has a favorable sign.

Let $f$ be a scalar function,
and for
$Z \in
\lbrace \argLrough{\muxmulevelsetvalue}, \Rtransarg{\muxmulevelsetvalue}
\rbrace$, let $\mytr_{\gtorusroughfirstfund} \deform{Z}$ be the $\gtorusroughfirstfund$-trace
(see Def. <ref>)
of the deformation tensor $\deform{Z}$ of $Z$.
Then the following integral identities hold:

\begin{align}
\roughgeop{\timefunction}
\left(
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\, \volroughtorus
\right)
& =
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\left\lbrace
\argLrough{\muxmulevelsetvalue} f
\frac{1}{2} f \mytr_{\gtorusroughfirstfund} \deform{\argLrough{\muxmulevelsetvalue}}
\right\rbrace \, \volroughtorus,
\label{E:IDENTITYROUGHTIMEDERIVATIVEOFROUGHTORUSINTEGRAL}
\\
\roughgeop{u} \left(  \int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}} f \, \volroughtorus\right) & = \int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\left\lbrace
\Rtransarg{\muxmulevelsetvalue} f
\frac{1}{2} f \mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
\right\rbrace \, \volroughtorus.
\label{E:IDENTITYROUGHEIKONALDERIVATIVEOFROUGHTORUSINTEGRAL}
\end{align}

Moreover, for $u_1 \leq u_2$, we have:

\begin{align} \label{E:RTRANSINTEGRALONROUGHTORI}
\int_{\hypthreearg{\timefunction}{[u_1,u_2]}{\muxmulevelsetvalue}}
\left\lbrace
\Rtransarg{\muxmulevelsetvalue} f
\frac{1}{2} f \mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
\right\rbrace
\, \volRoughHypersurface
& =
\int_{\twoargroughtori{\timefunction,u_2}{\muxmulevelsetvalue}}
\, \volroughtorus
\int_{\twoargroughtori{\timefunction,u_1}{\muxmulevelsetvalue}}
\, \volroughtorus.
\end{align}

We prove only
(<ref>) and
(<ref>) because
was proved in <cit.>[Note that since $\argLrough{\muxmulevelsetvalue}$ is $\gfour$-orthogonal to
$\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$ and $\argLrough{\muxmulevelsetvalue} \timefunction = 1$, the vectorfield $\argLrough{\muxmulevelsetvalue}$ agrees with the vectorfield denoted by “$\breve{\underline{H}}$” in <cit.>.]
Let $\Phi_{(u')} = \Phi_{(u')}(\timefunction,u,x^2,x^3)$
be the flow map of $\Rtransarg{\muxmulevelsetvalue}$, normalized by $\Phi_{(0)}(\timefunction,u,x^2,x^3) = (\timefunction,u,x^2,x^3)$.
Since $\Rtransarg{\muxmulevelsetvalue} \timefunction = 0$ and $\Rtransarg{\muxmulevelsetvalue} u = 1$,
it follows that if $\timefunction \in [\timefunction_0,\timefunctionboot)$,
$u \in (- \rightu,\leftu)$, and $|u'|$ is sufficiently small (depending on $u$),
then $\Phi_{(u')}$ is a diffeomorphism from the rough torus
$\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$ onto the rough torus
$\twoargroughtori{\timefunction,u + u'}{\muxmulevelsetvalue}
\subset
\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
Hence, in view of (<ref>),
and the standard formula for change of variables in an integral,
we deduce the following identity,
where $\Phi_{(u')}^*$ denotes pullback by $\Phi_{(u')}$
(in particular, $\Phi_{(u')}^* f = f \circ \Phi_{(u')}$),
and throughout this proof, determinants are taken relative to the coordinates $(x^2,x^3)$ on the rough tori:

\begin{align} \label{E:RTRANSDIFFINTINTEGRALONROUGHTORIINTERMEDIATE}
\int_{\twoargroughtori{\timefunction,u + u'}{\muxmulevelsetvalue}}
\, \volroughtorus
& =
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
[\Phi_{(u')}^* f]
\volroughtorusarg{\Phi_{(u')}^* \gtorusroughfirstfund}
\int_{\mathbb{T}^2}
f \circ \Phi_{(u')}(\timefunction,u,x^2,x^3)
\sqrt{\mbox{\upshape det} [\Phi_{(u')}^* \gtorusroughfirstfund}]
\, \mathrm{d} x^2 \mathrm{d} x^3.
\end{align}

Next, using that $\Rtransarg{\muxmulevelsetvalue}$ is the infinitesimal generator of
the flow map $\Phi_{(u')}$,
and using (<ref>),
we note the following differentiation identities:
$\frac{d}{du'}|_{u' = 0} \Phi_{(u')}^* f = \Rtransarg{\muxmulevelsetvalue} f$
\frac{d}{du'}|_{u' = 0} \sqrt{\mbox{\upshape det} [\Phi_{(u')}^* \gtorusroughfirstfund}]
\frac{1}{2} \sqrt{\mbox{\upshape det} \gtorusroughfirstfund}
\,
\mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
Using these identities,
we differentiate (<ref>) under the integral on the RHS
to obtain:

\begin{align} \label{E:RTRANSDIFFINTINTEGRALONROUGHTORISECONDINTERMEDIATE}
\roughgeop{u}
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\, \volroughtorus
\left. \frac{d}{d u'}\right|_{u' = 0}
\int_{\twoargroughtori{\timefunction,u + u'}{\muxmulevelsetvalue}}
\, \volroughtorus
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\left\lbrace
\Rtransarg{\muxmulevelsetvalue} f
\frac{1}{2} f \mytr_{\gtorusroughfirstfund} \deform{\Rtransarg{\muxmulevelsetvalue}}
\right\rbrace
\, \volroughtorus.
\end{align}

We have therefore proved (<ref>).

(<ref>) then follows from integrating (<ref>)
with respect to $u$ and using the fundamental theorem of calculus,
and (<ref>).

§.§ Sobolev embedding and fundamental theorem of calculus-type estimates on the rough tori

In this section,
on the rough tori $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$,
we derive $L^{\infty}$ Sobolev embedding estimates as well as simple $L^2$ estimates
that rely on the fundamental theorem of calculus.

Let $f$ be a scalar function on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
Then the following estimates hold for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$:

\begin{align}
\| \tander^N f \|_{L^2\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}^2
\lesssim
\| \tander^N f  \|_{L^2\left(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}\right)}^2
\int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction]}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} |\Lunit \tander^N f|^2
\, \volPuRoughCoordinates,
\label{E:ROUGHTORUSINNULLHYPERSURFACEL2FUNDAMENTALTHEOREMOFCALCULUSESTIMATE}
\end{align}

\begin{align}
\| f \|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
& \lesssim
\| \tander^{\leq 2} f \|_{L^2\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)},
\label{E:H2LINFINITYSOBOLEVEMBEDDINGROUGHTORUS}
\\
\| f\|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}^2
& \lesssim
\|
\tander^{\leq 2} f
\|_{L^2\left(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue}\right)}^2
\int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction]}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} |\Lunit \tander^{\leq 2} f|^2 \, 		\volPuRoughCoordinates.
\label{E:H2LINFINITYFUNDAMENTALTHEOREMOFCALCULUSPLUSSOBOLEVEMBEDDINGONROUGHTORUS}
\end{align}

Using (<ref>) with $f^2$ in place of $f$,
the pointwise estimate (<ref>),
and Young's inequality, we deduce
\left|
\roughgeop{\timefunction}
\| \tander^N f \|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
\right|
\leq
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \Lunit \tander^N f
\right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
C \| \tander^N f\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
Integrating this inequality with respect to $\timefunction$,
applying Grönwall's inequality,
and also using the identity
$\int_{\timefunction' = \timefunction_0}^{\timefunction}
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \tander^Nf
\right\|_{L^2(\twoargroughtori{\timefunction',u}{\muxmulevelsetvalue})}^2
\, \mathrm{d}\timefunction'
= \left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \Lunit \tander^N f
\right \|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction]})}^2
we conclude the inequality (<ref>).

We now prove
We begin with the following estimate, which holds at fixed $(\timefunction,u)$ by virtue of
the standard Sobolev embedding result $H^2(\T^2) \hookrightarrow L^{\infty}(\T^2)$:

\begin{align} \label{E:STANDARDH2LINFINITYSOBOLEVEMBEDDINGONT2}
\| f \|_{L^{\infty}\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
& \lesssim
\sum_{I + J \leq 2}
\left\lbrace
\int_{\T^2}
\left|
\left(\roughgeop{x^2}\right)^I \left(\roughgeop{x^3}\right)^J
\right|^2
\,
\mathrm{d} x^2 \mathrm{d} x^3
\right\rbrace^{1/2}.
\end{align}

Using (<ref>),
Lemma <ref> to estimate derivatives of
$\timefunction$ in (<ref>),
the identities (<ref>)–(<ref>),
Lemma <ref>,
the bootstrap assumptions,
and the area form element comparison estimate
we deduce, in view of definitions (<ref>) and (<ref>),
that $\mbox{RHS~\eqref{E:STANDARDH2LINFINITYSOBOLEVEMBEDDINGONT2}}
\lesssim
\| \tander^{\leq 2} f \|_{L^2\left(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}\right)}
We have therefore proved (<ref>).
The estimate
(<ref>) then
then follows from
(<ref>) with $N = 0,1,2$.

§.§ Integration by parts identities

In this section, we establish several integration by parts identities that
we will exploit in our top-order energy estimates.
We start with the following lemma,
which provides a useful expression for the covariant divergence of a spacetime vectorfield.

Let $\mathscr{J}$ be a spacetime vectorfield.
Consider the decomposition $\upmu \mathscr{J}
- \upmu \mathscr{J}_{\Lunit} \Lunit
\mathscr{J}_{\muX} \Lunit
\mathscr{J}_{\Lunit} \muX
\upmu \smoothtorusproject \mathscr{J}$ of $\upmu \mathscr{J}$
afforded by Lemma <ref> ,
where $\mathscr{J}_{\Lunit} = \gfour(\mathscr{J},\Lunit)$, $\mathscr{J}_{\muX} = \gfour(\mathscr{J},\muX)$,
and $\smoothtorusproject \mathscr{J}$ is the $\ell_{t,u}$-projection of $\mathscr{J}$ (see Def. <ref>).
Then the following identity holds:

\begin{align} \label{E:COVARIANTDIVERGENCEOFSPACETIMEVECTORFIELDINTERMSOFRESCALEDFRAME}
\upmu \Dfour_\alpha \mathscr{J}^{\alpha}= - L(\upmu \mathscr{J}_{\Lunit}) - L(\mathscr{J}_{\muX}) - \muX (\mathscr{J}_{\Lunit}) + \angdiv(\upmu \smoothtorusproject \mathscr{J}) - \upmu \mytr_{\gtorus} \angk \mathscr{J}_{\Lunit} - \mytr_{\gtorus} \upchi \mathscr{J}_{\muX}.
\end{align}

The same proof of <cit.> holds with minor modifications to account for the third space dimension.

The following lemma provides some preliminary integration by parts identities.

Let $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
and let $\upupsilon$ and $\upzeta$ be scalar functions on $\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}$.
Then the following integration by parts identities hold,
where $A=2,3$ in (<ref>):

\begin{align}
\begin{split} \label{E:PRELIMINARYIBPINLROUGH}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
(\argLrough{\muxmulevelsetvalue} \upupsilon)
\upzeta
\, \volMRoughCoordinates
& =
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\upupsilon
(\argLrough{\muxmulevelsetvalue} \upzeta)
\, \volMRoughCoordinates
\\
& \ \
\frac{1}{2}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\mytr_{\gtorusroughfirstfund} \deform{\argLrough{\muxmulevelsetvalue}}
\upupsilon \upzeta
\, \volMRoughCoordinates
\\
& \ \
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\upupsilon \upzeta
\, \volRoughHypersurface
\int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}}
\upupsilon \upzeta
\, \volRoughHypersurface,
\end{split}
\\
\begin{split}	 \label{E:PRELIMINARYIBPINY}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \upupsilon)
\upzeta
\, \volMRoughCoordinates
& =
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\upupsilon ( \Yvf{A} \upzeta )
\, \volMRoughCoordinates
\\
& \ \
\frac{1}{2}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\mytr_{\gtorus} \, \angdeform{\Yvf{A}}
\upupsilon \upzeta
\, \volMRoughCoordinates
\\
& \ \
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \timefunction)
\upupsilon \upzeta
\, \volRoughHypersurface
\int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \timefunction_0)
\upupsilon \upzeta
\, \volRoughHypersurface.
\end{split}
\end{align}

The identity (<ref>) follows from integrating (<ref>)
first with respect to $u$ and then with respect to $\timefunction$ with $f \eqdef \upupsilon \upzeta$.

