

We first prove (<ref>). We set
\eqdef
\hypersurfacecontrolVort_{\leq \Ntop-8}(\timefunction,u)
\hypersurfacecontrolGradEnt_{\leq \Ntop-8}(\timefunction,u)$.
the wave energy bootstrap assumptions (<ref>),
and (<ref>),
we find that
\mathbb{T}(\timefunction,u)
\leq
C \initialsmall^2
\int_{u' = - \rightu}^u
\mathbb{T}(\timefunction,u')
\, \mathrm{d} u'
Applying Grönwall's inequality, we conclude that $\mathbb{T}(\timefunction,u)
\leq
C \initialsmall^2$, which yields (<ref>).

Similarly, to prove (<ref>),
we set
\eqdef
\hypersurfacecontrolVortVort_{\leq \Ntop-9}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-9}(\timefunction,u)$.
the wave energy bootstrap assumptions (<ref>),
and the already proved estimates (<ref>),
we find that
\mathbb{T}(\timefunction,u)
\leq
C \initialsmall^2
\int_{u' = - \rightu}^u
\mathbb{T}(\timefunction,u')
\, \mathrm{d} u'
Applying Grönwall's inequality, we conclude that $\mathbb{T}(\timefunction,u)
\leq
C \initialsmall^2$, which yields (<ref>).

The estimates
can be proved by combining similar arguments with the wave energy bootstrap assumptions

§.§ Proof of the estimates

Fix any integer $N$ with $0 \leq N \leq \Ntop-1$.
Using (<ref>),
and the data estimates
we deduce that:

\begin{align}
\begin{split} \label{E:FIRSTSTEPPROOFOFBELOWTOPORDERESTIMATESP:ROUGHTORIENERGYESTIMATES}
\left\|
\tander^N (\Omega,\GradEnt)
\right\|_{L^2\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}^2
\lesssim
\left\|
\tander^N (\Omega,\GradEnt)
\right\|_{L^2\left(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}\right)}^2
\int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction]}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \left|\Lunit \tander^N (\Omega,\GradEnt) \right|^2
\, \volPuRoughCoordinates
\\
\lesssim
\initialsmall^2
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}}
\tander^{\leq N}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction]})}^2
\lesssim
\initialsmall^2
\hypersurfacecontrolVort_{N+1}(\timefunction,u)
\hypersurfacecontrolGradEnt_{N+1}(\timefunction,u).
\end{split}
\end{align}

From (<ref>)
and the already proven estimates
(<ref>) and (<ref>)
\hypersurfacecontrolVort_{N+1}(\timefunction,u),
\hypersurfacecontrolGradEnt_{N+1}(\timefunction,u)
we conclude,
in view of definitions (<ref>)–(<ref>),
the desired estimates (<ref>) and (<ref>).

The estimates
can be proved via similar arguments based on the data estimates
the already proven estimates
(<ref>) and (<ref>),
and definitions (<ref>)–(<ref>).

§ TOP-ORDER ELLIPTIC-HYPERBOLIC $L^2$ ESTIMATES FOR THE SPECIFIC VORTICITY AND ENTROPY GRADIENT

Our main goal in this section is to derive the top-order $L^2$ estimate
for the modified fluid variables $\VortVort$ and $\DivGradEnt$.
It turns out that by virtue of the elliptic-hyperbolic
integral identity (<ref>),
the proof of (<ref>)
is coupled to the proof of the top-order energy estimates
for $\vortrenormalized$ and $\GradEnt$
along the rough tori. Hence, we also prove (<ref>)
in this section. Finally, as a simple consequence of (<ref>),
we will also prove the top-order
rough tori energy estimate (<ref>) for
$\VortVort$ and $\DivGradEnt$.
In Sect. <ref>, we will use the estimates for $\VortVort$ and $\DivGradEnt$
that we derive in this section in our proof of the wave a priori estimates,
which we stated as Prop. <ref>.
Hence, we highlight that for the logic of the paper, it is important that
the estimates we derive in this section do not rely on the wave estimates of Prop. <ref>;
our proofs of (<ref>)
and (<ref>)
instead rely on the bootstrap assumptions
for the wave energies, which are weaker than the estimates that we derive in Prop. <ref>.

To explain the main challenges in the analysis,
we recall that the transport equations (<ref>)
satisfied by
$\VortVort$ and $\DivGradEnt$
feature some difficult source terms,
denoted by $\mainnullform_{(\VortVort)}^i$ and $\mainnullform_{(\DivGradEnt)}$,
that depend on the general first-order derivatives
of $\vortrenormalized$ and $\GradEnt$.
These source terms have the potential to cause the loss of a derivative
at the top-order because they cannot be bounded using pure transport estimates.
In the below-top-order estimates of
Sect. <ref>, we allowed
the loss of a derivative, as is signified by the terms
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVort_{\leq N+1}(\timefunction,u')
\hypersurfacecontrolGradEnt_{\leq N+1}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
on RHS (<ref>).
To avoid the loss at the top-order, we handle the difficult source terms in a different way,
one that is based on combining the elliptic-hyperbolic integral identity provided by
Prop. <ref>
with pointwise estimates that take into account the special structure of the equations
of Theorem <ref>, sharp estimates for the acoustic geometry
and the rough time function, and the below-top-order estimates that we already derived in
Sect. <ref>.

We organize this section this as follows:

* In Sect. <ref>,
we derive some preliminary energy integral inequalities for $\VortVort$ and $\DivGradEnt$,
which feature the difficult source terms; this part of the proof is not more difficult than the
proof of the below-top-order energy integral inequalities we derived in
Lemma <ref>.

* In Sect. <ref>,
we use the preliminary energy integral inequalities to derive the main energy integral inequalities for the top-order
derivatives of $\VortVort$ and $\DivGradEnt$. These main energy integral inequalities
are conditional on having $L^2$ estimates for the difficult source terms, which we derive independently as
Prop. <ref>
in Sect. <ref>.

* In Sect. <ref>,
we use the main energy integral inequalities
to prove the top-order $L^2$ estimate (<ref>).

* In Sect. <ref>,
to initiate the proof of Prop. <ref>,
we derive estimates for the rough tori error integrals
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$ for $\timefunction \in [\timefunction_0,\timefunctionboot]$,
which appear on the right-hand side of the elliptic-hyperbolic integral identity
(with $\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)$ in the role of $\SigmatTan$)
when we use the identity in our proof of Prop. <ref>.
It might be tempting to think of these rough tori integrals as “data terms” since
the rough tori $\roughtori{\timefunction,- \rightu}$ of interest
are contained in the
“data null hypersurface” portion
In particular, if the data on $\Sigma_0$ are compactly supported in
$\Sigma_0 \cap \lbrace - \rightu \leq u \leq \leftu \rbrace$, then standard domain of dependence considerations
imply that $(\vortrenormalized,\GradEnt)$ vanish along $\roughtori{\timefunction,- \rightu}$.
However, in general,
the integrals
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$
are not true “data terms” because their size depends
on various norms of the rough time function $\timefunctionarg{\muxmulevelsetvalue}$,
which in turn depends on the behavior of the fluid near the singular boundary.
Hence, to bound the integrals
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$,
we combine slight extensions of the Cauchy-stability results
that we derive in Appendix <ref>
with suitable $C_{\textnormal{geo}}^{2,1}$ estimates for $\timefunctionarg{\muxmulevelsetvalue}$.

* In Sect. <ref>,
in service of the proof of Prop. <ref>,
we derive estimates for the error integrals appearing in
the elliptic-hyperbolic integral identity

* In Sect. <ref>,
we prove Prop. <ref>.

* Finally, in Sect. <ref>,
we prove the top-order estimate (<ref>)
for $\vortrenormalized$ and $\GradEnt$ along the rough tori
as well as the top-order estimate (<ref>)
for $\VortVort$ and $\DivGradEnt$ along the rough tori.