We now prove (<ref>).
We first expand $\Yvf{A}$ in terms of the rough adapted coordinate vectorfields as follows:

\begin{align} \label{E:YCOMMMUTATOREXPANDEDINROUGHADAPATEDCOORDIANTES}
\Yvf{A} = (\Yvf{A} \timefunction) \roughgeop{\timefunction} + \Yvf{A}^B \roughgeop{x^B}.
\end{align}

Next, using (<ref>)
and the fact that relative to the rough adapted coordinates, we have
$\sqrt{|\mbox{\upshape det} \gfour|}
\frac{\upmu}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \sqrt{\mbox{\upshape det} \gtorusroughfirstfund}$
(see (<ref>)),
we can expand the product of the covariant divergence of the vectorfield
$\frac{1}{\upmu} \upupsilon \upzeta \Yvf{A}$ and $\sqrt{|\mbox{\upshape det} \gfour|}$ as follows
(again, relative to the rough adapted coordinates):

\begin{align} \label{E:IBPINYINTERMEDIATESTEP}
\sqrt{|\mbox{\upshape det} \gfour|}
\Dfour_\alpha \left( \frac{1}{\upmu} \upupsilon \upzeta \Yvf{A}\right)^{\alpha}
& =
\roughgeop{\timefunction}
\left(
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \timefunction)
\upupsilon
\upzeta \sqrt{\mbox{\upshape det} \gtorusroughfirstfund} \right)
\roughgeop{x^B}
\left( \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Yvf{A}^B \upupsilon \upzeta \sqrt{\mbox{\upshape det} \gtorusroughfirstfund} \right).
\end{align}

Through a straightforward Leibniz rule-based expansion, we also have:

\begin{align} \label{E:IBPINYSECONDINTERMEDIATESTEP}
\sqrt{|\mbox{\upshape det} \gfour|}
\Dfour_\alpha \left( \frac{1}{\upmu} \upupsilon \upzeta \Yvf{A}\right)^{\alpha}
\left\lbrace
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \upupsilon) \upzeta
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\upupsilon
(\Yvf{A} \upzeta)
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\frac{\Yvf{A}\upmu}{\upmu} \upupsilon \upzeta
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \upupsilon \upzeta
\left(\Dfour_\alpha \Yvf{A}^{\alpha}\right)
\right\rbrace
\sqrt{\mbox{\upshape det} \gtorusroughfirstfund}.
\end{align}

Next we use (<ref>), the fact that
$\Yvf{A}$ is $\ell_{t,u}$-tangent, and Lemma <ref>
to express the term $\Dfour_\alpha \Yvf{A}^{\alpha}$ on RHS (<ref>)
as follows:

\begin{align} \label{E:SPACETIMECOVARIANTDIVERGENCEOFYCOMMUTATOR}
\Dfour_\alpha \Yvf{A}^{\alpha}
\frac{1}{\upmu} \Yvf{A} \upmu
\angdiv \Yvf{A}
\frac{1}{\upmu} \Yvf{A} \upmu
\frac{1}{2} \mytr_{\gtorus} \angdeform{\Yvf{A}}.
\end{align}

Using (<ref>) to substitute for the
factor $\Dfour_\alpha \Yvf{A}^{\alpha}$ on RHS (<ref>)
and noting that the factor $\frac{1}{\upmu} \Yvf{A} \upmu$ on RHS (<ref>)
cancels the singular $\frac{\Yvf{A}\upmu}{\upmu} \upupsilon \upzeta$ term in (<ref>),
we deduce that:

\begin{align} \label{E:IBPINYTHIRDINTERMEDIATESTEP}
\sqrt{|\mbox{\upshape det} \gfour|}
\Dfour_\alpha \left( \frac{1}{\upmu} \upupsilon \upzeta \Yvf{A}\right)^{\alpha}
\left\lbrace
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \upupsilon) \upzeta
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \upupsilon
(\Yvf{A} \upzeta)
\frac{1}{2}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \upupsilon \upzeta
\mytr_{\gtorus} \angdeform{\Yvf{A}}
\right\rbrace
\sqrt{\mbox{\upshape det} \gtorusroughfirstfund}.
\end{align}

Next we integrate RHS (<ref>)
over $[\timefunction_0,\timefunction) \times [- \rightu,u] \times \mathbb{T}^2$
with respect to $\mathrm{d} x^2 \, \mathrm{d}x^3 \mathrm{d} u' \mathrm{d} \timefunction'$,
equate it to the integral of RHS (<ref>),
and use Fubini's theorem. In view of definitions (<ref>) and (<ref>),
we see that the spacetime integrals of the terms on RHS (<ref>)
appear in (<ref>).
The integral of the last term on RHS (<ref>) vanishes because $\mathbb{T}^2$
is a closed manifold. Finally, we note that the integral of the first term on RHS (<ref>)
yields, in view of (<ref>) and the fundamental theorem of calculus,
the two hypersurface integrals in (<ref>).

We are now ready to establish the
integration by parts identity
that we will use to control top-order wave equation error terms
that are tied to the partially modified quantities
defined in (<ref>).

Let $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
and let $\varphi$ and $\upeta$ be scalar functions on $\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}$.
Let $N \geq 1$.
Then the following integration by parts identity holds:

\begin{align}
\begin{split} \label{E:KEYIBPIDENTIFYFORWAVEEQUATIONENERGYESTIMATES}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
(1 + 2 \upmu)
(\muX \varphi)
(\argLrough{\muxmulevelsetvalue} \tander^N \varphi)
\Yvf{A} \upeta
\, \volMRoughCoordinates
\\
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
(1 + 2 \upmu) (\muX \varphi)
(\Yvf{A} \tander^N \varphi)
\argLrough{\muxmulevelsetvalue} \upeta
\, \volMRoughCoordinates
\\
& \ \
- \int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue} }
(1 + 2 \upmu) \muX \varphi (\Yvf{A}\tander^N \varphi)\upeta
\, \volRoughHypersurface
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
(\Yvf{A} \timefunction)
(1 + 2 \upmu)
(\muX \varphi)
(\argLrough{\muxmulevelsetvalue} \tander^N \varphi)
\upeta
\, \volRoughHypersurface
\\
&  \ \
\int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}}
(1 + 2 \upmu) \muX \varphi (\Yvf{A}\tander^N \varphi) \upeta
\, \volRoughHypersurface
\int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}}
(\Yvf{A} \timefunction)
(1 + 2 \upmu)
(\muX \varphi)
(\argLrough{\muxmulevelsetvalue} \tander^N \varphi) \upeta
\, \volRoughHypersurface
\\
& \ \
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\ErrorIBP[\tander^N\varphi;\upeta]
\, \volMRoughCoordinates,
\end{split}
\end{align}

\begin{align}
\begin{split} \label{E:ERRORTERM1KEYIBPIDENTIFYFORWAVEEQUATIONENERGYESTIMATES}
\ErrorIBP[\tander^N \varphi;\upeta](\timefunction,u,x^2,x^3)
\eqdef
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(1 + 2 \upmu)
(\muX \varphi)
([\Lunit, \Yvf{A}] \tander^N \varphi)
\upeta
\\
& \ \
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Lunit \upmu)
(\muX \varphi)
(\Yvf{A} \tander^N \varphi)
\upeta
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(1 + 2 \upmu)
(\Lunit \muX \varphi)
(\Yvf{A} \tander^N \varphi)
\upeta
\\
& \ \
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Yvf{A} \upmu)
(\muX \varphi)
(\Lunit \tander^N \varphi)
\upeta
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(1 + 2 \upmu)
(\Yvf{A} \muX \varphi)
(\Lunit \tander^N \varphi)
\upeta.
\end{split}
\end{align}

Moreover, the following integration by parts identity holds, where $T = L + 2 \upmu \Transport = (1+2 \upmu) \Lunit + 2\muX$:

\begin{align}
& \int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (\multipliervectorfield \tander^N \varphi ) \Yvf{A} \upeta \, \volMRoughCoordinates \notag \\
& =  \int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (\Yvf{A} \tander^N \varphi ) \left( (1+2 \upmu)\Lunit \upeta + 2\muX \upeta\right) \, \volMRoughCoordinates \notag \\
& - \int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue} } (1 + 2 \upmu) (\Yvf{A}\tander^N \varphi)\upeta \, \volRoughHypersurface + \int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}} \Yvf{A} \timefunction(1 + 2 \upmu) (\argLrough{\muxmulevelsetvalue} \tander^N \varphi) \upeta\, \volRoughHypersurface \notag \\
&  + \int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}}  (1 + 2 \upmu)  (\Yvf{A}\tander^N \varphi)\upeta \, \volRoughHypersurface \notag - \int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}} \Yvf{A} \timefunction(1 + 2 \upmu)  (\argLrough{\muxmulevelsetvalue} \tander^N \varphi) \upeta\, \volRoughHypersurface \notag \\
& - 2 \int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}} \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu} ( \Yvf{A} \tander^N \varphi ) \upeta \, \volRoughHypersurface + 2 \int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}} \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu} (\Yvf{A} \tander^N \varphi) \upeta \, \volRoughHypersurface \notag \\
& \ \ + 2 \int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (\Yvf{A} \tander^N \varphi) \upeta \, \volPuRoughCoordinates - 2 \int_{\nullhypthreearg{\muxmulevelsetvalue}{- \rightu}{\timefunction}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (\Yvf{A} \tander^N \varphi) \upeta \, \volPuRoughCoordinates \notag \\
& \ \ + \int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}} \ErrorIBP^T_1[\tander^N\varphi;\upeta] \, \volMRoughCoordinates + \int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}} \ErrorIBP^T_2[\tander^N \varphi;\upeta] \, \volRoughHypersurface \notag \\
& - \int_{\hypthreearg{\timefunction_0}{u}{\muxmulevelsetvalue}} \ErrorIBP^T_2[\tander^N \varphi;\upeta] \, \volRoughHypersurface, \label{eq:IBPidentitymodifiedfluidquant}
\end{align}

\begin{align*}
& \ErrorIBP_1^T[\tander^N\varphi;\upeta](\timefunction,u,x^2,x^3) \\
&  \eqdef \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} (1 + 2 \upmu)  [ \Lunit,\Yvf{A}] \tander^N \varphi \upeta + 2 (\argLrough{\muxmulevelsetvalue} \upmu)  \Yvf{A}\tander^N \varphi \upeta   + \frac{1}{2} (1 + 2 \upmu)  \Yvf{A} \tander^N \varphi \upeta (\gtorusroughinversefirstfund)^{AB} \, \deformarg{\argLrough{\muxmulevelsetvalue}}{\widetilde A \,}{\widetilde{B}} \\
& \ \ + 2 \argLrough{\muxmulevelsetvalue} \Yvf{A} \upmu \tander^N \varphi \upeta + 2 \Yvf{A} \upmu  \tander^N \varphi \argLrough{\muxmulevelsetvalue} \upeta + \argLrough{\muxmulevelsetvalue} \upmu \tander^N \varphi \upeta \mytr_{\gtorus} \angdeform{\Yvf{A}} +  \frac{1}{2} (1+2 \upmu)   \tander^N \varphi( \argLrough{\muxmulevelsetvalue} \upeta)  \mytr_{\gtorus} \angdeform{\Yvf{A}} \\
& \ \ + \frac{1}{2} (1 + 2 \upmu)  \tander^N \varphi \upeta \argLrough{\muxmulevelsetvalue} \mytr_{\gtorus} \angdeform{\Yvf{A}} + \frac{1}{4} (1 + 2 \upmu) \tander^N \varphi \upeta  \mytr_{\gtorus} \angdeform{\Yvf{A}} (\gtorusroughinversefirstfund)^{AB} \, \deformarg{\argLrough{\muxmulevelsetvalue}}{\widetilde A \,}{\widetilde{B}} \\
& - 2 \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Yvf{A} \tander^N \varphi \upeta \upmu \mytr_{\gtorus} \upchi + 2\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Yvf{A} \tander^N \varphi \upeta \upmu \mytr_{\gtorus} \angk  \\
& \ \ + 2 \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} [\muX, \Yvf{A} ] \tander^N \varphi \upeta  - \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \muX \tander^N \varphi \upeta \mytr_{\gtorus} \angdeform{\Yvf{A}},
\end{align*}

\begin{align*}
& \ErrorIBP_2^T[\tander^N\varphi;\upeta](\timefunction,u,x^2,x^3)
\\
& \eqdef  - 2 \Yvf{A}\upmu \tander^N \varphi \upeta  - \frac{1}{2}(1 + 2 \upmu) \tander^N \varphi \upeta \mytr_{\gtorus} \angdeform{\Yvf{A}} + \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Yvf{A} \timefunctionarg{\muxmulevelsetvalue} \muX \tander^N \varphi \upeta.
\end{align*}

We now summarize the tedious but straightforward calculations that yield (<ref>).
we use definition (<ref>)
and the integration by parts identity (<ref>)
with $\upupsilon \eqdef \upeta$ and
$\upzeta \eqdef (1 + 2 \upmu)
(\muX \varphi)
(\Lunit \tander^N \varphi)$
to remove the $\Yvf{A}$ operator from $\upeta$ on LHS (<ref>).
This produces the main integral
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(1+2 \upmu) (\muX \varphi)
(\Yvf{A} \Lunit \tander^N \varphi)
\upeta
\, \volMRoughCoordinates$
as well as many error integrals.
We then commute $\Yvf{A}$ and $\Lunit$ and appeal to definition (<ref>)
to rewrite the main integral as follows:

\begin{align}
\begin{split} \label{E:PROOFSTEPKEYIBPIDENTIFYFORWAVEEQUATIONENERGYESTIMATES}
- \int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
(1+2 \upmu)
(\muX \varphi)
(\argLrough{\muxmulevelsetvalue} \Yvf{A} \tander^N \varphi)
\upeta
\, \volMRoughCoordinates
\\
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(1+2 \upmu)
(\muX \varphi)
([\Lunit,\Yvf{A}] \tander^N \varphi)
\upeta
\, \volMRoughCoordinates.
\end{split}
\end{align}

Finally, we use the integration by parts identity (<ref>)
$\upupsilon \eqdef \Yvf{A} \tander^N \varphi$
$\upzeta \eqdef
(1+2 \upmu)
(\muX \varphi)
\upeta
to remove the factor of $\argLrough{\muxmulevelsetvalue}$ from the factor $\argLrough{\muxmulevelsetvalue} \Yvf{A}\tander^N \varphi$
in the first integral in (<ref>).