§.§ Preliminary energy integral inequalities for the top-order derivatives of $\VortVort$ and $\DivGradEnt$

For any $\varsigma \in (0,1]$,
the following integral inequalities hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$,
where the pointwise norm $|\cdot|_{\hfour}$ is defined in (<ref>)
and the implicit constants are independent of $\varsigma$:

\begin{align}
\begin{split} \label{E:PRELIMINARYINTEGRALINEQUALITIESFORTOPDERIVATIVESOFMODIFIEDFLUID}
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\lesssim
\initialsmall^2
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}^2
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}^2
\right\rbrace
\, \volMRoughCoordinates
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\int_{u' = - \rightu}^u
\left\lbrace \hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction,u') + \hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u')\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u')
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\\
& \ \
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction',u)
\, \mathrm{d} \timefunction'
\varepsilon^2
\int_{u' = - \rightu}^u
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u')
\, \mathrm{d} u'.
\end{split}
\end{align}

The proof is almost identical to the proof of (<ref>)
with $\Ntop$ in the role of $N$,
except we separate the spacetime error integrals
generated by the top-order derivatives of $(\vortrenormalized,\GradEnt)$.
More precisely, with $N \eqdef \Ntop$ in the pointwise estimates
for $|\upmu \Transport (\tander^{\Ntop} \VortVort,\tander^{\Ntop} \DivGradEnt)|$,
we isolate the contribution of the error terms
$|\tander^{\Ntop + 1} (\vortrenormalized,\GradEnt)|$ on the RHSs.
Since (schematically)
$\tander^{\Ntop + 1} (\vortrenormalized^i,\GradEnt^i) =
\Singletan^{\alpha} \partial_{\alpha} \tander^{\Ntop} (\vortrenormalized^i,\GradEnt^i)
$ for some $\Singletan \in \{\Lunit,\Yvf{2},\Yvf{3}\}$,
we can use the simple Cartesian component bound $|\Singletan^{\alpha}| \lesssim 1$
(which follows from Lemma <ref> and the bootstrap assumptions)
and (<ref>)
to pointwise bound these error terms in magnitude by
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}$.
Thus, in the energy identity
(that is, (<ref>) with
$f \eqdef (\tander^{\Ntop}\VortVort,\tander^{\Ntop}\DivGradEnt)$),
the spacetime error integral corresponding to these terms is bounded by:

\begin{align} \label{E:MAINERRORINTEGRALINPROOFOFPRELIMINARYINTEGRALINEQUALITIESFORTOPDERIVATIVESOFMODIFIEDFLUID}
\lesssim
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\left|
\tander^{\Ntop} \VortVort
\right|_{\hfour}
\left|
\tander^{\Ntop} \DivGradEnt
\right|_{\hfour}
\right\rbrace
\left\lbrace
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}
\right\rbrace
\, \volMRoughCoordinates.
\end{align}

Using the estimate (<ref>),
and Young's inequality,
for any $\varsigma \in (0,1]$,
we find that
\mbox{RHS~\eqref{E:MAINERRORINTEGRALINPROOFOFPRELIMINARYINTEGRALINEQUALITIESFORTOPDERIVATIVESOFMODIFIEDFLUID}}
\lesssim
\frac{1}{\varsigma}
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}^2
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}^2
\right\rbrace
\, \volMRoughCoordinates
which is bounded by RHS (<ref>)
as desired.

§.§ The main energy integral inequalities for the top-order derivatives of $\VortVort$ and $\DivGradEnt$, conditional on
Prop. <ref>

Most of our effort in Sect. <ref>
is dedicated towards bounding the
spacetime integrals:

\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\left\lbrace
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}^2
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}^2
\right\rbrace
\, \volMRoughCoordinates,
which appear on RHS (<ref>).
We derive the needed estimates in Prop. <ref>.
Given Prop. <ref>
and Lemma <ref>,
it is easy to derive energy integral inequalities
that can be used to obtain the desired top-order energy estimates
for $\VortVort$ and $\DivGradEnt$; we derive these integral inequalities in the next lemma.

For any $\varsigma \in (0,1]$,
the following integral inequalities hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$,
where the implicit constants are independent of $\varsigma$:

\begin{align}
\begin{split}  \label{E:MAININTEGRALINEQUALITIESFORTOPDERIVATIVESOFMODIFIEDFLUID}
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
& \lesssim
\varsigma
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\right\rbrace
\frac{\initialsmall^2}{|\timefunction|}
\\
& \ \
\left(1 + \frac{1}{\varsigma} \right)
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
\varepsilon^2
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u)
\\
&  \ \
\frac{1}{|\timefunction|^2}
\left\lbrace
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u)
\right\rbrace
\frac{1}{|\timefunction|^{5/2}}
\left\lbrace
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u)
\right\rbrace.
\end{split}
\end{align}

We start with inequality (<ref>).
Using Lemma <ref>
and the estimate
$\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \approx 1$
implied by (<ref>),
we deduce the pointwise bounds
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}^2
\lesssim
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}^2
\lesssim
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
for the error integrands on the first line of RHS (<ref>).
Hence, thanks to the estimates
which we prove independently in Sect. <ref>,
the spacetime integral
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\left\lbrace
\left|
\pmb{\partial} \tander^{\Ntop} \vortrenormalized
\right|_{\hfour}^2
\left|
\pmb{\partial} \tander^{\Ntop} \GradEnt
\right|_{\hfour}^2
\right\rbrace
\, \volMRoughCoordinates
on the first line of RHS (<ref>)
is bounded by

\[	\varsigma
\left\lbrace
\mbox{RHS~\eqref{E:ELLIPTICHYPERBOLICINTEGRALINEQUALITYFORSPECIFICVORTICITY}}
\mbox{RHS~\eqref{E:ELLIPTICHYPERBOLICINTEGRALINEQUALITYFORENTROPYGRADIENT}}
\right\rbrace.\]

The RHS of the resulting inequality features the
error integrals
$\left(1 + \frac{1}{\varsigma} \right)
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
which we place directly on RHS (<ref>).
Finally, we further bound the remaining error integrals on the RHS
by using the monotonicity of the controlling quantities
(with respect to their arguments $\timefunction, u$) to pull the controlling quantities out of the integrals and to gain a power of $|\timefunction|$ upon
integration with respect to $\mathrm{d} \timefunction'$,
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^3}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction',u)
\, \mathrm{d} \timefunction'
\lesssim
\frac{1}{|\timefunction|^2}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\int_{u' = - \rightu}^u
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\, \mathrm{d} u'
\lesssim
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
In total, these arguments yield (<ref>).

§.§ Proof of the main top-order energy estimate (<ref>)

Given Lemma <ref>,
we are now ready to prove our main top-order a priori energy estimates (<ref>)
for the modified fluid variables.
We again emphasize that Lemma <ref>
is conditional on the estimates of Prop. <ref>,
which we prove independently below.

To proceed, we choose and fix $\varsigma > 0$ sufficiently small such that the term
\varsigma
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\right\rbrace
on RHS (<ref>)
can be absorbed back into the LHS at the expense of increasing the implicit
constants. Next, we
use the already proven estimates
(<ref>) and
the bootstrap assumptions (<ref>)–(<ref>),
and (<ref>)
to bound all terms on RHS (<ref>)
except for the integral
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'
In total, this leads to the following inequality:

\begin{align} \label{E:GRONWALLREADYMAINTOPORDERENERGYESTIMATESMODIFIEDFLUIDVARIABLESBLOWUP}
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\leq
\initialsmall^2 |\timefunction|^{-17.1}
\int_{u' = - \rightu}^u
\left\lbrace
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\right\rbrace
\, \mathrm{d} u'.
\end{align}

From (<ref>) and
Grönwall's inequality,
we conclude
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\leq
\initialsmall^2 |\timefunction|^{-17.1}
which yields the desired bound (<ref>).

§.§ Control of $\left\|
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$
for $\timefunction \in [\timefunction_0,\timefunctionboot]$

Recall that our proof of the top-order $L^2$ estimates
relies on Prop. <ref>,
whose proof relies on
the integral identity
In order to exploit the identity (<ref>)
with $u_1 = - \rightu$,
we in particular have to first control the rough tori error integrals
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$ for $\timefunction \in [\timefunction_0,\timefunctionboot]$;
in view of (<ref>)
with $\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)$ in the role of $\SigmatTan$,
we see that control of
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}
$ is sufficient to bound the
first integral $\int_{\twoargroughtori{\timefunction_2,u_1}{\muxmulevelsetvalue}} \cdots$ on
RHS (<ref>).
As we explained at the beginning of Sect. <ref>,
these integrals are not pure “data terms” because their size depends
on various norms of the rough time function, which in turn
depends on the behavior of the fluid near the singular boundary.
Hence, in the next lemma,
we derive estimates for
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$.
The proof relies on the Cauchy stability-type estimates proved in
Appendix <ref>,
which also rely on various applications of the integral identity
We emphasize that our bounds for
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$
cannot be proved by combining
the data estimates (<ref>)
along the null hypersurface $\nullhyparg{- \rightu}$
with trace estimates because of the usual loss of differentiability incurred by trace estimates.
However, if we had assumed that the data are one degree more differentiable, more precisely that
(<ref>) holds
with $\Ntop$ replaced by $\Ntop+1$, then we could have used trace estimates to give a simpler proof
of the desired estimates for
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}$.
Though simpler, we avoided that approach because it would have led to estimates such that
the solution is less differentiable than the data.

\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1}
be the norm of the perturbation of the data away from the background solution,
as defined in (<ref>).
If $
\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1}
is sufficiently small, then the following estimates hold for
$\timefunction \in [\timefunction_0,\timefunctionboot]$:

\begin{align} \label{E:SMALLDATAOFVORTICITYANDENTORPYGRADIENTONOUTERROUGHTORI}
\left\|
\tander^{\leq \Ntop}(\vortrenormalized,\GradEnt)
\right\|_{L^2(\roughtori{\timefunction,- \rightu})}
\leq \initialsmall,
\end{align}

$\initialsmall = \mathcal{O}(\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1})$,
and the implicit constants depend on the background solution.

The bootstrap assumption (<ref>)
implies that for $\timefunction \in [\timefunction_0,\timefunctionboot]$,
the rough tori $\roughtori{\timefunction,- \rightu}$ are contained in the
“data null hypersurface” $\nullhyparg{- \rightu}^{[0,4 \mathring{\updelta}_*]}$.
For this reason,
the proof of (<ref>)
relies on the smallness results we derived for the solution
on $\nullhyparg{- \rightu}^{[0,4 \mathring{\updelta}_*]}$ in
Appendix <ref>.
In particular, in our proof here, we will refer to various steps in the
proof of Prop. <ref>.
We clarify that although the proof of Prop. <ref>
relies on ideas from the bulk of the paper,
its proof is independent of the results of Lemma <ref>.
In the rest of the proof, we will
silently assume that $\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1}$
is sufficiently small.

There are two broad steps in the proof of (<ref>):
I) extend
the rough time function $\timefunctionarg{\muxmulevelsetvalue}$ into a subset of the “smallness region” $\CSregion_{Small}^{[0,5 \blowuptimePS]}$
from Prop. <ref> (see Fig. <ref>)
and derive standard $C_{\textnormal{geo}}^{2,1}$ estimates for $\timefunctionarg{\muxmulevelsetvalue}$ in the extended region;
and II) combine a version of the integral identity
(<ref>) in a well-chosen subset of
$\CSregion_{Small}^{[0,5 \blowuptimePS]}$
with the estimates for $\timefunctionarg{\muxmulevelsetvalue}$ from step I),
and use some results from the proof of Prop. <ref>
to conclude (<ref>).

Step I: Extending and controlling $\timefunctionarg{\muxmulevelsetvalue}$.
In Steps 1 and 2 of the proof of Prop. <ref>,
we show that up to the top-order derivative level
(i.e., the derivative level corresponding to the energy estimates of
Props. <ref>,
the fluid variables, $\upmu - 1$, $\Lsmall^i$, and $\upchi$
are bounded by
$\lesssim \mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1}$
on the region
$\CSregion_{Small}^{[0,5 \blowuptimePS]}$ defined in (<ref>),
which, by (<ref>),
contains $\nullhyparg{- \rightu}^{[0,4 \mathring{\updelta}_*]}$.
That is, in $\CSregion_{Small}^{[0,5 \blowuptimePS]}$,
the solution is close to the trivial fluid solution with Euclidean acoustic geometry.
Unlike in
Props. <ref>,
in Steps 1 and 2 of the proof of Prop. <ref>
we derive the smallness
in $\CSregion_{Small}^{[0,5 \blowuptimePS]}$
with respect to foliations by portions
of Cartesian time slices $\Sigma_t$,
null hypersurfaces $\nullhyparg{u}$,
and smooth tori $\ell_{t,u}$.
We will now explain how we can combine this smallness with
the transport equation (<ref>)
and the data of $\timefunctionarg{\muxmulevelsetvalue}$ on
\nullhypthreearg{\muxmulevelsetvalue}{-\rightu}{[\timefunction_0,\timefunctionboot]}
(which is already “known” to us in view of the last item in Lemma <ref>)
to extend $\timefunctionarg{\muxmulevelsetvalue}$
to a larger domain with the following two properties:
a) $\timefunctionarg{\muxmulevelsetvalue} = \timefunctionarg{\muxmulevelsetvalue}(t,u,x^2,x^3)$ is defined on
\twoargMrough{[\timefunction_0,\timefunctionboot],[-U_*,\leftu]}{\muxmulevelsetvalue}
where $U_* > 0$ is the number in (<ref>),
and b)
\twoargMrough{[\timefunction_0,\timefunctionboot],[-U_*,-\rightu]}{\muxmulevelsetvalue}
\subset
\twoargMrough{[\timefunction_0,\timefunctionboot],[-\rightu,-\leftu]}{\muxmulevelsetvalue}
\cup
\CSregion_{Small}^{[0,5 \blowuptimePS]}
To carry out this extension, we first note that
and (<ref>)
imply that
the data null hypersurface
\nullhypthreearg{\muxmulevelsetvalue}{- \rightu}{[\timefunction_0,\timefunctionboot]}
is contained in $\CSregion_{Small}^{[0,5 \blowuptimePS]}$
and that the distance (with respect to the standard flat metric on geometric coordinate space)
\nullhypthreearg{\muxmulevelsetvalue}{- \rightu}{[\timefunction_0,\timefunctionboot]}
and the top boundary of $\CSregion_{Small}^{[0,5 \blowuptimePS]}$
(which we denote by “$\Sigma_{5 \blowuptimePS}^{[15 \blowuptimePS - \farrightu,- \rightu]}$”
in Fig. <ref>) is at least $\frac{1}{2} \blowuptimePS$.
We next note that Def. <ref> implies that in $\CSregion_{Small}^{[0,5 \blowuptimePS]}$
(a region in which $u < \interestingu$),
the transport equation (<ref>)
takes the form $\muX \timefunctionarg{\muxmulevelsetvalue} = 0$,
where we recall that
$\geop{u} = \muX - \muX^A \geop{x^A}$.
The smallness provided by Steps 1 and 2
of the proof of Prop. <ref>
implies that
\sum_{A=2,3}
\| \muX^A \|_{C_{\textnormal{geo}}^{2,1}\left(\CSregion_{Small}^{[0,5 \blowuptimePS]} \right)}
\lesssim
\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1}
Using this smallness
and deriving standard $C_{\textnormal{geo}}^{2,1}$-estimates for solutions
to $\muX \timefunctionarg{\muxmulevelsetvalue} = 0$
starting from the data of $\timefunctionarg{\muxmulevelsetvalue}$ on
the data null hypersurface portion
\nullhypthreearg{\muxmulevelsetvalue}{- \rightu}{[\timefunction_0,\timefunctionboot]}
(which is contained in $\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue}$),
we find that:

\begin{align} \label{E:C21GEOBOUNDFORROUGHTIMEFUNCTIONINEXTENDEDREGION}
\| \timefunctionarg{\muxmulevelsetvalue} \|_{C_{\textnormal{geo}}^{2,1}\left(\twoargMrough{[\timefunction_0,\timefunctionboot],[-U_*,-\rightu]}{\muxmulevelsetvalue} \right)}
\leq
\| \timefunctionarg{\muxmulevelsetvalue} \|_{C_{\textnormal{geo}}^{2,1}\left(\twoargMrough{[\timefunction_0,\timefunctionboot],[- \rightu,\leftu]}{\muxmulevelsetvalue} \right)}
\mathcal{O}\left(\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1} \right).
\end{align}

From (<ref>)
and (<ref>),
we deduce the following bound, which we will use below:

\begin{align} \label{E:C21GEOESTIMATEFORROUGHTIMEFUNCTIONINSMALLCACUHYSTABILITYREGION}
\| \timefunctionarg{\muxmulevelsetvalue} \|_{C_{\textnormal{geo}}^{2,1}\left(\twoargMrough{[\timefunction_0,\timefunctionboot],[-U_*,\leftu]}{\muxmulevelsetvalue} \right)}
\leq C.
\end{align}

Step II: Using energy estimates and applications of the integral identity
(<ref>) to finish the proof.
In Step 5 of the proof of Prop. <ref>,
we derived – independently of Lemma <ref>
(see Remark <ref>) –
geometric energy estimates in subsets
\twoargMrough{[\timefunction_*,\timefunction_0/2],[-U_*,\leftu]}{\muxmulevelsetvalue}
of $\CSregion_{Small}^{[0,5 \blowuptimePS]}$,
where $\timefunction_* \in [2 \timefunction_0, (3/2) \timefunction_0]$ is the number from (<ref>),
$U_* > 0$ is the number from (<ref>),
and $\timefunction_2$ and $u_2$ are any numbers satisfying
$\timefunction_2 \in [\timefunction_*,\timefunction_0/2]$
$u_2 \in [-U_*,\leftu]$.
These estimates in particular yield the rough tori $L^2$ estimates stated in
The proof of (<ref>)
relies on applying the integral identity (<ref>) on the region
i.e., with $\timefunction_1 = \timefunction_*$
$u_1 = -U_*$ in (<ref>).
We highlight two key ingredients that are needed
to control the error terms in that application of (<ref>):
i) bounds for the rough tori integrals
arising from RHS (<ref>)
(see Remark <ref>),
i.e., the integrals
$\int_{\twoargroughtori{\timefunction_2,-U_*}{\muxmulevelsetvalue}} \cdots$,
\int_{\twoargroughtori{\timefunction_*,u_2}{\muxmulevelsetvalue}}
\cdots
\int_{\twoargroughtori{\timefunction_*,-U_*}{\muxmulevelsetvalue}}
\cdots
which we bound in the arguments given above
and ii) the bound
\| \timefunctionarg{\muxmulevelsetvalue} \|_{C_{\textnormal{geo}}^{2,1}\left(\twoargMrough{[\timefunction_*,\timefunction_0/2],[-U_*,\leftu]}{\muxmulevelsetvalue} \right)}
\leq C
which is needed to bound various error terms on RHS (<ref>)
that arise when we apply it on the region
The key point is that the estimate (<ref>)
shows that the same bound for $\timefunctionarg{\muxmulevelsetvalue}$ holds on the larger
subset $\twoargMrough{[\timefunction_0,\timefunctionboot],[-U_*,-\rightu]}{\muxmulevelsetvalue}$
of $\CSregion_{Small}^{[0,5 \blowuptimePS]}$.
Hence, the same arguments that yield
can be used to show that:

\begin{align} \label{E:ENERGIESAREBOUNDEDONALLTORIINTEGRALSALONDATANULLHYPERSURFACEUPTOBOOSTRAPTIME}
\max_{\timefunction \in [\timefunction_*,\timefunctionboot]}
\max_{u \in [-U_*,\leftu]}
\int_{\roughtori{\timefunction,u}}
|\tander^{\leq \Ntop} (\Omega,\GradEnt)|^2
\, \volroughtorus
\lesssim
\left(\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1} \right)^2.
\end{align}