§.§ Fundamental energy identity

In this section,
we derive the fundamental energy–null-flux integral identities used to derive estimates.

§.§.§ Energy-momentum tensor, energy currents, and the multiplier vectorfield for the wave variables

To derive energy identities for the wave variables $\wavearray$, which solve the quasilinear wave equations
we rely on the well-known multiplier method, which we now introduce.

Let $f$ be a scalar function (in our applications, $f$ will be some vectorfield derivative of one of the wave variables).
We define the energy-momentum tensor associated to $f$ to be the following symmetric type $\binom{0}{2}$ tensorfield,
where we recall that $\Dfour$ is the Levi-Civita connection of the acoustical metric $\gfour$:

\begin{align} \label{E:DEFOFENERGYMOMENTUM}
\enmomem_{\alpha \beta}
&	=
\enmomem_{\alpha \beta}[f]
\eqdef
(\Dfour_\alpha f)
\Dfour_\beta f
\frac{1}{2}
\gfour_{\alpha \beta}
(\gfour^{-1})^{\kappa \lambda}
(\Dfour_{\kappa} f)
\Dfour_{\lambda} f.
\end{align}

Given any scalar function $f$ and any multiplier vectorfield $Z$,
we define the corresponding energy current vectorfield as follows:

\begin{align}  \label{E:WAVEEQUATIONENERGYCURRENT}
\Jenarg{Z}{\alpha}[f]
\eqdef \enmomem^{\alpha \beta}[f] Z_{\beta}.
\end{align}

Recall that the deformation tensor of $Z$ is the following symmetric type $\binom{0}{2}$ tensorfield:

\begin{align} \label{E:AGAINDEFORMATIONTENSORDEF}
\deformarg{Z}{\alpha}{\beta}
& \eqdef \Dfour_\alpha Z_\beta + \Dfour_\beta Z_\alpha.
\end{align}

The starting point for our derivation of our $L^2$-type integral identities
for solutions of covariant wave equations is the following well-known identity,
which follows easily from the definitions:

\begin{align} \label{E:DIVERGENCEOFWAVEENERGYCURRENT}
\Dfour_\alpha \Jenarg{Z}{\alpha}[f]
& =
(\square_{\gfour(\wavearray)} f) Z f
\frac{1}{2} \enmomem^{\alpha \beta} \deformarg{Z}{\alpha}{\beta}.
\end{align}

In order to obtain wave equation energy estimates that are sufficient to allow us
to track the solution up to the singular boundary,
we use the multiplier vectorfield from the next definition.

We define $\multipliervectorfield$ to be the following vectorfield:

\begin{align} \label{E:MULTIPLIERVECTORFIELD}
\multipliervectorfield
& \eqdef (1 + 2 \upmu) \Lunit
2 \muX.
\end{align}

Simple calculations based on Lemma <ref>
imply that $\gfour(\multipliervectorfield,\multipliervectorfield) = -4\upmu(1 + \upmu)$.
Hence, $\multipliervectorfield$ is $\gfour$-timelike whenever $\upmu > 0$.
This property is important because it leads to coercive energy identities.

§.§.§ Building block energies, null-fluxes, and spacetime integrals

We are now ready to define our building block energies and null-fluxes for the fluid variables.
We also define spacetime integrals that play a crucial role in our energy estimates;
see Def. <ref>.

Let $f$ be a scalar function on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$.
Recall that $\hypunitnormalarg{\muxmulevelsetvalue}$ denotes the future-directed $\gfour$-timelike unit normal of
$\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$,
that $\multipliervectorfield$ denotes the multiplier vectorfield defined in (<ref>),
and that the area forms $\volRoughHypersurface$ and $\volPuRoughCoordinates$ are defined in
Def. <ref>.
For $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
we respectively define the wave energy and the null-flux associated to $f$ as follows:

\begin{align}
\mathbb{E}_{(\textnormal{Wave})}[f](\timefunction,u)
\eqdef
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\enmomem[f](\multipliervectorfield,\hypunitnormalarg{\muxmulevelsetvalue})|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}
\, \volRoughHypersurface,
\label{E:WAVEENERGYDEF}
\\
\mathbb{F}_{(\textnormal{Wave})}[f] (\timefunction,u)
\eqdef \int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \enmomem[f](\multipliervectorfield,\Lunit)
\, \volPuRoughCoordinates.
\label{E:WAVENULLFLUXDEF}
\end{align}

If $f$ is a scalar function on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$
and $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
then we respectively define the transport energy and the null-flux associated to $f$
as follows, where $\phi = \phi(u)$ is the cut-off function introduced in Def. <ref>:

\begin{align}
\mathbb{E}_{(\textnormal{Transport})}[f](\timefunction,u)
\eqdef
\int_{\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}}
\left(\upmu - \frac{\muxmulevelsetvalue \phi(u')}{\Lunit \upmu}\right)
\, \volRoughHypersurface, \label{E:TRANSPORTENERGYDEF}
\\
\mathbb{F}_{(\textnormal{Transport})}[f] (\timefunction,u)
\eqdef
\int_{\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\, \volPuRoughCoordinates.
\label{E:TRANSPORTNULLFLUXDEF}
\end{align}

The integrals appearing in the next definition yield spacetime
$L^2$ control of $\angrmD f$ without degenerate $\upmu$ weights.
These integrals arise from favorable terms in the wave equation energy identities (see Prop. <ref>).
They are of crucial importance for our energy estimates.

If $f$ is a scalar function on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$
and $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
then we define the following spacetime integral,
where $\angrmD f$ is as in Def. <ref>,
$\mathbf{1}_{[-\interestingu,\interestingu]} = \mathbf{1}_{[-\interestingu,\interestingu]}(u')$
denotes the characteristic function
of the interval $[-\interestingu,\interestingu]$,
$\phi = \phi(u')$ is the cut-off function from Def. <ref>,
= \volMRoughCoordinates(\timefunction',u',x^2,x^3)
is defined in (<ref>):

\begin{align} \label{E:COERCIVESPACETIMEWAVENERGYINTEGRAL}
\spacetimeintegralcontrolwave[f](\timefunction,u)
\eqdef
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\frac{1}{2}
\mathbf{1}_{[-\interestingu,\interestingu]}(u')
(\argLrough{\muxmulevelsetvalue} \upmu)
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi
\right\rbrace
|\angrmD f|_{\gtorus}^2
\,
\volMRoughCoordinates.
\end{align}

§.§.§ The fundamental energy–null-flux integral identities

In the next proposition,
we provide the fundamental energy–null-flux integral identities that form the foundation
of our hyperbolic energy estimates for the fluid variables.

Wave equation energy–null-flux identity.
Suppose that on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
the scalar function $f$ is a solution to the inhomogeneous covariant wave equation
$\upmu \Box_{\gfour} f = \mathfrak{G}$. Then for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
the following identity holds,
where $\spacetimeintegralcontrolwave[f](\timefunction,u)$ is defined in

\begin{align}
\begin{split} \label{E:FUNDAMENTALENERGYINTEGRALIDENTITCOVARIANTWAVES}
\mathbb{E}[f]_{(\textnormal{Wave})}(\timefunction,u)
\mathbb{F}[f]_{(\textnormal{Wave})}(\timefunction,u)
\spacetimeintegralcontrolwave[f](\timefunction,u)
\mathbb{E}[f]_{(\textnormal{Wave})}(\timefunction_0,u)
+ \mathbb{F}[f]_{(\textnormal{Wave})}(\timefunction,- \rightu)
\\
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}} \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \left\lbrace (1 + 2 \upmu)\Lunit f + 2\muX f\right\rbrace \mathfrak{G} \,  \volMRoughCoordinates
\\
& \ \
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\, \volMRoughCoordinates.
\end{split}
\end{align}

The error term ${^{(\multipliervectorfield)}\mathfrak{B}}[f]$ appearing
in the last integral on RHS (<ref>)
can be decomposed as follows,
where $\mathbf{1}_{[-\interestingu,\interestingu]^c} = \mathbf{1}_{[-\interestingu,\interestingu]^c}(u')$
denotes the characteristic function
of $[-\interestingu,\interestingu]^c = (-\infty,-\interestingu) \cup (\interestingu,\infty)$:

\begin{align}  \label{E:WAVEENERGYIDENTITYBULKERRORTERM}
& \eqdef
\frac{1}{2} \mathbf{1}_{[-\interestingu,\interestingu]^c} (\Lunit \upmu) |\angrmD f|_{\gtorus}^2
\sum_{i=1}^6 {^{(\multipliervectorfield)}\mathfrak{B}_{(i)}[f]},
\end{align}

\begin{align}
\eqdef (\Lunit f)^2
\left\lbrace
- \frac{1}{2} \Lunit \upmu
\muX \upmu
\frac{1}{2} \upmu \mytr_{\gtorus} \upchi
2 \upmu \mytr_{\gtorus} \angktan
2 \mytr_{\gtorus} \angktrans
\right\rbrace,
\label{E:WAVEENERGYIDENTITYBULKERRORTERM1}
\\
\eqdef
- (\Lunit f) (\muX f)
\left\lbrace
\mytr_{\gtorus} \upchi
2 \upmu \mytr_{\gtorus} \angktan
2 \mytr_{\gtorus} \angktrans
\right\rbrace,
\label{E:WAVEENERGYIDENTITYBULKERRORTERM2}
\\
& \eqdef
|\angrmD f|_{\gtorus}^2
\left\lbrace
\Rtransarg{\muxmulevelsetvalue} \upmu
\upmu \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue} \upmu
2 \upmu  \Lunit \upmu
\frac{1}{2}
\upmu \mytr_{\gtorus}\upchi
\upmu^2 \mytr_{\gtorus} \angktan
\upmu \mytr_{\gtorus} \angktrans
\right\rbrace,
\label{E:WAVEENERGYIDENTITYBULKERRORTERM3}
\\
\eqdef
(\Lunit f) (\angrmD^{\#} f)
\cdot
\left\lbrace
(1 - 2 \upmu) \angrmD \upmu
2 \upmu \zetatan
2 \zetatrans
\right\rbrace,
\label{E:WAVEENERGYIDENTITYBULKERRORTERM4}
\\
& \eqdef
(\muX f)
(\angrmD^{\#} f)
\cdot
\left\lbrace
\angrmD \upmu
2 \upmu \zetatan
2 \zetatrans
\right\rbrace, \label{E:WAVEENERGYIDENTITYBULKERRORTERM5}
\\
& \eqdef
\upmu \angrmD^{\#} f \otimes \angrmD^{\#} f
\cdot
\left\lbrace
\upchi + 2 \upmu \angktan + 2 \angktrans
\right\rbrace. \label{E:WAVEENERGYIDENTITYBULKERRORTERM6}
\end{align}

In (<ref>)–(<ref>)
and below in (<ref>),
the vectorfields $\Rtransarg{\muxmulevelsetvalue}$ and $\Roughtoritangentvectorfieldarg{\muxmulevelsetvalue}$ are as in Def. <ref>
and the $\ell_{t,u}$-tangent tensorfields $\upchi$,
$\angktan$, $\angktrans$, $\zetatan$, and $\zetatrans$
are as in Lemma <ref>.