Finally, we note that since
$[\timefunction_0,\timefunctionboot] \subset [\timefunction_*,\timefunctionboot]$
and $- \rightu \in [-U_*,\leftu]$,
implies (<ref>)
with $\initialsmall = \mathcal{O}(\mathring{\Delta}_{\Sigma_0^{[-\farrightu,\leftu]}}^{\Ntop+1})$.

§.§ $L^2$ estimates for the error terms

In the next lemma, we derive $L^2$ estimates for the error integrals
in the elliptic-hyperbolic identities provided by Prop. <ref>.

Let $\varsigma \in [0,1)$,
let $(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$,
and let
\EllipticHyperbolicCurrentIntegralIdentityTotalSpacetimeErrorTerm[\pmb{\partial} \tander^{\Ntop} \vortrenormalized, \pmb{\partial} 			\tander^{\Ntop} \vortrenormalized]
\EllipticHyperbolicCurrentIntegralIdentityTotalSpacetimeErrorTerm[\pmb{\partial} \tander^{\Ntop} \GradEnt,
\pmb{\partial} \tander^{\Ntop} \GradEnt]
be the error terms defined by (<ref>).
Then the following spacetime integral estimates hold,
where the implicit constants are independent of $\varsigma$:

\begin{align}
\begin{split} \label{E:SPACETIMEERRORTERMESTIMATEFORSPECIFICVORTICITYELLIPTICHYPERBOLICIDENTITY}
\left|
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\EllipticHyperbolicCurrentIntegralIdentityTotalSpacetimeErrorTerm[ \tander^{\Ntop} \vortrenormalized, \pmb{\partial}
\tander^{\Ntop} \vortrenormalized]
\, \volMRoughCoordinates
\right|
& \lesssim
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\, \volMRoughCoordinates
\\
& \ \
\int_{u' = - \rightu}^u
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\, \mathrm{d} u'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction',u)
\, \mathrm{d} \timefunction'
\\
&  \ \
\left(1 + \frac{1}{\varsigma} \right)
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\left\lbrace
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction',u)
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction',u)
\right\rbrace
\, \mathrm{d} \timefunction'
\\
& \ \
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction',u)
\, \mathrm{d} \timefunction',
\end{split}
\\
\begin{split}  \label{E:SPACETIMEERRORTERMESTIMATEFORENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITY}
\left|
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\EllipticHyperbolicCurrentIntegralIdentityTotalSpacetimeErrorTerm[ \tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
\, \volMRoughCoordinates
\right|
& \lesssim
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
\, \volMRoughCoordinates
\\
& \ \
\int_{u' = - \rightu}^u
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\, \mathrm{d} u'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction',u)
\, \mathrm{d} \timefunction'
\\
&  \ \
\left(1 + \frac{1}{\varsigma} \right)
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\left\lbrace
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction',u)
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction',u)
\right\rbrace
\, \mathrm{d} \timefunction'
\\
& \ \
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction',u)
\, \mathrm{d} \timefunction'.
\end{split}
\end{align}

Moreover, the error terms
\Currentboundaryerrorhavetocontrolprincipal[\tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized],
\cdots,
\Currentboundaryerrorhavetocontrollowerorder[\tander^{\Ntop} \GradEnt, \tander^{\Ntop} \GradEnt]
defined by
verify the following rough hypersurface integral estimates:

\begin{align}
\begin{split} \label{E:PRINCIPALSPATIALERRORTERMESTIMATEFORSPECIFICVORTICITYELLIPTICHYPERBOLICIDENTITY}
& \left|
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\Currentboundaryerrorhavetocontrolprincipal
[\tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\, \volRoughHypersurface
\right|
\\
& \ \
\lesssim
\initialsmall^2
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u)
\\
& \ \
\varepsilon^2
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u),
\end{split}
\\
\begin{split} \label{E:PRINCIPALSPATIALERRORTERMESTIMATEFORENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITY}
& \left|
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\Currentboundaryerrorhavetocontrolprincipal[\tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
\, \volRoughHypersurface
\right|
\\
\ \ \lesssim
\initialsmall^2
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolVortVort_{\leq
\Ntop-1}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u)
\\
& \ \
\varepsilon^2
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u),
\end{split}
\end{align}

\begin{align} \label{E:LOWERORDERSPATIALERRORTERMESTIMATEFORSPECIFICVORTICITYELLIPTICHYPERBOLICIDENTITY}
\left|
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\Currentboundaryerrorhavetocontrollowerorder[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized]
\, \volRoughHypersurface
\right|
& \lesssim
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolVort_{\Ntop}(\timefunction,u),
\\
\left|
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\Currentboundaryerrorhavetocontrollowerorder[\tander^{\Ntop} \GradEnt, \tander^{\Ntop} \GradEnt]
\, \volRoughHypersurface
\right|
& \lesssim
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolGradEnt_{\Ntop}(\timefunction,u).
\label{E:LOWERORDERSPATIALERRORTERMESTIMATEFORENTROPYGRADIENTELLIPTICHYPERBOLICIDENTITY}
\end{align}

Finally, the error terms
$\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized],
\cdots,
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \GradEnt, \tander^{\Ntop} \GradEnt]
defined by (<ref>)
verify the following rough tori integral estimates:

\begin{align}
\label{E:TOPORDERSMALLNESSOFVORTICITYANDENTROPYGRADIENTDATAALONGNULLHYPTORI}
\int_{\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized]
\, \volroughtorus,
\,
\int_{\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \GradEnt, \tander^{\Ntop} \GradEnt]
\, \volroughtorus
& \lesssim
\initialsmall^2,
\\
\int_{\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized]
\, \volroughtorus,
\,
\int_{\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \GradEnt, \tander^{\Ntop} \GradEnt]
\, \volroughtorus
& \lesssim
\frac{\initialsmall^2}{|\timefunction_0|}.
\label{E:TOPORDERSMALLNESSOFVORTICITYANDENTROPYGRADIENTDATAALONGROUGHHYPTORI}
\end{align}

We first prove (<ref>).
We provide the details only for the $\tander^{\Ntop} \vortrenormalized$-dependent integral
on LHS (<ref>)
since the $\tander^{\Ntop} \GradEnt$-dependent integral can be treated using identical arguments.
To proceed, we first use (<ref>) to deduce that $\upmu$ is bounded from below by $\gtrsim 1$ on
$\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}$.
using (<ref>),
and (<ref>),
we conclude that
\int_{\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized]
\, \volroughtorus
\lesssim
\int_{\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}}
|\tander^{\Ntop} \vortrenormalized|_g^2
\, \volroughtorus
\lesssim
\initialsmall^2
as desired.
Similarly, to prove (<ref>)
for the $\tander^{\Ntop} \vortrenormalized$-dependent integral on the LHS,
we use (<ref>) with $\timefunction \eqdef \timefunction_0$,
and (<ref>)
to conclude that
\int_{\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized, \tander^{\Ntop} \vortrenormalized]
\, \volroughtorus
\lesssim
\frac{1}{|\timefunction_0|}
\int_{\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}}
|\tander^{\Ntop} \vortrenormalized|_g^2
\, \volroughtorus
\lesssim
\frac{1}{|\timefunction|_0}
\initialsmall^2
as desired. The $\tander^{\Ntop} \GradEnt$-dependent integral
on LHS (<ref>)
can be bounded using identical arguments.

The remaining estimates
are straightforward consequences of
the pointwise estimates provided by Prop. <ref>,
the pointwise estimates
|\upmu| \lesssim 1
\left|
\phi \frac{\muxmulevelsetvalue}{\Lunit \upmu}
\right|
\lesssim 1
(which follow from the bootstrap assumptions),
Def. <ref> of the $L^2$-controlling quantities,
the coerciveness guaranteed by Lemma <ref>
together with
the estimates (<ref>) and
the already proven $L^2$ estimates (<ref>),
and the fact that the $L^2$-controlling quantities
$\hypersurfacecontrolwave_M(\timefunction,u)$, $\hypersurfacecontrolVortVort_M(\timefunction,u)$, etc.	are increasing in their arguments.

§.§ The main elliptic-hyperbolic integral inequalities

Thanks to the availability of
Lemma <ref>,
we are now ready to prove Prop. <ref>.

Let $\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]$
be the quadratic form from Def. <ref>,
and let
be the quadratic form defined by (<ref>).
Then the following spacetime integral estimates hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$:

\begin{align}
\begin{split} \label{E:ELLIPTICHYPERBOLICINTEGRALINEQUALITYFORSPECIFICVORTICITY}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\, \volMRoughCoordinates
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \vortrenormalized,\tander^{\Ntop} \vortrenormalized]
\, \volroughtorus
\\
& \lesssim
\frac{\initialsmall^2}{|\timefunction|}
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{2}}
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction,u)
\frac{1}{|\timefunction|^{2}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u)
\\
& \ \
\varepsilon^2
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u)
\\
& \ \
\int_{u' = - \rightu}^u
\hypersurfacecontrolVortVort_{\Ntop}(\timefunction,u')
\, \mathrm{d} u'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction',u)
\, \mathrm{d} \timefunction'
\\
&  \ \
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction',u)
\, \mathrm{d} \timefunction'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction',u)
\, \mathrm{d} \timefunction'
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction',u)
\, \mathrm{d} \timefunction',
\end{split}
\\
\begin{split} \label{E:ELLIPTICHYPERBOLICINTEGRALINEQUALITYFORENTROPYGRADIENT}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
\, \volMRoughCoordinates
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\tander^{\Ntop} \GradEnt,\tander^{\Ntop} \GradEnt]
\, \volroughtorus
\\
& \lesssim
\frac{\initialsmall^2}{|\timefunction|}
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolVortVort_{\leq \Ntop-1}(\timefunction,u)
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction,u)
\\
& \ \
\varepsilon^2
\frac{1}{|\timefunction|^{3/2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction,u)
\frac{1}{|\timefunction|^{5/2}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction,u)
\\
& \ \
\int_{u' = - \rightu}^u
\hypersurfacecontrolDivGradEnt_{\Ntop}(\timefunction,u')
\, \mathrm{d} u'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolDivGradEnt_{\leq \Ntop-1}(\timefunction',u)
\, \mathrm{d} \timefunction'
\\
&  \ \
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolVort_{\leq \Ntop}(\timefunction',u)
\, \mathrm{d} \timefunction'
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{3}}
\hypersurfacecontrolGradEnt_{\leq \Ntop}(\timefunction',u)
\, \mathrm{d} \timefunction'
\varepsilon^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{2}}
\hypersurfacecontrolwave_{[1,\Ntop]}(\timefunction',u)
\, \mathrm{d} \timefunction'.
\end{split}
\end{align}

We consider the integral identity
with $\tander^{\Ntop} \vortrenormalized$ and $\tander^{\Ntop} \GradEnt$ in the role of $\SigmatTan$.
Using Lemma <ref> with
$\timefunction_1 \eqdef \timefunction_0$,
$\timefunction_2 \eqdef \timefunction$,
$u_1 \eqdef - \rightu$,
$u_2 \eqdef u$,
we bound the integrals on RHS (<ref>),
where we can discard the integrals
\int_{\twoargroughtori{\timefunction_1,u_1}{\muxmulevelsetvalue}}
\CurrentboundaryerrorperfectRderivative[\SigmatTan,\SigmatTan]
\, \volroughtorus
because they are non-positive in view of
Finally, by choosing and fixing $\varsigma > 0$ to be sufficiently small,
we can absorb the first term
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \vortrenormalized, \pmb{\partial} \tander^{\Ntop} \vortrenormalized]
\, \volMRoughCoordinates
on RHS (<ref>)
the first term
\varsigma
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\ellipticCoerciveQuadratic[\pmb{\partial} \tander^{\Ntop} \GradEnt, \pmb{\partial} \tander^{\Ntop} \GradEnt]
\, \volMRoughCoordinates
on RHS (<ref>)
into LHS (<ref>)
LHS (<ref>) respectively.
This yields the desired estimates

§.§ Proof of the top-order rough tori energy estimates (<ref>) and

We first prove the estimate (<ref>).
We consider the estimates
In view of
definitions (<ref>)–(<ref>),
the quantitative positive definiteness estimate
the estimate (<ref>),
the fact that the cutoff function $\phi$ on
RHS (<ref>)
is supported in the $u$-interval $[-\interestingu,\interestingu]$ (see Def. <ref>),
and the estimate $|\upmu| \lesssim 1$ (which follows from the bootstrap assumptions),
we see that up to $\mathcal{O}(1)$ factors, the rough tori integrals on
LHSs (<ref>)–(<ref>)
bound the terms on LHS (<ref>) from above.
Moreover, using the already proven estimates
the wave energy bootstrap assumptions (<ref>)–(<ref>),
and (<ref>),
we see that all terms on
RHSs (<ref>)–(<ref>)
are bounded by $\lesssim \initialsmall^2 |\timefunction|^{-17.1}$,
which yields the desired result.

The estimate (<ref>) follows from combining the same
arguments we used to prove (<ref>)
with the data estimates
the already proven estimate (<ref>),
and definitions (<ref>)–(<ref>).

§ ELLIPTIC ESTIMATES FOR THE ACOUSTIC GEOMETRY ON THE ROUGH TORI ELL

In this section, we derive elliptic
$L^2$ estimates for symmetric $\binom{0}{2}$-type tensorfields that are tangent to the smooth tori $\ell_{t,u}$.
As we will explain, our analysis fundamentally relies on the rough tori
$\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$, which, unlike the smooth tori,
are adapted to our foliations by level sets of $\timefunctionarg{\muxmulevelsetvalue}$.
We provide the main estimate in Prop. <ref>.
In Sect. <ref>,
we combine the elliptic estimates of Prop. <ref>
with hyperbolic $L^2$ estimates for the
fully modified quantities defined in Sect. <ref>
to obtain top-order $L^2$ estimates for the null second fundamental form $\upchi$,
which is tangent to $\ell_{t,u}$.
We fundamentally need these top-order estimates for $\upchi$ to avoid the loss of a derivative
in the top-order commuted wave equations.
Specifically, in the top-order case $N = \Ntop$,
we need these $L^2$ estimates to handle the terms
on RHSs (<ref>)–(<ref>)
that explicitly depend on the order $N$ derivatives $\mytr_{\gtorus}\upchi$.
The point is that when $N = \Ntop$, $\tanderY^N \mytr_{\gtorus}\upchi$ cannot be controlled in $L^2$ through
pure transport estimates; using only transport estimates at the top order would result in the loss of one derivative
due to the presence of the source term
$|\upchi|_{\gtorus}^2$ on the RHS of the transport equation (<ref>)
satisfied by $\mytr_{\gtorus}\upchi$,
which depends not only on
$\mytr_{\gtorus}\upchi$, but also on its trace-free part $\hat{\upchi}$.
The strategy of avoiding derivative loss in $\upchi$ via a combination of elliptic $L^2$ estimates
on co-dimension $2$ surfaces and hyperbolic $L^2$ estimates
was originally employed in the context of Einstein's equations in <cit.>.
Later, this strategy was used in many other works on wave and wave-like equations,
for example, in the context of low regularity local well-posedness for quasilinear wave equations in <cit.>,
in the context of irrotational shock formation in <cit.>,
and in the context of shock formation in $3D$ with vorticity and entropy in <cit.>.
What is new here compared to these works is our reliance
on the rough tori to obtain the needed top-order estimates, even though the operators
$\tanderY^N$ and $\upchi$ are adapted to the smooth tori $\ell_{t,u} = \Sigma_t \cap \nullhyparg{u}$.

To obtain the desired elliptic estimates, we decompose
symmetric type $\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfields $\upxi$ into
a main piece that is tangent to $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$,
which we control with standard elliptic estimates on the rough tori (see Lemma <ref>),
and error terms, which we must control with separate (easier) arguments.
Our primary application will be to apply the main elliptic estimate (<ref>)
with $\angLie_{\tander}^{\Ntop-1} \upchi$ in the role of $\upxi$,
which will yield $L^2$ control of $\angLie_{\tander}^{\Ntop} \upchi$;
see the proof of (<ref>).
There are many equivalent ways we could have carried out the decompositions and analysis
of this section.
We have chosen to use orthonormal frames on the smooth tori and the rough tori
and to quantitatively control the relationship between the two frames;
see Sect. <ref>.

§.§ Statement of the main elliptic estimates

In this section, we state the proposition that yields the main elliptic estimates of interest.
Its proof is located in Sect. <ref>.