Transport equation energy–null-flux identity.
Suppose that on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
$f$ is a solution to the inhomogeneous transport equation $\upmu \Transport f = \mathfrak{G}$.
Then for $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
the following identity holds:

\begin{align}
\begin{split}
\label{E:ENERGYIDENTITYFORBTRANSPORTEQUATIONS}
\mathbb{E}[f]_{(\textnormal{Transport})}(\timefunction,u)
\mathbb{F}[f]_{(\textnormal{Transport})}(\timefunction,u)
\mathbb{E}[f]_{(\textnormal{Transport})}(\timefunction_0,u) + \mathbb{F}[f]_{(\textnormal{Transport})}(\timefunction,-\rightu)
\\
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} f \cdot G
\, \volMRoughCoordinates
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \left\lbrace
\Lunit \upmu + \upmu \mytr_{\gtorus} \angk
\right\rbrace f^2
\, \volMRoughCoordinates.
\end{split}
\end{align}

Proof of (<ref>):

We will apply the divergence identity (<ref>) with $Z \eqdef \multipliervectorfield$,
where $\multipliervectorfield$ is defined in (<ref>).
Throughout, we often use the abbreviated notation
$\mathbf{J} \eqdef \Jen{\multipliervectorfield}[f]$
to denote the energy current defined by (<ref>).
To proceed, we fist write $\mathbf{J}$
in terms of the rough adapted coordinate partial derivative vectorfields
as follows:

\begin{align} \label{E:WAVEENERGYCURRENTINTERMSOFROUGHGEOMETRICCOORDINATES}
\mathbf{J}
& =
\mathbf{J}^{\timefunction} \roughgeop{\timefunction}
\mathbf{J}^u \roughgeop{u} + \mathbf{J}^A \roughgeop{x^A}.
\end{align}

Next, we claim that the following identities hold, where $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$
is as in (<ref>):

\begin{align}
\mathbf{J}^{\timefunction}
& =
- \frac{(\Lunit \timefunctionarg{\muxmulevelsetvalue}) |\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}{\upmu} \enmomem[f](\multipliervectorfield,\hypunitnormalarg{\muxmulevelsetvalue}),
\label{E:WAVECURRENTTIMEFUNCTIONCOMPONENT}
\\
\mathbf{J}^u
& = - \frac{1}{\upmu} \enmomem[f](\multipliervectorfield,\Lunit).
\label{E:WAVECURRENTUCOMPONENT}
\end{align}

To prove (<ref>), we first note that since
$\hypunitnormalarg{\muxmulevelsetvalue}$ is $\gfour$-orthogonal to $\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$
while $\roughgeop{u}$ and $\roughgeop{x^A}$ are tangent to $\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$,
we have, in view of (<ref>) and (<ref>):

\begin{align}  \label{E:ENMOMENTUMCOMPONENTMULTIPLIERHYPNORMAL}
\enmomem[f](\multipliervectorfield,\hypunitnormalarg{\muxmulevelsetvalue})
& =
\gfour\left(\Jen{\multipliervectorfield}[f],\hypunitnormalarg{\muxmulevelsetvalue}\right)
\mathbf{J}^{\timefunction} \gfour\left(\roughgeop{\timefunction}, \hypunitnormalarg{\muxmulevelsetvalue} \right).
\end{align}

Next, we use
and the fact that $\Lunit u = \roughgeop{\timefunction}  u = 0$
to deduce the identity
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit^C \roughgeop{x^C}$,
and we use Lemma <ref>,
(<ref>) to deduce the identity
$\gfour(\Lunit,\hypunitnormalarg{\muxmulevelsetvalue}) = - \frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}$.
Combining these two identities with (<ref>)
and using that $\gfour(\roughgeop{x^C},\hypunitnormalarg{\muxmulevelsetvalue}) = 0$,
we find that:

\begin{align} \label{E:SECONDENMOMENTUMCOMPONENTMULTIPLIERHYPNORMAL}
\enmomem[f](\multipliervectorfield,\hypunitnormalarg{\muxmulevelsetvalue})
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\mathbf{J}^{\timefunction} \gfour (\Lunit,\hypunitnormalarg{\muxmulevelsetvalue})
- \frac{\upmu}{(\Lunit \timefunctionarg{\muxmulevelsetvalue}) |\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} \mathbf{J}^{\timefunction},
\end{align}

which yields (<ref>).
The identity (<ref>) can be proved using similar arguments
based on taking the $\gfour$-inner product of $\Jen{\multipliervectorfield}[f]$ with $\Lunit$ and using that
$\Lunit$ is $\gfour$-orthogonal to $\nullhyparg{u}$ as well as the identity
$\roughgeop{u} = \Rtransarg{\muxmulevelsetvalue} - \Rtransarg{\muxmulevelsetvalue}^C \roughgeop{x^C}$, which follows from
Lemma <ref> and

Next, we note the following formula,
which follows by combining the standard identity for the divergence of a vectorfield expressed relative to the rough adapted coordinates
with the identities

\begin{align}
\begin{split}  \label{E:IDFORDETWEIGHTEDCOVARIANTDIVERGENCEOFWAVEENERGYCURRENT}
\sqrt{|\mbox{\upshape det} \gfour|} \, \Dfour_\alpha \Jenarg{\multipliervectorfield}{\alpha}[f]
& =
\roughgeop{\timefunction}
\left( \sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^{\timefunction} \right)
\roughgeop{u} \left( \sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^u\right)
\roughgeop{x^A} \left( \sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^A \right)
\\
& =
\roughgeop{\timefunction}
\left(
\enmomem[f](\multipliervectorfield,\hypunitnormalarg{\muxmulevelsetvalue}) |\Rtransarg{\muxmulevelsetvalue}|_{\gfour}
\sqrt{ \mbox{\upshape det} \gtorusroughfirstfund}
\right)
\roughgeop{u}
\left(
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\enmomem[f](\multipliervectorfield,\Lunit)
\sqrt{ \mbox{\upshape det} \gtorusroughfirstfund}
\right)
\roughgeop{x^A}
\left(
\sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^A
\right).
\end{split}
\end{align}

Integrating (<ref>) over
$\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}$
with respect to $\mathrm{d} x^2 \, \mathrm{d}x^3 \mathrm{d} u' \mathrm{d} \timefunction'$,
using (<ref>) to relate the canonical volume form $\volcanonical_{\gfour}$ to $\volMRoughCoordinates$, using (<ref>) to substitute for $\Dfour_\alpha \Jenarg{\multipliervectorfield}{\alpha}[f]$,
applying Fubini's theorem,
and using that the integral of the last term on RHS (<ref>)
over $\T^2$ vanishes (since $\T^2$ is a closed manifold), we deduce,
in view of definitions (<ref>)–(<ref>) and
(<ref>), the following identity:

\begin{align}
\begin{split} \label{E:FUNDAMENTALENERGYINTEGRALIDENTITCOVARIANTWAVESINTERMEDIATE}
\mathbb{E}[f]_{(\textnormal{Wave})}(\timefunction,u)
\mathbb{F}[f]_{(\textnormal{Wave})}(\timefunction,u)
\mathbb{E}[f]_{(\textnormal{Wave})}(\timefunction_0,u)
\mathbb{F}[f]_{(\textnormal{Wave})}(\timefunction,- \rightu)
\\
& \ \
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\left\lbrace
(1 + 2 \upmu) \Lunit f + 2 \muX f
\right\rbrace
\mathfrak{G}
\,  \volMRoughCoordinates
\\
& \ \
\frac{1}{2}
\int_{\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\upmu \enmomem^{\alpha \beta}[f] \deformarg{\multipliervectorfield}{\alpha}{\beta}
\, \volMRoughCoordinates.
\end{split}
\end{align}

Next, we decompose the integrand in the last integral on RHS (<ref>)
\frac{1}{2}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\upmu  (\gfour^{-1})^{\alpha \delta} (\gfour^{-1})^{\beta \sigma} \enmomem_{\delta\sigma}[f]
\deformarg{\multipliervectorfield}{\alpha}{\beta}
definitions (<ref>),
and (<ref>)
the identity
$(\gfour^{-1})^{\alpha \delta}
\Lunit^{\alpha} \Lunit^{\delta}
\Lunit^{\alpha} X^{\delta}
- X^{\alpha}\Lunit^{\delta}
(\gtorus^{-1})^{\alpha \delta}
which follows from (<ref>),
and the analogous identity for $(\gfour^{-1})^{\beta \sigma}$.
Among the many terms that arise from the expansion
are the following two:[The precise origin of these terms is $- \enmomem[f](\Lunit,X) \deformarg{\multipliervectorfield}{L}{\muX}$, which is found in the expansion of $-\frac{1}{2} \upmu \enmomem^{\alpha \beta}[f] \deformarg{\multipliervectorfield}{\alpha}{\beta}$.]

\begin{align} \label{E:FUNDAMENTALENERGYINTEGRALIDENTITCOVARIANTWAVESINTERMEDIATE2}
\frac{1}{2}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Lunit \upmu)
|\angrmD f|_{\gtorus}^2
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\muX \upmu) |\angrmD f|_{\gtorus}^2.
\end{align}

We now decompose the first product in (<ref>) as follows,
where we use definition (<ref>):

\begin{align} \label{E:DECOMPOSITIONOFLMUWAVENERGYTERM}
\frac{1}{2}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Lunit \upmu)
|\angrmD f|_{\gtorus}^2
& =
\frac{1}{2}
\mathbf{1}_{[-\interestingu,\interestingu]}
(\argLrough{\muxmulevelsetvalue} \upmu)
|\angrmD f|_{\gtorus}^2
\frac{1}{2}
\mathbf{1}_{[-\interestingu,\interestingu]^c}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Lunit \upmu)
|\angrmD f|_{\gtorus}^2.
\end{align}

Since the spacetime integral of
\mathbf{1}_{[-\interestingu,\interestingu]}
(\argLrough{\muxmulevelsetvalue} \upmu)
|\angrmD f|_{\gtorus}^2
is found in the negative of the coercive spacetime integral defined in (<ref>)
(i.e., $- \spacetimeintegralcontrolwave[f](\timefunction,u)$),
we bring this term to LHS (<ref>)
as part of $\spacetimeintegralcontrolwave[f](\timefunction,u)$.
The remaining term
\frac{1}{2}
\mathbf{1}_{[-\interestingu,\interestingu]}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\Lunit \upmu)
|\angrmD f|_{\gtorus}^2
in the decomposition (<ref>)
is manifestly present on RHS (<ref>)
as the first term in RHS (<ref>).
Next, we examine the term
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\muX \upmu) |\angrmD f|_{\gtorus}^2$ present in (<ref>).
Using (<ref>), we express this term as follows:

\begin{align} \label{E:MUXMUTIMESANGULARDERIVATIVEENERGYESTIMATEINTEGRANDFACTORDECOMPOSITION}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
(\muX \upmu)
|\angrmD f|_{\gtorus}^2
& =
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi
|\angrmD f|_{\gtorus}^2
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
|\angrmD f|_{\gtorus}^2
\left\lbrace
\Rtransarg{\muxmulevelsetvalue} \upmu
\upmu \Roughtoritangentvectorfieldarg{\muxmulevelsetvalue} \upmu
\right\rbrace
|\angrmD f|_{\gtorus}^2.
\end{align}

We bring the integral of the first product
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi |\angrmD f|_{\gtorus}^2$
on RHS (<ref>)
over to LHS (<ref>)
as the remaining part of $\spacetimeintegralcontrolwave[f](\timefunction,u)$
(see definition (<ref>)).
These terms in the last product on RHS (<ref>)
are manifestly present in the term ${^{(\multipliervectorfield)}\mathfrak{B}_{(3)}[f]}$
defined in (<ref>).
The remaining terms in the decomposition of
\frac{1}{2}
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\upmu  (\gfour^{-1})^{\alpha \delta} (\gfour^{-1})^{\beta \sigma} \enmomem_{\delta\sigma}[f]
\deformarg{\multipliervectorfield}{\alpha}{\beta}
can be derived using the same arguments,
based on straightforward but tedious calculations,
given in the proof of <cit.>.
We have therefore proved (<ref>).