Let $\upxi$ be a symmetric type-$\binom{0}{2}$
$\ell_{t,u}$-tangent tensorfield.
Then the following estimate holds
for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [-\rightu,\leftu]$:

\begin{align}
\begin{split} \label{E:ELLIPTICESTIMATECHI}
\sum_{\Singletan \in \{\Lunit,\Yvf{2},\Yvf{3}\}}
\int_{\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}}
\upmu^2 |\angLie_{\Singletan} \upxi|_{\gtorus}^2
\, \volRoughHypersurface
\leq
C \int_{\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}}
\upmu^2 |\angLie_{\Lunit} \upxi|_{\gtorus}^2
\, \volRoughHypersurface
\int_{\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}}
\upmu^2 |\angdiv \upxi|_{\gtorus}^2
\, \volRoughHypersurface
\\
& \ \
\sum_{A=2,3}
\int_{\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}}
\upmu^2 (\Yvf{A} \mytr_{\gtorus}\upxi)^2
\, \volRoughHypersurface
\fundbootsmall
\int_{\hypthreearg{\timefunction}{[-\rightu,u]}{\muxmulevelsetvalue}}
\, \volRoughHypersurface.
\end{split}
\end{align}

§.§ Orthonormal frames on the smooth tori and the rough tori

For use throughout Sect. <ref>, we recall that
$\gtorus$ denotes the first fundamental form of the smooth tori $\ell_{t,u}$
and $\gtorusroughfirstfund$
denotes the first fundamental form of the rough tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$.
In our ensuing analysis, we will use the pairs of orthonormal frames featured in the next definition.

$\{e_A\}_{A = 2,3}$ is defined to be the orthonormal frame on the smooth torus $\ell_{t,u}$
obtained from applying the Gram–Schmidt process to the geometric coordinate partial derivative vectorfields
$\left\{\geop{x^A} \right\}_{A = 2,3}$ with respect to $\gtorus$, starting with
$e_2 \eqdef \frac{1}{\sqrt{\gtorus(\geop{x^2},\geop{x^2})}}\geop{x^2}$.
Similarly, $\left\{f_A \right\}_{A = 2,3}$ is defined to be
the orthonormal frame on the rough torus $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$
obtained from applying the Gram–Schmidt process to the rough adapted coordinate partial derivative vectorfields
$\left\{\roughgeop{x^A} \right\}_{A = 2,3}$ with respect to $\gtorusroughfirstfund$,
starting with
$f_2 \eqdef \frac{1}{\sqrt{\gtorusroughfirstfund(\roughgeop{x^2},\roughgeop{x^2})}}\roughgeop{x^2}$.

In the next lemma, we provide standard expressions for
$\gtorus^{-1}$ and $\gtorusroughinversefirstfund$ relative to the orthonormal frames.

Let $\gtorus^{-1}$ be the inverse first fundamental form of $\ell_{t,u}$ from Def. <ref>,
let $\gtorusroughinversefirstfund$ be the inverse first fundamental form of $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$
from Def. <ref>,
and let $\{e_A \}_{A = 2,3}$ and $\{f_A\}_{A = 2,3}$ be the orthonormal frames from
Def. <ref>.
Then the following identities hold, where $\updelta^{AB}$ is the Kronecker delta:

\begin{align}
\gtorus^{-1}
& = \updelta^{AB} e_A \otimes e_B,
\label{E:INVERSEFIRSTFUNDOFSMOOTHTORIRELATIVETOORTHONORMALFRAME}
\\
\gtorusroughinversefirstfund
& = \updelta^{AB} f_A \otimes f_B.
\label{E:INVERSEFIRSTFUNDOFROUGHTORIRELATIVETOORTHONORMALFRAME}
\end{align}

are standard identities for inverse metrics
relative to orthonormal frames.

In the next lemma, we exhibit the relationships between
the two frames $\{e_A\}_{A = 2,3}$ and $\{f_A\}_{A = 2,3}$.

On $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
there exists a $2 \times 2$ orthogonal-matrix-valued function $\COVframe$ with components $\{\COVframe_{AB}\}_{A,B = 2,3}$
and scalar functions $\{\COVL_A \}_{A = 2,3}$ such that:

\begin{align}
e_A & =
\COVframe_{AB} f_B
\COVL_A \Lunit
= \COVframe_{AB}
\left\lbrace
f_B + \COVframe_{CB} \COVL_C \Lunit
\right\rbrace,
\label{E:COVGEOMETRICTORIFRAMETOROUGHTORIFRAME}
\\
f_A & = (\COVframe^{-1})_{AB}
\left\lbrace
e_B - \COVL_B \Lunit
\right\rbrace
\COVframe_{BA}
\left\lbrace
e_B - \COVL_B \Lunit
\right\rbrace. \label{E:COVROUGHTORIFRAMETOGEOMETRICTORIFRAME}
\end{align}

Moreover, $\gtorusroughinversefirstfund$ and $\gtorus^{-1}$ are related
through the following identity:

\begin{align} \label{E:RELATIONSHIPBETWEENINVERSEFIRSTFUNDSOFSMOOTHANDROUGHTORI}
\gtorusroughinversefirstfund
& =
\gtorus^{-1}
\COVL_A e_A \otimes \Lunit
\COVL_A \Lunit \otimes e_A
\COVL_A
\COVL_A
\Lunit \otimes \Lunit.
\end{align}

Finally, the following estimates
hold on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$:

\begin{align}
& \leq 1, \label{E:COVFRAMEABSIZE}
\\
|\COVL_A| & \leq C \fundbootsmall.
\label{E:COVFRAMELSIZE}
\end{align}

The existence of a matrix $\COVframe$ and scalar functions $\{\COVL_A\}_{A = 2,3}$ satisfying
(<ref>) follows
from the fact that the frames $\{e_2,e_3,\Lunit\}$ and $\{f_2,f_3,\Lunit\}$ both span the tangent space
of $\nullhyparg{u}$ at any of its points.
To see that $\COVframe$ is orthogonal,
we use the fact that $\Lunit$ is $\gfour$-orthogonal to the tangent space of $\nullhyparg{u}$
and the fact that both frames are $\gfour$-orthonormal
to deduce, with $\updelta_{AB}$ denoting the Kronecker delta, that:

\begin{align} \label{E:CHANGEOFFRAMEMATRIXISORTHOGONAL}
\updelta_{AB}
\gtorus(e_A,e_B) = \gfour(e_A,e_B) =
\COVframe_{AC} \COVframe_{BD}
\gfour(f_C,f_D) =
\COVframe_{AC}\COVframe_{BD} \gtorusroughfirstfund (f_C,f_D)
\COVframe_{AC}\COVframe_{BD} \updelta_{CD}
\COVframe_{AC} \COVframe_{BC}.
\end{align}

From (<ref>),
we see that
$(\COVframe^{-1})_{AB} = \COVframe_{BA}$ (i.e., $\COVframe$ is an orthogonal matrix)
and thus (<ref>) follows.

follows from using
to substitute for the frame vectorfields $\lbrace f_A \rbrace_{A=2,3}$ in
using the orthogonality of the matrix $\COVframe_{AB}$,
and comparing with (<ref>).

The estimate (<ref>) follows trivially since any orthogonal matrix has Euclidean-orthonormal rows
and thus its entries are $\leq 1$ in magnitude.

To derive the estimate (<ref>),
we first use (<ref>) and
Lemma <ref>
to deduce that
0 = \gfour(e_A,X)
\COVframe_{AB} \gfour(f_B,X)
\COVL_A
$. From this identity and (<ref>),
we see that $|\COVL_A| \lesssim |\gfour(f_B,X)|$.
(<ref>) will follow once we show that:

\begin{align} \label{E:INNERPRODUCTOFFBANDXISSMALL}
|\gfour(f_B,X)| \lesssim \fundbootsmall.
\end{align}

To prove (<ref>),
we first use
Lemma <ref>,
the bootstrap assumptions, and Cor. <ref>
to deduce that
\Speed^{-2} \updelta_{AB} + \mathcal{O}(\fundbootsmall)
\left\lbrace
1 + \mathcal{O}(\mathring{\upalpha})
\right\rbrace
\updelta_{AB}
\mathcal{O}(\fundbootsmall)
where $\updelta_{AB}$ is the Kronecker delta.
Also using (<ref>)
and the estimates of Lemma <ref>
for $\timefunctionarg{\muxmulevelsetvalue}$,
we find that
\left\lbrace
1 + \mathcal{O}(\mathring{\upalpha})
\right\rbrace
\updelta_{AB}
\mathcal{O}(\fundbootsmall)$.
Next, since $\left\lbrace \roughgeop{x^2}, \roughgeop{x^3} \right\rbrace$
spans the tangent space of
the rough tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$,
for $A=2,3$,
there exist scalar functions $\upalpha_A$ and $\upbeta_A$
such that
$f_A = \upalpha_A \roughgeop{x^2} + \upbeta_A \roughgeop{x^3}$.
$1 = \gtorusroughfirstfund(f_A,f_A)$
(with no summation over $A$) by assumption,
it follows from the estimate
\left\lbrace
1 + \mathcal{O}(\mathring{\upalpha})
\right\rbrace
\updelta_{AB}
\mathcal{O}(\fundbootsmall)
$|\upalpha_A|, \, |\upbeta_A| \lesssim 1$,
i.e., $f_A = \mathcal{O}(1) \roughgeop{x^2} +\mathcal{O}(1) \roughgeop{x^3}$.
From this relation,
Lemma <ref>,
the bootstrap assumptions,
Lemma <ref>,
and Cor. <ref>,
we deduce that
$f_A = \mathcal{O}(1) \geop{x^2} + \mathcal{O}(1) \geop{x^3}
\mathcal{O}\left(\frac{\geop{x^A} \timefunction}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} \right) \Lunit
\mathcal{O}(1) \geop{x^2} + \mathcal{O}(1) \geop{x^3}
\mathcal{O}(\fundbootsmall) \Lunit
From this identity and Lemma <ref>, we conclude