Proof of (<ref>):
We will apply the divergence theorem to the vectorfield $\mathbf{J} \eqdef f^2 \Transport$.
We begin by expressing this vectorfield in terms of the rough geometric coordinate vectorfields as follows:

\begin{align}
\mathbf{J}
& =
\mathbf{J}^{\timefunction} \roughgeop{\timefunction}
\mathbf{J}^u \roughgeop{u}
\mathbf{J}^A \roughgeop{x^A}.
\end{align}

Next, we claim that the following identities hold:

\begin{align}
\mathbf{J}^{\timefunction}
= f^2
\left(\Lunit \timefunctionarg{\muxmulevelsetvalue}
\frac{\muxmulevelsetvalue \phi}{\upmu \Lunit \upmu}
\Lunit \timefunctionarg{\muxmulevelsetvalue}
\right)
\label{E:TRANSPORTENERGYCURRENTROUGHTIMECOMPONENT}
\\
\mathbf{J}^u
& = \frac{1}{\upmu} f^2.
\label{E:TRANSPORTENERGYCURRENTROUGHUCOMPONENT}
\end{align}

To prove (<ref>),
we note that
$\mathbf{J}^{\timefunction} = \mathbf{J} \timefunctionarg{\muxmulevelsetvalue} = f^2 \Transport \timefunctionarg{\muxmulevelsetvalue}$
and use the equations
$0 = \Wtrans \timefunctionarg{\muxmulevelsetvalue}
\muX \timefunctionarg{\muxmulevelsetvalue}
\frac{\muxmulevelsetvalue\phi}{\Lunit \upmu}
\Lunit \timefunctionarg{\muxmulevelsetvalue}$
(see (<ref>) and (<ref>))
$\Transport = \Lunit + X$ (see (<ref>)).
Similarly, to prove (<ref>),
we note that
$\mathbf{J}^u = \mathbf{J} u = f^2 \Transport u$
and then use Lemma <ref>.

Next, using the divergence theorem,
we deduce that:

\begin{align}
\begin{split} \label{E:EXPRESSIONFORDETWEIGHTEDCOVARIANTDIVERGENCEOFTRANSPORTENERGYCURRENTINROUGHADATPEDCOORDINATES}
\sqrt{|\mbox{\upshape det} \gfour|} \Dfour_\alpha \mathbf{J}^{\alpha}
\sqrt{|\mbox{\upshape det} \gfour|} \Dfour_\alpha (f^2 \Transport^{\alpha})
\roughgeop{\timefunction} \left( \sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^{\timefunction}\right)
\roughgeop{u}
\left(
\sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^u
\right)
\roughgeop{x^A}
\left(
\sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^A
\right)
\\
& = \roughgeop{\timefunction} \left( f^2\left\lbrace \upmu - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}\right\rbrace\sqrt{\mbox{\upshape det} \gtorusroughfirstfund} \right) + \roughgeop{u} \left( \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} f^2 \sqrt{\mbox{\upshape det} \gtorusroughfirstfund}\right) + \roughgeop{x^A} \left(\sqrt{|\mbox{\upshape det} \gfour|} \mathbf{J}^A\right).
\end{split}
\end{align}

Next, we note the following covariant divergence identity,
which follows from the relation $\mathbf{J} = f^2 \Transport$,
the Leibniz rule,
Lemma <ref>,

\begin{align} \label{E:COVARIANTDIVERGENCEOFTRANSPORTENERGYCURRENT}
\Dfour_\alpha \mathbf{J}^{\alpha}
\frac{1}{\upmu} (\Lunit \upmu) f^2
\mytr_{\gtorus} \angk f^2
2 f \Transport f.
\end{align}

Substituting (<ref>) into
LHS (<ref>)
and then integrating both sides of the resulting identity
over $\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}$
with respect to $\mathrm{d} x^2 \, \mathrm{d}x^3 \mathrm{d} u' \mathrm{d} \timefunction'$,
using (<ref>) to relate the canonical volume form
$\volcanonical_{\gfour}$ to $\volMRoughCoordinates$,
applying Fubini's theorem,
and using that the integral of the last term on RHS (<ref>)
over $\T^2$ vanishes (since $\T^2$ is a closed manifold)
we conclude,
in view of definitions
the desired identity (<ref>).

§.§ The fundamental $L^2$-controlling quantities

In this section, we use the building block quantities
from Sect. <ref> to
construct the “fundamental $L^2$-controlling quantities” that we use to control
$\wavearray$, $\vortrenormalized$, $\GradEnt$, $\VortVort$, $\DivGradEnt$,
and their derivatives in $L^2$ in various regions.

In terms of the energy-null-flux quantities of Def. <ref>
and the differentiated conventions of Def. <ref>,
we define the following $L^2$-controlling quantities:

* Total wave-controlling quantities.

\begin{align}
\hypersurfacecontrolwave_N(\timefunction,u)
\eqdef
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{ \RRiemann, \LRiemann, v^2, v^3, \Ent\}}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Wave})}[\tander^N \Psi](\timefunction',u')
\mathbb{F}_{(\textnormal{Wave})}[\tander^N \Psi](\timefunction',u')
\right\rbrace, \label{E:WAVESPACELIKEANDNULLHYPERSURFACEL2CONTROLLINGQUANTITY}
\\
\spacetimeintegralcontrolwave_N(\timefunction,u)
\eqdef
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)}
\\ \Psi \in \{ \RRiemann, \LRiemann, v^2, v^3, \Ent\}}}
\spacetimeintegralcontrolwave[\tander^N \Psi] (\timefunction,u),
\label{E:WAVESPACETIMEL2CONTROLLINGQUANTITY}
\\
\totalcontrolwave_N(\timefunction,u)
\eqdef
\max
\left\lbrace
\hypersurfacecontrolwave_N(\timefunction,u),
\,
\spacetimeintegralcontrolwave_N(\timefunction,u)
\right\rbrace,
\label{E:WAVETOTALL2CONTROLLINGQUANTITY}
\end{align}

* Partial wave-controlling quantities.

\begin{align}
\hypersurfacecontrolwavepartial_N(\timefunction,u)
\eqdef
\max_{\substack{ \tander^N \in \mathfrak{P}^{(N)}
\\ \Psi \in \{\LRiemann, v^2, v^3, \Ent\}}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Wave})}[ \tander^N\Psi](\timefunction',u')
\mathbb{F}_{(\textnormal{Wave})}[\tander^N \Psi](\timefunction',u')
\right\rbrace, \label{E:WAVEPARTIALSPACELIKEANDNULLHYPERSURFACEL2CONTROLLINGQUANTITY}
\\
\spacetimeintegralcontrolwavepartial_N(\timefunction,u) & \eqdef \max_{\substack{ \tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{\LRiemann, v^2, v^3, \Ent\}}} \spacetimeintegralcontrolwave[\tander^N \Psi] (\timefunction,u),
\label{E:PARTIALWAVESPACETIMEL2CONTROLLINGQUANTITY}
\\
\totalcontrolwavepartial_N(\timefunction,u)
\eqdef
\max
\left\lbrace
\hypersurfacecontrolwavepartial_N(\timefunction,u),
\,
\spacetimeintegralcontrolwavepartial_N(\timefunction,u)
\right\rbrace.
\label{E:WAVEPARTIALL2CONTROLLINGQUANTITY}
\end{align}

* Specific vorticity- and entropy-controlling quantities.

\begin{align}
\hypersurfacecontrolVort_N(\timefunction,u)
& \eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Transport})}[\tander^N \vortrenormalized] (\timefunction',u')
\mathbb{F}_{(\textnormal{Transport})}[\tander^N \vortrenormalized] (\timefunction',u')
\right\rbrace,
\label{E:VORTICITYL2CONTROLLINGQUANTITY}
\\
\hypersurfacecontrolGradEnt_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Transport})}[\tander^N \GradEnt] (\timefunction',u')
\mathbb{F}_{(\textnormal{Transport})}[\tander^N \GradEnt] (\timefunction',u')
\right\rbrace, \label{E:ENTROPYGRADIENTL2CONTROLLINGQUANTITY}
\end{align}

\begin{align}
\toricontrolVort_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\left\|
\tander^N \vortrenormalized
\right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2,
\label{E:TORIVORTICITYL2CONTROLLINGQUANTITY}
\\
\toricontrolGradEnt_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\left\|
\tander^N \GradEnt
\right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2.
\label{E:TORIENTROPYGRADIENTL2CONTROLLINGQUANTITY}
\end{align}

* Modified fluid variable-controlling quantities.

\begin{align}
\hypersurfacecontrolVortVort_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Transport})}[\tander^N \VortVort] (\timefunction',u')
\mathbb{F}_{(\textnormal{Transport})}[\tander^N \VortVort] (\timefunction',u')
\right\rbrace, \label{E:MODIFIEDVORTICITYVORTICITYL2CONTROLLINGQUANTITY}
\\
\hypersurfacecontrolDivGradEnt_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\sup_{\substack{\timefunction' \in [\timefunction_0,\timefunction]
\\
u' \in [- \rightu,u]}}
\left\lbrace
\mathbb{E}_{(\textnormal{Transport})}[\tander^N \DivGradEnt] (\timefunction',u')
\mathbb{F}_{(\textnormal{Transport})}[\tander^N \DivGradEnt] (\timefunction',u')
\right\rbrace,
\label{E:MODIFIEDDIVGRADENTL2CONTROLLINGQUANTITY}
\end{align}

\begin{align}
\toricontrolVortVort_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\left\|
\tander^N \VortVort
\right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2,
\label{E:TORIMODIFIEDVORTICITYL2CONTROLLINGQUANTITY}
\\
\toricontrolDivGradEnt_N(\timefunction,u)
\eqdef
\max_{\tander^N \in \mathfrak{P}^{(N)}}
\left\|
\tander^N \DivGradEnt
\right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2.
\label{E:TORIDIVGRADENTL2CONTROLLINGQUANTITY}
\end{align}

Although $\hypersurfacecontrolwavepartial_N$ might seem to be a redundant quantity, it plays an important role in our energy estimates since in the top-order case,
the partial energy $\hypersurfacecontrolwavepartial_N$
is only weakly influenced by the full energy $\hypersurfacecontrolwave_N$.
Similar remarks apply to

For positive integers $N_1 < N_2$ and non-negative integers $N$,
we define the following summed $L^2$-controlling quantities:

\begin{align} \label{D:ENERGYSUMCONVENTIONS}
\hypersurfacecontrolwave_{[N_1,N_2]}(\timefunction,u)
\eqdef \sum_{M = N_1}^{N_2} \hypersurfacecontrolwave_M(\timefunction,u),
\hypersurfacecontrolVort_{\leq N}(\timefunction,u)
& =
\sum_{M = 0}^N \hypersurfacecontrolVort_M(\timefunction,u),
\end{align}

and similarly for the other controlling quantities. When $N = 0$,
we often omit the subscript, e.g., we write
instead of

§.§ The coerciveness of the fundamental $L^2$-controlling quantities

In this section, we exhibit the coerciveness properties of the $L^2$ controlling quantities
from Def. <ref>.

§.§.§ Decomposition of components of the energy-momentum tensor

We start with the following lemma,
which yields identities for various components of the energy-momentum tensor.
The components $\enmomem[f](\Transport,\Lunit)$ and $\enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\multipliervectorfield)$
are of particular interest since they appear in our energy–null-flux identity

Let $f$ be a scalar function, and let $\enmomem$ be the corresponding energy-momentum tensor
defined in (<ref>).
Then the following identities hold,
where $|\angD f|_{\gtorus}^2 = (\gtorus^{-1})^{AB} (\geop{x^A} f) \geop{x^B} f$:

\begin{align}
\enmomem[f](\Lunit,\Lunit) & = (\Lunit f)^2,
\label{E:QLL}
\\
\enmomem[f](X,\Lunit)
& =
- \frac{1}{2} (\Lunit f)^2 + \frac{1}{2} | \angD f|_{\gtorus}^2,
\label{E:QLX}
\\
\enmomem[f](\Transport,\Lunit)
= \frac{1}{2} (\Lunit f)^2 + \frac{1}{2}|\angD f|_{\gtorus}^2
\label{E:QTRANSPORTL},
\\
\enmomem[f](X,\Transport)
& = (\Lunit f) X f + (X f)^2 \label{E:QXTRANSPORT},
\\
\enmomem[f] \left(\Lunit,\roughgeop{x^A}\right)
= (\Lunit f) \roughgeop{x^A} f, \label{E:QLROUGHGEOPA}
\\
\enmomem[f]\left(X,\roughgeop{x^A}\right)
& =
(X f) \roughgeop{x^A} f
\frac{1}{2} \frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} (\Lunit f)^2
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} (\Lunit f) Xf
\frac{1}{2} \frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} | \angD f|_{\gtorus}^2,
\label{E:QXROUGHGEOPA}
\\
\enmomem[f]\left(\Transport, \roughgeop{x^A}\right)
= (\Lunit f) \roughgeop{x^A} f
(Xf) \roughgeop{x^A} f
\frac{1}{2} \frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} (\Lunit f)^2
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} (\Lunit f) Xf
\frac{1}{2}
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
| \angD f|_{\gtorus}^2.
\label{E:QTRANSPORTROUGHGEOOPA}
\end{align}