§.§ An alternate representation of $\gfour$-orthogonal projection onto the rough tori

Let $\upxi$ be a symmetric type-$\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfield.
We define $\roughangxi$ to be the symmetric type $\binom{0}{2}$ $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfield
whose type $\binom{2}{0}$ $\gfour$-dual, which we denote by $\roughangxitwouparg{\#}{\#}$,
has the following components relative to the orthonormal frame $\lbrace f_2, f_3 \rbrace$
on $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$
from Def. <ref>:

\begin{align} \label{E:ROUGHTORICOMPONENTSOFFLATTORITENSORS}
\roughangxitwouparg{\#}{\#}
\eqdef \upxi(f_A,f_B) f_A \otimes f_B.
\end{align}

It is straightforward to check that the
tensorfield $\roughangxi$ defined by (<ref>)
is the $\gfour$-orthogonal projection of $\upxi$ onto
$\roughangxi = \roughtorusproject \upxi$,
where $\roughtorusproject \upxi$ is defined by (<ref>).

§.§ Identities involving symmetric type $\binom{0}{2}$ tensorfields

In the remainder of Sect. <ref>,
we will work with $\nullhyparg{u}$-tangent tensorfields, as defined in
Def. <ref>.
It is straightforward to check that $\upeta$ is
$\nullhyparg{u}$-tangent if and only if
any contraction of it with $\Lunit$ (which is $\gfour$-orthogonal to $\nullhyparg{u}$)

Let $\upxi$ be a symmetric type $\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfield,
and let $\roughangxi$ be the corresponding $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfield from
Def. <ref>.
Then the following identity holds:

\begin{align}
\begin{split} \label{E:ROUGHTORICOMPONENTSINTERMSOFSMOOTHTORITANGENTTENSORANDLERRORS}
\widetilde{\upxi}^{\# \#}
\upxi^{\# \#}
\COVL_A
\upxi(e_A,e_B) \Lunit
\otimes
\COVL_B
\upxi(e_A,e_B)
\otimes
\Lunit
\COVL_A
\COVL_B
\upxi(e_A,e_B)
\Lunit
\otimes
\Lunit.
\end{split}
\end{align}

Moreover, if $\upeta$ is any symmetric type $\binom{0}{2}$ $\nullhyparg{u}$-tangent tensorfield,
then the following identities hold,
where $\Dfour$ is the Levi-Civita connection of $\gfour$
and $\angLie_{\Lunit}$ is $\ell_{t,u}$-projected Lie derivative operator from
Def. <ref>:

\begin{align}
[\Dfour \upeta](\Lunit,\Lunit)
& = 0,
\label{E:LLCOMPONENTOFFULLCOVARIANTDERIVATIVEOFLNORMALTENSORFIELDVANISHES}
\\
[\Dfour_{\Lunit} \upeta](\Lunit,\cdot)
& = 0,
\label{E:LCOMPONENTOFLCOVARIANTDERIVATIVEOFLNORMALTENSORFIELDVANISHES}
\\
[\Dfour_{e_A} \upeta](\Lunit,e_B)
& =
\upeta(e_C,e_B) \upchi(e_A,e_C),
\label{E:LCOMPONENTOFSMOOTHTORUSCOMPONENTCOVARIANTDERIVATIVEOFLNORMALTENSORFIELDEXPRESSIBLEINTERMSOFNULLSECONDFUND}
\end{align}

\begin{align} \label{E:SMOOTHTORUSCOMPONENTOFLCOVARIANTDERIVATIVEOFLNORMALTENSORFIELDEXPRESSIBLEINTERMSOFNULLSECONDFUND}
[\Dfour_{\Lunit} \upeta](e_A,e_B)
& =
[\angLie_{\Lunit} \upeta](e_A,e_B)
\upeta[e_C,e_B] \upchi(e_A,e_C)
\upeta[e_A,e_C] \upchi(e_B,e_C).
\end{align}

The identity (<ref>)
follows from using (<ref>)
to substitute for the frame vectorfields $\lbrace f_A \rbrace_{A=2,3}$ in (<ref>)
and using that $\upxi(\Lunit,\cdot) = 0$.

To prove (<ref>),
we differentiate the identity $\upeta(\Lunit,\Lunit) = 0$ with $\Dfour$, use the Leibniz rule,
and use that $\upeta(\Lunit,\cdot) = 0$.

To prove (<ref>),
we differentiate the identity $\upeta(\Lunit,\cdot) = 0$ with $\Dfour_{\Lunit}$,
use the Leibniz rule and the identity (<ref>),
and use that $\upeta(\Lunit,\cdot) = 0$.

To prove (<ref>),
we differentiate the identity $\upeta(\Lunit,\cdot) = 0$ with $\Dfour_{e_A}$,
use the Leibniz rule and the identity (<ref>),
and use that $\upeta(\Lunit,\cdot) = 0$.

To prove (<ref>),
we contract the Lie differentiation identity
\Lie_{\Lunit} \upxi_{\alpha \beta}
\Dfour_{\Lunit} \upxi_{\alpha \beta}
\upxi_{\kappa \beta} \Dfour_{\alpha} \Lunit^{\kappa}
\upxi_{\alpha \kappa} \Dfour_{\beta} \Lunit^{\kappa}
against $e_A^{\alpha} e_B^{\beta}$,
use that $\upeta(\Lunit,\cdot) = 0$,
and use the identity $\Dfour_{e_A} \Lunit = \upchi(e_A,e_C) e_C - \upzeta_A \Lunit$,
which follows from
Lemma <ref>,

§.§ Identities connecting $\gtorus$, $\gtorusroughfirstfund$, and their corresponding differential operators

Let $\upeta$ be a $\nullhyparg{u}$-tangent tensorfield.
Recall that $|\cdot|_{\gtorus}$ and $|\cdot|_{\gtorusroughfirstfund}$
are defined in (<ref>) and
respectively. The following identity holds:

\begin{align} \label{E:SEMINNORMSOFPUTANGENTTENSORSINDEPENDENTOFMETRIC}
= |\upeta|_{\gtorusroughfirstfund}.
\end{align}

Moreover, if $\upxi$ and $\roughangxi$ are as in Def. <ref>, then:

\begin{align}
|\upxi|_{\gtorus} & = |\roughangxi|_{\gtorusroughfirstfund}.
\label{E:NORMSOFROUGHTORICOMPONENTSANDFLATTORIAREEQUAL}
\end{align}

In addition, if $\upeta$ is a symmetric type $\binom{0}{2}$ $\nullhyparg{u}$-tangent tensorfield, then
with $\mytr_{\gtorus}$ and $\mytr_{\gtorusroughfirstfund}$ as defined in
(<ref>) respectively, the following identity holds:

\begin{align} \label{E:TRACESOFPUTANGENTTENSORINDEPENDENTOFMETRIC}
\mytr_{\gtorus} \upeta
& =
\mytr_{\gtorusroughfirstfund} \upeta.
\end{align}

Finally, if $\upxi$ and $\roughangxi$ are as in Def. <ref>, then:

\begin{align} \label{E:TRACESOFROUGHTORICOMPONENTSANDFLATTORIAREEQUAL}
\mytr_{\gtorus} \upxi
& =
\mytr_{\gtorusroughfirstfund} \roughangxi.
\end{align}

The lemma follows easily from the fact that
any $\nullhyparg{u}$-tangent tensorfield
(including $\gtorus$
vanishes upon any contraction with $\Lunit$,
and the identities

§.§ Differential operator pointwise comparison estimates needed for the elliptic estimates

Let $\varphi$ be a scalar function,
let $\angrmD \varphi$ be the $\ell_{t,u}$-tangent one-form from Def. <ref>,
and let $\roughangrmD \varphi$ be the $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent one-form
from Def. <ref>.
Let $\upxi$ be a symmetric type $\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfield,
and let $\roughangxi$
be the corresponding symmetric type $\binom{0}{2}$ $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfield
from Def. <ref>.
Then the following pointwise estimates hold
on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$:

\begin{align}
|\roughangrmD \varphi|_{\gtorusroughfirstfund}
\left\lbrace
1 + \mathcal{O}(\fundbootsmall)
\right\rbrace
|\angrmD \varphi|_{\gtorus}
\mathcal{O}(\fundbootsmall)
|\Lunit \varphi|,
\label{E:ROUGHTORUSDIFFERENTIALAPPROXIMATEDBYSMOOTHTORUSDIFFERENTIAL}
\\
|\roughangdiv \roughangxi|_{\gtorusroughfirstfund}
& =
\left\lbrace
1 + \mathcal{O}(\fundbootsmall)
\right\rbrace
|\angdiv \upxi|_{\gtorus}
\mathcal{O}(\fundbootsmall)
|\angLie_{\Lunit} \upxi|_{\gtorus}
\mathcal{O}(\fundbootsmall)
\label{E:ROUGHTORUSDIVERGENCEAPPROXIMATEDBYSMOOTHTORUSDIVERGENCE}
\\
|\roughangD \roughangxi|_{\gtorusroughfirstfund}
& =
\left\lbrace
1 + \mathcal{O}(\fundbootsmall)
\right\rbrace
|\angD \upxi|_{\gtorus}
\mathcal{O}(\fundbootsmall)
|\angLie_{\Lunit} \upxi|_{\gtorus}
\mathcal{O}(\fundbootsmall)
\label{E:ROUGHTORUSCOVARIANTDERIVATIVEAPPROXIMATEDBYSMOOTHTORUSCOVARIANTDERIVATIVE}
\end{align}

We prove only (<ref>)
since (<ref>)
and (<ref>)
can be proved by similar arguments.

We start by noting the following identities,
where $\COVframe_{AB}$ is the orthogonal matrix from Lemma <ref>:

\begin{align} \label{E:ANGDIVSMOOTHTORITANGENTTENSORNORMSQUAREDINTERMSOFSMOOTHORTHONORMALFRAME}
|\angdiv \upxi|_{\gtorus}^2
\\
|\roughangdiv \roughangxi|_{\gtorusroughfirstfund}^2
[\Dfour_{f_A} \roughangxi](f_A,f_C)
[\Dfour_{f_B} \roughangxi](f_B,f_C)
\left\lbrace
\COVframe_{CD}
[\Dfour_{f_A} \roughangxi](f_A,f_D)
\right\rbrace
\left\lbrace
\COVframe_{CE}
[\Dfour_{f_B} \roughangxi](f_B,f_E)
\right\rbrace.
\label{E:ANGDIVROUGHTORITANGENTTENSORNORMSQUAREDINTERMSOFROUGHORTHONORMALFRAME}
\end{align}

The identity (<ref>)
follows from (<ref>)
and the fact that $\angD \upxi = \smoothtorusproject \Dfour \upxi$,
where $\smoothtorusproject$ is the $\gfour$-orthogonal projection onto $\ell_{t,u}$
from Def. <ref>.
Similarly, the first equality in (<ref>)
follows from (<ref>)
and the fact that  $\roughangD \roughangxi = \roughtorusproject \Dfour \roughangxi$,
where $\roughtorusproject$ is the $\gfour$-orthogonal projection onto $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$
from Def. <ref>.
The second equality in (<ref>)
follows from the orthogonality of $\COVframe$.

We now note the following identity, which we derive just below:

\begin{align}
\begin{split} \label{E:FDERIVATIVESOFROUGHTORITANGENTTENSORINTERMSOFEDERIVATIVESOFSMOOTHTORITANGENTTENSOR}
\COVframe_{CD}
[\Dfour_{f_A} \roughangxi](f_A,f_D)
[\Dfour_{e_A} \upxi](e_A,e_C)
\COVL_A
[\angLie_{\Lunit} \upxi](e_A,e_C)
\\
\	\
\COVL_D
\upxi(e_D,e_C)
\mytr_{\gtorus} \upchi
\COVL_C
\upxi(e_A,e_D)\upchi(e_A,e_D)
\COVL_A
\upxi(e_C,e_D)\upchi(e_A,e_D).
\end{split}
\end{align}

From (<ref>),
the identities
the orthogonality of the matrix $\COVframe$,
the Cauchy–Schwarz inequality with respect to $\gtorus$,
Lemma <ref>,
the estimate (<ref>),
and the pointwise estimate $|\upchi|_{\gtorus} \lesssim \fundbootsmall$
noted below (<ref>),
we conclude (<ref>).

It remains for us to prove (<ref>).
To proceed, we first use
the orthogonality of the matrix $\COVframe_{AB}$,
the fact that $\Lunit$ is null and $\gfour$-orthogonal to $\nullhyparg{u}$,
the fact that $\roughangxi$ and $\upxi$ are symmetric and satisfy
$\roughangxi(\Lunit,\cdot) = \upxi(\Lunit,\cdot) = 0$,
and the fact that $\angLie_{\Lunit} \Lunit = 0$
to compute that:

\begin{align}
\begin{split} \label{E:FIRSTPROOFSTEPROUGHTORUSDIVERGENCEAPPROXIMATEDBYSMOOTHTORUSDIVERGENCE}
[\Dfour_{e_A} \roughangxi](e_A,e_C)
& =
\COVframe_{CD}
[\Dfour_{f_A} \roughangxi](f_A,f_D)
\\
& \ \
\COVL_C
[\Dfour_{e_A} \roughangxi](e_A,\Lunit)
\COVL_A
[\Dfour_{e_A} \roughangxi](\Lunit,e_C)
\COVL_C
\COVL_A
[\Dfour_{e_A} \roughangxi](\Lunit,\Lunit)
\\
& \ \
\COVL_A
[\Dfour_{\Lunit} \roughangxi](e_A,e_C)
\COVL_A
\COVL_A
[\Dfour_{\Lunit} \roughangxi](\Lunit,e_C)
\COVL_C
\COVL_A
[\Dfour_{\Lunit} \roughangxi](e_A,\Lunit)
\COVL_C
\COVL_A
\COVL_A
[\Dfour_{\Lunit} \roughangxi](\Lunit,\Lunit)
\\
& =
\COVframe_{CD}
[\Dfour_{f_A} \roughangxi](f_A,f_D)
\COVL_A
[\angLie_{\Lunit} \upxi](e_A,e_C)
\\
& \ \
\COVL_C
\upxi(e_A,e_D) \upchi(e_A,e_D)
\COVL_A
\upxi(e_C,e_D)\upchi(e_A,e_D)
\COVL_A
\upxi(e_A,e_D)\upchi(e_C,e_D).
\end{split}
\end{align}

Next, using
the fact that $\Lunit$ is $\gfour$-orthogonal to $\nullhyparg{u}$,
the fact that $\roughangxi(\Lunit,\cdot) = 0$,
the identity $\gfour(\Dfour_{e_A}\Lunit,e_B) = \upchi(e_A,e_B)$
(which follows from (<ref>)),
and the identity $\upchi(e_A,e_A) = \mytr_{\gtorus} \upchi$,
we deduce:

\begin{align} \label{E:SECONDPROOFSTEPROUGHTORUSDIVERGENCEAPPROXIMATEDBYSMOOTHTORUSDIVERGENCE}
[\Dfour_{e_A} \roughangxi](e_A,e_C)
& =
[\Dfour_{e_A} \upxi](e_A,e_C)
\COVL_D
\upxi(e_D,e_C)
\mytr_{\gtorus} \upchi
\COVL_D
\upxi(e_A,e_D)
\upchi(e_A,e_C).
\end{align}

Using (<ref>)
to substitute for LHS (<ref>)
and rearranging terms,
we conclude (<ref>).

§.§ Standard elliptic estimates for symmetric type $\binom{0}{2}$ tensorfields on the rough tori

In the next lemma, we provide standard elliptic estimates for
symmetric type $\binom{0}{2}$ $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfields.
Its proof is located in Sect. <ref>.

Let $\Upxi$ be a symmetric type $\binom{0}{2}$ $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfield.
Then the following estimate
holds for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu]$:

\begin{align} \label{E:STANDARDELLIPTICONROUGHTORI}
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\upmu^2 |\roughangD \Upxi|_{\gtorusroughfirstfund}^2
\, \volroughtorus
& \leq
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\upmu^2 |\roughangdiv \Upxi|_{\gtorusroughfirstfund}^2
\, \volroughtorus
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\upmu^2 |\roughangD \mytr_{\gtorusroughfirstfund} \Upxi|_{\gtorusroughfirstfund}^2
\, \volroughtorus
\fundbootsmall
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\, \volroughtorus.
\end{align}

§.§.§ The Gauss curvature of $\gtorusroughfirstfund$

Our proof of Lemma <ref> relies on the next lemma,
which provides $L^{\infty}$ estimates for the Gauss curvature
$\Gausstorus$ of $(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue},\gtorusroughfirstfund)$.

Recall that $\Gausstorus$ denotes the Gauss curvature of $(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue},\gtorusroughfirstfund)$
(see Sect. <ref>).
Then the following estimate holds for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot] \times [-\rightu,\leftu]$:

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

Recall that $\Gausstorus$ is equal to half the scalar curvature of $\gtorusroughfirstfund$
(see (<ref>)).
Hence, at fixed $(\timefunction,u)$, relative to the coordinates $(x^2,x^3)$ on the rough tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$,
using the standard expression for curvature in terms of the
coordinate components of $\gtorusroughfirstfund$ and their partial derivatives,