Moreover, the following identities also hold, where $\multipliervectorfield$ is the multiplier vectorfield (<ref>),
$\phi$ is the cut-off function from Def. <ref>,
$\Rtransarg{\muxmulevelsetvalue}$ is as in (<ref>),
$\hypunitnormalarg{\muxmulevelsetvalue}$ is as in (<ref>),
$|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ is as in (<ref>),
$\Rtransnormsmallfactorarg{\muxmulevelsetvalue}$ is defined by (<ref>):

\begin{align}
\enmomem[f](\multipliervectorfield,\Lunit)
= (1 + \upmu)(\Lunit f)^2 + \upmu |\angD f|_{\gtorus}^2,
\label{E:QTL}
\\
\enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\Lunit)
\frac{\upmu(1 - \Rtransnormsmallfactorarg{\muxmulevelsetvalue})
\frac{\muxmulevelsetvalue\phi }{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} \enmomem[f](\Lunit,\Lunit)
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}\enmomem[f](X,\Lunit)
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
\enmomem[f](\roughgeop{x^B},\Lunit), \label{E:QNL}
\\
\enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue}, \Transport)
\frac{\upmu(1 - \Rtransnormsmallfactorarg{\muxmulevelsetvalue})
\frac{\muxmulevelsetvalue\phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} \enmomem[f](\Transport,\Lunit)
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}\enmomem[f](\Transport,X)
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
\enmomem[f](\roughgeop{x^B},\Transport),
\label{E:QNB}
\\
\begin{split} \label{E:QNMULTIPLIER}
\enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\multipliervectorfield)
\frac{\upmu^2(1 - \Rtransnormsmallfactorarg{\muxmulevelsetvalue}) - \upmu\frac{\muxmulevelsetvalue\phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\left\lbrace
(\Lunit f)^2
|\angD f|_{\gtorus}^2
\right\rbrace
\frac{2 \upmu^2}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} \left\lbrace (\Lunit f) Xf + (Xf)^2 \right\rbrace
\\
& \ \
\frac{\upmu^2}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\Big\lbrace
- 2 (\Lunit f)
\frac{\gtorusroughinversefirstfund\left(dx^A,dx^B \right) \geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}
{ \geop{t} \timefunctionarg{\muxmulevelsetvalue}} \roughgeop{x^B} f
2 (Xf)
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}
{ \geop{t} \timefunctionarg{\muxmulevelsetvalue}} \roughgeop{x^B} f
\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\Rtransnormsmallfactorarg{\muxmulevelsetvalue} (\Lunit f)^2
2 \Rtransnormsmallfactorarg{\muxmulevelsetvalue} (\Lunit f) Xf
\Rtransnormsmallfactorarg{\muxmulevelsetvalue} |\angD f|_{\gtorus}^2
\Big\rbrace
\\
& \ \
\frac{\upmu
\left( \frac{1}{2} - \Rtransnormsmallfactorarg{\muxmulevelsetvalue} \right)
\frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} (\Lunit f)^2
\frac{\upmu}{2 |\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} |\angD f|_{\gtorus}^2
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue}}
{\geop{t} \timefunctionarg{\muxmulevelsetvalue}} (\Lunit f) \roughgeop{x^B} f.
\end{split}
\end{align}

(<ref>)–(<ref>) are straightforward consequences of the definition
(<ref>) of $\enmomem$,
Lemma <ref>,
and the decompositions of $\gfour^{-1}$ implied by (<ref>).

(<ref>)–(<ref>) follow from combining similar
arguments with the identity

(<ref>) follows from the identity
$\multipliervectorfield = \Lunit + 2 \upmu \Transport$
(which is a consequence of (<ref>) and (<ref>)),
and (<ref>).

To derive (<ref>)–(<ref>),
we first decompose $\hypunitnormalarg{\muxmulevelsetvalue}$ as follows using
and (<ref>):

\begin{align}
\begin{split} \label{E:USEFULHYPUNITNORMALDECOMPOSITION}
\hypunitnormalarg{\muxmulevelsetvalue}
= \frac{\upmu(1 - \Rtransnormsmallfactorarg{\muxmulevelsetvalue}) - \frac{2\muxmulevelsetvalue \phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}\Lunit
\frac{1}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} \muX
\frac{\muxmulevelsetvalue \phi}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour} \Lunit \upmu} \Lunit
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue} \roughgeop{x^B}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}
\\
& =
\frac{\upmu(1 - \Rtransnormsmallfactorarg{\muxmulevelsetvalue}) - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}\Lunit
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} X
\frac{\upmu}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
\gtorusroughinversefirstfund\left(dx^A,dx^B \right)
\frac{\geop{x^A} \timefunctionarg{\muxmulevelsetvalue} \roughgeop{x^B}}{\geop{t} \timefunctionarg{\muxmulevelsetvalue}}.
\end{split}
\end{align}

(<ref>)–(<ref>) now follow from
the linearity of the map $Z \rightarrow \enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},Z)$.

Finally, (<ref>) follows from
and the following identity identity,
which follows from the identity $\multipliervectorfield = \Lunit + 2 \upmu \Transport$  mentioned above:
$\enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\multipliervectorfield) = \enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\Lunit) + 2 \upmu \enmomem[f](\hypunitnormalarg{\muxmulevelsetvalue},\Transport)$.

§.§.§ Coerciveness estimates on submanifolds

In the next lemma,
we exhibit the coerciveness of
the $L^2$-controlling quantities from Def. <ref>.
In particular, we exhibit their $L^2$ coerciveness properties on
the submanifolds
$\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$,
We reveal the coerciveness of the spacetime integrals
in a separate lemma, namely Lemma <ref>.

Let $\hypersurfacecontrolwave_N(\timefunction,u)$,
be the $L^2$-controlling quantities from Def. <ref>,
and let $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$.
Then for $1 \leq N \leq \Ntop$,
the following lower bounds hold,
where $\mathfrak{P}^{(N)}$ is the set of order $N$ $\nullhyparg{u}$-tangential commutator operators from
Sect. <ref>:

\begin{align}
\begin{split} \label{E:COERCIVENESSOFHYPERSURFACECONTROLWAVE}
\hypersurfacecontrolwave_N(\timefunction,u)
& \geq
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{ \RRiemann,\LRiemann,v^2,v^3,\Ent \}}}
\left\lbrace
\left\|
\sqrt{\upmu - \frac{2 \muxmulevelsetvalue \phi}{\Lunit \upmu}} \Lunit \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\sqrt{\upmu} \left| \angrmD \tander^N \Psi \right|_{\gtorus}
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\right.
\\
\left.
\left\|
\muX \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \Lunit \tander^N \Psi
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2,
\,
\left\|
\frac{\sqrt{\upmu}}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}}
\left| \angrmD\tander^N \Psi \right|_{\gtorus}
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\right\rbrace,
\end{split}
\\
\begin{split} 	\label{E:COERCIVENESSOFHYPERSURFACECONTROLWAVEPARTIAL}
\hypersurfacecontrolwavepartial_N(\timefunction,u)
& \geq
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{ \LRiemann,v^2,v^3,\Ent \}}}
\left\lbrace
\left\|
\sqrt{\upmu - \frac{2 \muxmulevelsetvalue \phi}{\Lunit \upmu}} \Lunit \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\sqrt{\upmu} \left| \angrmD \tander^N \Psi \right|_{\gtorus}
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\right.
\\
& \left.
\left\|
\muX \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \Lunit \tander^N \Psi
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2,
\,
\left\|
\frac{\sqrt{\upmu}}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}}
\left| \angrmD\tander^N \Psi\right|_{\gtorus}
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\right\rbrace.
\end{split}
\end{align}

Moreover, for $N \leq \Ntop$, the following lower bounds hold:

\begin{align}
\hypersurfacecontrolVort_N(\timefunction,u)
\geq
\max
\left\lbrace
\left\|
\sqrt{\upmu - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}} \tander^N \vortrenormalized
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \tander^N \vortrenormalized
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2,
\right\rbrace \label{E:COERCIVENESSOFCONTROLVORT}
\\
\hypersurfacecontrolGradEnt_N(\timefunction,u)
& \geq
\max
\left\lbrace
\left\|
\sqrt{\upmu - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}} \tander^N \GradEnt
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \tander^N \GradEnt
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\right\rbrace,
\label{E:COERCIVENESSOFCONTROLGRADENT}
\end{align}

\begin{align}
\hypersurfacecontrolVortVort_N (\timefunction,u)
& \geq
\max
\left\lbrace
\left\|
\sqrt{\upmu - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}} \tander^N \VortVort
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \tander^N \VortVort
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\right\rbrace \label{E:COERCIVENESSHYPERSURFACECONTROLVORTVORT}
\\
\hypersurfacecontrolDivGradEnt_N (\timefunction,u)
& \geq
\max
\left\lbrace
\left\|
\sqrt{\upmu - \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}} \tander^N \DivGradEnt
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2,
\,
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \tander^N \DivGradEnt
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2 \right\rbrace.
\label{E:COERCIVENESSHYPERSURFACEDIVGRADENT}
\end{align}

In addition, for $1 \leq N \leq \Ntop$ and $N' \leq \Ntop$,
we have:

\begin{align}
\left\| \tander^N \Psi \right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
\leq
C \initialsmall^2
C \hypersurfacecontrolwave_N(\timefunction,u),
\label{E:LOSSOFONEDERIVATIVEL2ESTIMATESFORWAVEVARIABLESONROUGHTORIINTERMSOFDATAANDCONTROLLING}
\\
\left\| \tander^{\leq N'}  \vortrenormalized \right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
& \leq C \initialsmall^2 + C \hypersurfacecontrolVort_{\leq N'+1}(\timefunction,u),
\label{E:LOSSOFONEDERIVATIVEL2ESTIMATESFORTRANSPORTVARIABLESONROUGHTORIINTERMSOFDATAANDCONTROLLING}
\\
\left\| \tander^{\leq N'}  \GradEnt \right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
& \leq C \initialsmall^2
C \hypersurfacecontrolGradEnt_{\leq N'+1}(\timefunction,u),
\label{E:LOSSOFONEDERIVATIVETORIL2ESTIMATESFORGRADENTONROUGHTORIINTERMSOFDATAANDCONTROLLING}
\\
\left\| \tander^{N'} \Psi \right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
& \leq
C \initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction',u)}{|\timefunction'|^{1/2}}
\, \mathrm{d} \timefunction'
\leq
C \initialsmall
\hypersurfacecontrolwave_{[1,N]}^{1/2}(\timefunction,u).
\label{E:L2ESTIMATESFORWAVEVARIABLESONROUGHHYPERSURFACELOSSOFONEDERIVATIVE}
\end{align}

Finally, if $1 \leq N \leq \Ntop$,
$\Psi \in \{ \RRiemann, \LRiemann, v^2, v^3, \Ent\}$,
$\tander^N \in \mathfrak{P}^{(N)}$,
and $\fundbootsmall$ and $\mupositive$ are sufficiently small,
then the following sharpened coerciveness estimates
hold whenever $0 \leq \upmu < \mupositive$:

\begin{align} \label{E:L2WAVECONTROLWITHBETTERCONSTANTIFMUSMALL}
\hypersurfacecontrolwave_N(\timefunction,u)
\geq
\mathbb{E}_{(\textnormal{Wave})}[\tander^N \Psi] (\timefunction,u)
\geq
\left\| \muX \tander^N \Psi \right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2.
\end{align}

Proofs of (<ref>)–(<ref>):
These estimates follow directly from the definitions (<ref>)–(<ref>),
and (<ref>)–(<ref>).

Proofs of (<ref>)–(<ref>):
We fix any $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
$\Psi \in \{ \RRiemann,\LRiemann,v^2,v^3,\Ent \}$,
$\tander^N \in \mathfrak{P}^{(N)}$.
First, using (<ref>)
and (<ref>) with $f \eqdef \tander^N \Psi$,
and (<ref>),
we find that
\left\|
\frac{1}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}} \Lunit \tander^N \Psi
\right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\leq
\hypersurfacecontrolwave_N(\timefunction,u)
\left\| \frac{\sqrt{\upmu}}{\sqrt{\Lunit \timefunctionarg{\muxmulevelsetvalue}}}
\left| \angrmD\tander^N \Psi\right|_{\gtorus} \right\|_{L^2(\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)})}^2
\leq
\hypersurfacecontrolwave_N(\timefunction,u)
as desired.
Next, we note that the product of $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ and the terms on first line of RHS (<ref>)
can be expressed as follows:

\begin{align}
\begin{split} \label{E:COERCIVESTEP1}
\left\lbrace
\upmu^2\left(\frac{0.001}{1.001} - \Rtransnormsmallfactorarg{\muxmulevelsetvalue}\right)
\upmu \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}
\right\rbrace
(\Lunit \tander^N \Psi)^2
\left\lbrace
\upmu^2 (1-\Rtransnormsmallfactorarg{\muxmulevelsetvalue})
\upmu \frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}
\right\rbrace |\angD \tander^N \Psi|_{\gtorus}^2  \\
\left( \frac{\upmu}{\sqrt{1.001}} \Lunit \tander^N \Psi + \sqrt{1.001} \muX \tander^N \Psi\right)^2
0.999(\muX \Psi)^2.
\end{split}
\end{align}

Next, we multiply the terms on the second and third lines of RHS (<ref>) by $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$
and use
Lemma <ref>,
the estimates of Prop. <ref>,
Cor. <ref>,
and Young's inequality to bound the magnitude of
the resulting terms as follows:

\begin{align} \label{E:ERRORTERMSQNT}
\leq
C \fundbootsmall \upmu^2 (\Lunit \tander^N \Psi)^2
C \fundbootsmall \upmu^2 |\angD \tander^N \Psi|_{\gtorus}^2
C \fundbootsmall (\muX \tander^N \Psi)^2.
\end{align}

The same reasoning, in conjunction with (<ref>),
yields that all the terms in (<ref>) are positive definite
and that the product of
$|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ and the last term on RHS (<ref>) can be absorbed by
the product of $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ and the two terms
\frac{\upmu
\left( \frac{1}{2} - \Rtransnormsmallfactorarg{\muxmulevelsetvalue} \right)
\frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} (\Lunit f)^2
\frac{\upmu}{2 |\Rtransarg{\muxmulevelsetvalue}|_{\gfour}} |\angD f|_{\gtorus}^2
on the last line of (<ref>).
In total, we see that if $\fundbootsmall$ is small enough,
then in the product of $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ and (<ref>),
the overall coefficient of
$(\muX \tander^N \Psi)^2$
can be bounded from below by $0.99$,
the overall coefficient of
$\upmu (\Lunit \tander^N \Psi)^2$ can be bounded from below by $0.49 \left(\upmu - \frac{2 \muxmulevelsetvalue \phi}{\Lunit \upmu} \right)$,
and the overall coefficient of $\upmu |\angD \tander^N \Psi|_{\gtorus}^2$
can be bounded from below by $0.49$.
From these estimates,
(<ref>) with $f \eqdef \tander^N \Psi$,
and (<ref>),
we deduce that
\left\|
\muX \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2
\leq
\hypersurfacecontrolwave_N(\timefunction,u)
\left\|
\sqrt{\upmu - \frac{2 \muxmulevelsetvalue \phi}{\Lunit \upmu}} \Lunit \tander^N \Psi
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2
\leq
\hypersurfacecontrolwave_N(\timefunction,u)
\left\|
\sqrt{\upmu} \left| \angrmD \tander^N \Psi \right|_{\gtorus}
\right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}^2
\leq
\hypersurfacecontrolwave_N(\timefunction,u)
Combining all five of the $L^2$ bounds along $\nullhypthreearg{\muxmulevelsetvalue}{u}{[\timefunction_0,\timefunction)}$
and $\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$
that we derived in this paragraph,
we conclude the desired lower bounds (<ref>).

Taking into account definition (<ref>),
we can prove the lower bounds stated in (<ref>)
via exactly the same argument.

Proof of (<ref>):
We again fix any $(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [- \rightu,\leftu]$,
$\Psi \in \{ \RRiemann,\LRiemann,v^2,v^3,\Ent \}$,
$\tander^N \in \mathfrak{P}^{(N)}$.
We consider the product of $|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}$ and RHS (<ref>)
with $f \eqdef \tander^N \Psi$.
The terms in the second braces on the first line of RHS (<ref>) generate the terms
$2 \upmu (\Lunit \tander^N \Psi) \muX \tander^N \Psi + 2 (\muX \tander^N \Psi)^2$,
which by Young's inequality, can be pointwise bounded from below by
$1.999 (\muX \tander^N \Psi)^2 - 1000 \upmu^2 (\Lunit \tander^N \Psi)^2$.
If $\mupositive \leq 10^{-5}$,
then (<ref>) implies
that on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
we have the pointwise bound
$|1000 \upmu^2 (\Lunit \tander^N \Psi)^2| \leq \frac{1}{10} \upmu (\Lunit \tander^N \Psi)^2$.
Hence, if $\fundbootsmall$ is sufficiently small,
then the same arguments we used in the proof of (<ref>)
imply that we can absorb the term
$- 1000 \upmu^2 (\Lunit \tander^N \Psi)^2$ into
the terms
\frac{\upmu
\left( \frac{1}{2} - \Rtransnormsmallfactorarg{\muxmulevelsetvalue} \right)
\frac{\muxmulevelsetvalue \phi}{\Lunit \upmu}}{|\Rtransarg{\muxmulevelsetvalue}|_{\gfour}}
(\Lunit \tander^N \Psi)^2
on RHS (<ref>).
The arguments we used in the proof of (<ref>)
also imply that all remaining terms on RHS (<ref>)
are either positive definite or can be absorbed into the positive definite terms by
exploiting the smallness of $\fundbootsmall$.
In total, these arguments yield the pointwise estimate
|\Rtransarg{\muxmulevelsetvalue}|_{\gfour} \times \mbox{RHS~\eqref{E:QNMULTIPLIER}} \geq 1.99 (\muX \tander^N \Psi)^2
From this estimate and the same arguments we used to prove (<ref>),
we conclude the desired lower bound (<ref>).

Proofs of (<ref>)–(<ref>):
To prove (<ref>),
we first use (<ref>) with
$f \eqdef \tander^N \Psi$
as well as the already proved coerciveness result (<ref>)
to deduce that
$\left\| \tander^N \Psi \right\|_{L^2(\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue})}^2
\leq
\left\| \tander^N \Psi \right\|_{L^2(\twoargroughtori{\timefunction_0,u}{\muxmulevelsetvalue})}^2
\hypersurfacecontrolwave_{N}(\timefunction,u)$.
From this estimate and the data estimate (<ref>),
we conclude (<ref>).
The estimates
(<ref>)–(<ref>) follow from a nearly identical argument based on the coerciveness results
and the data estimates (<ref>).

Proof of (<ref>):
We first use
to deduce that:

\begin{align} \label{E:FIRSTPROOFSTEPL2ESTIMATESFORWAVEVARIABLESONROUGHHYPERSURFACELOSSOFONEDERIVATIVE}
\left\| \tander^{N'} \Psi \right\|_{L^2\left(\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\lesssim
\initialsmall
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\left\| \argLrough{\muxmulevelsetvalue} \tander^{N'} \Psi \right\|_{L^2\left(\hypthreearg{\timefunction'}{[- \rightu,u]}{\muxmulevelsetvalue}\right)}
\, \mathrm{d} \timefunction.
\end{align}

using (<ref>),
we bound the time integral on RHS (<ref>) by
\lesssim
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{\hypersurfacecontrolwave_{N'}^{1/2}(\timefunction',u)}{|\timefunction'|^{1/2}}
\, \mathrm{d} \timefunction
which yields the first inequality stated in (<ref>).
The second inequality stated in (<ref>)
follows from the fact that $\hypersurfacecontrolwave_N(\timefunction,u)$ is increasing in its arguments.

§.§.§ Coerciveness of the spacetime integrals $\spacetimeintegralcontrolwave$

The next lemma complements Lemma <ref>
by exhibiting the coerciveness of the spacetime integrals
defined in
(<ref>) and (<ref>)
respectively; recall that these spacetime integrals
appear on the left-hand side of our energy identity (<ref>)
for the wave variables.
The key point is that the integrands on
RHSs (<ref>)–(<ref>)
are quantitatively positive in the region $\lbrace |u| \leq \interestingu \rbrace$
where the shock can form
and in particular, these integrands do not contain any degenerate factor of $\upmu$.
We fundamentally need the non-degenerate coerciveness
guaranteed by (<ref>)–(<ref>)
in order to control some of the error integrals that arise in our energy estimates.

Let $1 \leq N \leq \Ntop$ be an integer,
let $\spacetimeintegralcontrolwave_N(\timefunction,u)$
be the spacetime integrals defined in
(<ref>) and (<ref>),
and let $\mathfrak{P}^{(N)}$ be the set of order $N$ $\nullhyparg{u}$-tangential commutator operators from
Sect. <ref>.
Then the following lower bounds hold for
$(\timefunction,u) \in [\timefunction_0,\timefunctionboot)\times [-\rightu,\leftu]$,
where $\mathbf{1}_{[-\interestingu,\interestingu]} = \mathbf{1}_{[-\interestingu,\interestingu]}(u')$
denotes the characteristic function of the interval $[-\interestingu,\interestingu]$
and $\phi$ is the cut-off from Def. <ref>:

\begin{align} \label{E:COERCIVENESSOFSPACETIMEINTEGRAL}
\spacetimeintegralcontrolwave_N(\timefunction,u)
\geq
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{ \RRiemann,\LRiemann,v^2,v^3,\Ent \}}}
\frac{1}{4}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\mathbf{1}_{[-\interestingu,\interestingu]}(u')
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi
\right\rbrace
\left|\angrmD \tander^N \Psi \right|_{\gtorus}^2
\, \volMRoughCoordinates,
\\
\spacetimeintegralcontrolwavepartial_N(\timefunction,u)
\geq
\max_{\substack{\tander^N \in \mathfrak{P}^{(N)} \\ \Psi \in \{\LRiemann,v^2,v^3,\Ent \}}}
\frac{1}{4}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\mathbf{1}_{[-\interestingu,\interestingu]}(u')
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi
\right\rbrace
\left|\angrmD \tander^N \Psi \right|_{\gtorus}^2
\, \volMRoughCoordinates.
\label{E:COERCIVENESSOFPARTIALSPACETIMEINTEGRAL}
\end{align}

Let $f$ be a scalar function, and let $\spacetimeintegralcontrolwave[f](\timefunction,u)$
be the corresponding spacetime integral defined in (<ref>).
Using the estimate (<ref>), we
deduce that:

\begin{align} \label{E:PROOFCOERCIVENESSOFSPACETIMEINTEGRAL}
\spacetimeintegralcontrolwave[f](\timefunction,u)
\geq
\frac{1}{4}
\int_{\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u]}{\muxmulevelsetvalue}}
\left\lbrace
\mathbf{1}_{[-\interestingu,\interestingu]}(u')
\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}}
\muxmulevelsetvalue \phi
\right\rbrace
\left|\angrmD f \right|_{\gtorus}^2
\, \volMRoughCoordinates.
\end{align}

The desired bounds
now follow from (<ref>) and definitions
(<ref>) and (<ref>).

§ THE ELLIPTIC-HYPERBOLIC INTEGRAL IDENTITIES

In this section, we set up the top-order
elliptic-hyperbolic regularity theory for the transport-div-curl system satisfied
by $\vortrenormalized$ and $\GradEnt$, i.e., for the top-order derivatives of solutions to
(<ref>), (<ref>),
More precisely, for any $\Sigma_t$-tangent vectorfield $\SigmatTan$,
we derive coercive integral identities –  featuring error terms –
that are localized to spacetime
regions of the form $\twoargMrough{[\timefunction_1,\timefunction_2),[u_1,u_2]}{\muxmulevelsetvalue}$;
see Prop. <ref> for the main identity,
which, in view of the coerciveness guaranteed by Lemma <ref>,
yields $L^2$ spacetime control of
$\pmb{\partial} \SigmatTan$ (see definition (<ref>))
in terms of error terms.
In our forthcoming applications, we will apply the identity with
$\tander^{\Ntop} \vortrenormalized$ and $\tander^{\Ntop} \GradEnt$
in the role of $\SigmatTan$,
and we will use the special structure of the equations of
Theorem <ref> and commutator estimates
to control the error terms. Ultimately, this will yield (see Prop. <ref>)
spacetime $L^2$ control over the top-order terms
$\pmb{\partial} \tander^{\Ntop} \vortrenormalized$ and $\pmb{\partial} \tander^{\Ntop} \GradEnt$.

The integral identities are adaptations of the framework we developed in <cit.>
to handle the structure of the singular boundary of shock-forming solutions.
Unlike in <cit.>, our setup here avoids boundary integrals along
the characteristics $\nullhyparg{u}$. This allows us to avoid error integrals on $\nullhyparg{u}$
that involve the top-order derivatives of $\upmu$, which would have been uncontrollable.
Our setup also yields error terms whose singularity strength is controllable under the scope of our approach.
To achieve these goals, we rely on the following key ingredients:

* New well-constructed characteristic currents (see Sect. <ref>),
the analysis of which incorporates both the elliptic and the hyperbolic sub-structures in the equations of
Theorem <ref>.

* A delicate integration by parts identity to handle some difficult
boundary integrals along $\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$,
which takes into account the rough acoustic geometry and
the precise structure of the equations of Theorem <ref>;
see Lemma <ref>
for a differential version of the identity.
Ultimately, this leads
to an integral identity (see Prop. <ref>)
that provides control (see Prop. <ref>)
of the spacetime integrals
\int_{\twoargMrough{[\timefunction_0,\timefunction),[-\rightu,u]}{\muxmulevelsetvalue}}
\, \volMRoughCoordinates
\int_{\twoargMrough{[\timefunction_0,\timefunction),[-\rightu,u]}{\muxmulevelsetvalue}}
|\pmb{\p}\tander^{\Ntop} \GradEnt|^2
\, \volMRoughCoordinates
as well as the rough tori integrals
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\, \volroughtorus
\int_{\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}}
\, \volroughtorus
We stress that the delicate integration by parts mentioned above yields rough tori integrals with favorable signs,
and that our proof would not have closed if the integrals had the wrong signs.
The availability of good signs for these terms is a key aspect of
the framework developed in <cit.>.

Throughout this section, we will use the observations provided by
Remark <ref>.

§.§ Basic geometric constructions and definitions

In this section, we define some basic geometric objects that play a role
in our derivation of the localized integral identities.

Let $\uLunit$ be the $\gfour$-null vectorfield defined in (<ref>)
(which, in view of (<ref>), is transversal to the characteristics $\nullhyparg{u}$).

* We define the type $\binom{1}{1}$ projection tensorfield $\Nullhypersurfaceproject$
onto the characteristic hypersurfaces $\nullhyparg{u}$ as follows,
where $\updelta_{\beta}^{\ \alpha}$ denotes the Kronecker delta:

\begin{align} \label{E:NULLHYPERSURFACEPROJECTION}
\Nullhypersurfaceproject_{\beta}^{\ \alpha}
\eqdef
\updelta_{\beta}^{\ \alpha}
\frac{1}{2}
\uLunit^{\alpha}
\Lunit_{\beta}.
\end{align}

* Given any type $\binom{m}{n}$ spacetime tensorfield $\upxi$,
we define its $\nullhyparg{u}$-projection $\Nullhypersurfaceproject \upxi$
as follows:

\begin{align}
(\Nullhypersurfaceproject \upxi)_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_m}
\eqdef
\Nullhypersurfaceproject_{\widetilde{\alpha}_1}^{\ \alpha_1}
\cdots
\Nullhypersurfaceproject_{\widetilde{\alpha}_m}^{\ \alpha_m}
(\gfour^{-1})^{\widetilde{\beta}_1 \widetilde{\gamma}_1}
\gfour_{\beta_1 \gamma_1}
\Nullhypersurfaceproject_{\widetilde{\beta}_1}^{\ \gamma_1}
\cdots
(\gfour^{-1})^{\widetilde{\beta}_n \widetilde{\gamma}_n}
\gfour_{\beta_n \gamma_n}
\Nullhypersurfaceproject_{\widetilde{\beta}_n}^{\ \gamma_n}
\upxi_{\widetilde{\gamma}_1 \cdots \widetilde{\gamma}_n}^{\widetilde{\alpha}_1 \cdots \widetilde{\alpha}_m}.
\label{E:PROJECTIONOFTENSORONTONULLHYPERSURFACE}
\end{align}

* We say that a spacetime tensorfield $\upxi$ is $\nullhyparg{u}$-tangent
if $\Nullhypersurfaceproject \upxi = \upxi$.

The type $\binom{0}{2}$ tensorfield
\gfour_{\alpha \widetilde{\alpha}}
\Nullhypersurfaceproject_{\beta}^{\ \widetilde{\alpha}}
$ is not symmetric.
This is the reason that in equation (<ref>),
there are factors of $\gfour^{-1}$ and $\gfour$ and we
were careful about the placement of the indices on $\Nullhypersurfaceproject$
that are contracted against the lower indices of $\upxi$,
unlike in equations

We respectively define $\hfour$ and $\hfour^{-1}$
to be the type $\binom{0}{2}$ and $\binom{2}{0}$ tensorfields with the following
Cartesian components:

\begin{align}
\hfour_{\alpha \beta}
& \eqdef
\gfour_{\alpha \beta}
2 \Transport_{\alpha} \Transport_{\beta},
\label{E:RIEMANNIANACOUSTICALMETRIC}
\\
(\hfour^{-1})^{\alpha \beta}
& \eqdef (\gfour^{-1})^{\alpha \beta}
2 \Transport^{\alpha} \Transport^{\beta}.
\label{E:INVERSERIEMANNIANACOUSTICALMETRIC}
\end{align}

By using $\hfour$ and $\hfour^{-1}$ to lower and raise the indices on the projection tensorfield $\Nullhypersurfaceproject$,
we also define the type $\binom{0}{2}$ tensorfield $\Nullhypersurfacemetric$
and the type $\binom{2}{0}$ tensorfield $\Nullhypersurfaceinversemetric$
as follows:

\begin{align}
\Nullhypersurfacemetric_{\alpha \beta}
& \eqdef \hfour_{\alpha \sigma} \Nullhypersurfaceproject_{\beta}^{\ \sigma},
\label{E:NULLHYPERSURFACESRIEMANNIANMETRIC}
\\
\Nullhypersurfaceinversemetric^{\alpha \beta}
& \eqdef (\hfour^{-1})^{\alpha \sigma} \Nullhypersurfaceproject_{\sigma}^{\ \beta}.
\label{E:NULLHYPERSURFACESINVERSERIEMANNIANMETRIC}
\end{align}

In the next lemma,
we exhibit some basic properties of the tensorfields from
Defs. <ref> and <ref>.
Later on, we will use the positive definiteness of
$\hfour$ and $\hfour^{-1}$ (which are revealed by the lemma)
to exhibit the coerciveness properties of various energies

The tensorfield $\Nullhypersurfaceproject_{\beta}^{\ \alpha}$ is a projection onto $\nullhyparg{u}$ in the following sense:
$\Nullhypersurfaceproject \uLunit =  0$,
while if $\Singletan$ is a $\nullhyparg{u}$-tangent vectorfield,
then $\Nullhypersurfaceproject \Singletan = \Singletan$.

Moreover, the tensorfields $\Nullhypersurfacemetric$ and $\Nullhypersurfaceinversemetric$ are
symmetric and positive semi-definite,
and $\Nullhypersurfacemetric$ restricts to a Riemannian metric on $\nullhyparg{u}$.

In addition, for all pairs of $\Sigma_t$-tangent vectorfields $(X,Y)$,
we have $\hfour(X,Y) = \gfour(X,Y)$, and if $\SigmatTan$ is a $\Sigma_t$-tangent vectorfield, then
$\hfour(\SigmatTan,\Transport) = 0$.
Moreover, we have $\hfour(\Transport,\Transport) = 1$,
and $\hfour$ is a Riemannian metric on spacetime.
Furthermore, following identity holds:

\begin{align} \label{E:HINVERSEISTHEINVERSOFH}
(\hfour^{-1})^{\alpha \gamma}
\hfour_{\gamma \beta}
& = \updelta_{\beta}^{\alpha},
\end{align}

where $\updelta_{\beta}^{\alpha}$ is the Kronecker delta. That is, $\hfour^{-1}$ is in fact
the inverse metric of $\hfour$.

In addition, the following identities hold, where
$g$ and $g^{-1}$ are respectively the first fundamental form of $\Sigma_t$
and inverse first fundamental form of $\Sigma_t$ from Def. <ref>:

\begin{align}
\hfour_{\alpha \beta}
g_{\alpha \beta}
\Transport_{\alpha} \Transport_{\beta},
\label{E:RIEMANNIANACOUSTICALMETRICINTERMSOFFIRSTFUNDOFSIGMATANDTRANSPORT}
\\
(\hfour^{-1})^{\alpha \beta}
(g^{-1})^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}.
\label{E:INVERSERIEMANNIANACOUSTICALMETRICINTERMSOFFIRSTFUNDOFSIGMATANDTRANSPORT}
\end{align}

Finally, the following identities hold:

\begin{align}
\Nullhypersurfacemetric_{\alpha \beta}
&  =
\hfour_{\alpha \beta}
\frac{1}{2}
\Lunit_{\alpha} \Lunit_{\beta}
\gtorus_{\alpha \beta}
\frac{1}{2}
\uLunit_{\alpha} \uLunit_{\beta},
\label{E:NULLHYPMETRICSINTERMSOFHFOUR}
\\
\Nullhypersurfaceinversemetric^{\alpha \beta}
= (\hfour^{-1})^{\alpha \beta}
\frac{1}{2}
\uLunit^{\alpha} \uLunit^{\beta}
(\gtorus^{-1})^{\alpha \beta}
\frac{1}{2}
\Lunit^{\alpha} \Lunit^{\beta},
\label{E:NULLHYPERSURFACEINVERSERIEMANNIANMETRICINTERMSOFNULLFRAME}
\\
\Nullhypersurfaceinversemetric^{\alpha \gamma}
\Nullhypersurfacemetric_{\gamma \beta}
& =
\updelta_{\beta}^{\alpha}
\frac{1}{2}
\uLunit^{\alpha}
\Lunit_{\beta},
\label{E:NULLHYPINVERSEMETRICCONTRACTEDWITHNULLHYPMETRIC}
\\
\Nullhypersurfacemetric_{\alpha \gamma}
\Nullhypersurfaceinversemetric^{\gamma \beta}
& =
\updelta_{\alpha}^{\beta}
\frac{1}{2}
\uLunit_{\alpha}
\Lunit^{\beta},
\label{E:NULLHYPMETRICCONTRACTEDWITHNULLHYPINVERSEMETRIC}
\\
\Nullhypersurfaceinversemetric^{\alpha \beta}
= (\hfour^{-1})^{\alpha \gamma}
(\hfour^{-1})^{\beta \delta}
\Nullhypersurfacemetric_{\gamma \delta},
\label{E:NULLHYPSURFACEINVERSEMETRICFORMEDBYRAISINGINDICESWITHHINVERSEMETRIC}
\end{align}

where $\gtorus$ and $\gtorus^{-1}$ are respectively the first fundamental form and inverse first fundamental form
of the smooth tori $\ell_{t,u} = \Sigma_t \cap \nullhyparg{u}$
from Def. <ref>.

The facts that
$\Nullhypersurfaceproject \uLunit =  0$,
while if $\Singletan$ is $\nullhyparg{u}$-tangent,
then $\Nullhypersurfaceproject \Singletan = \Singletan$
follow from (<ref>)–(<ref>)
and the fact that $\Lunit$ is $\gfour$-orthogonal to $\nullhyparg{u}$.

(<ref>) follows from a straightforward computation based on
definitions (<ref>)–(<ref>)
and the identity (<ref>).

(<ref>) follows from
definition (<ref>) and the identity (<ref>).
(<ref>) follows from
definition (<ref>) and
the identity (<ref>).

The remaining properties of $\hfour$ stated in the lemma follow in a straightforward fashion from
(<ref>) and (<ref>).

To prove the first equality in (<ref>),
we substitute the definition of
$\hfour_{\alpha \sigma}$ and the definition (<ref>)
$\Nullhypersurfaceproject_{\beta}^{\ \sigma}$ into
RHS (<ref>)
and carry out straightforward algebraic computations
using Lemmas <ref> and <ref>
(in particular, we use (<ref>)).
The second equality in (<ref>) follows from similar
arguments, where we in particular use
(<ref>) also yields the symmetry of $\Nullhypersurfacemetric$.
The identities in (<ref>)
as well as the symmetry of $\Nullhypersurfaceinversemetric$
follow from similar arguments based on definitions
and (<ref>).

The fact that $\Nullhypersurfacemetric$ restricts to a Riemannian metric on $\nullhyparg{u}$
follows from (<ref>),
Lemmas <ref> and <ref>,
the fact that the tangent space of $\nullhyparg{u}$ is the direct sum of
the tangent space of $\ell_{t,u}$ and the span of $\Lunit$,
the fact that $\gtorus_{\alpha \beta}$ is
positive definite on $\ell_{t,u}$-tangent vectorfields and satisfies $\gtorus(\Lunit,\cdot) = 0$.
The positive semi-definiteness of
$\Nullhypersurfacemetric$ and $\Nullhypersurfaceinversemetric$
follows from the second identities in
and the positive semi-definiteness of $\gtorus_{\alpha \beta}$ and $(\gtorus^{-1})^{\alpha \beta}$.

The identity (<ref>)
follows from definitions
the identity (<ref>),
and the symmetry of $\Nullhypersurfacemetric$ and $\Nullhypersurfaceinversemetric$.

§.§ Additional derivatives operators, pointwise norms, and comparison estimates

In this section, we define some additional derivative operators and pointwise norms,
and we establish some simple comparison estimates. Later, we will use them in our analysis of
the terms in the elliptic-hyperbolic integral identity provided by Prop. <ref>.

§.§.§ The Cartesian gradient of $\upxi$ and $|Z \SigmatTan|_g$

* Given a type $\binom{m}{n}$ spacetime tensorfield $\upxi$,
we define its Cartesian gradient
$\pmb{\partial} \upxi$ to be the type $\binom{m}{n+1}$ spacetime tensorfield with
the following Cartesian components:

\begin{align} \label{E:CARTESIANGRADIENTOFTENSORFIELD}
(\pmb{\partial} \upxi)_{\beta_1 \beta_2 \cdots \beta_{n+1}}^{\alpha_1 \cdots \alpha_m}
& \eqdef
\p_{\beta_1} \upxi_{\beta_2 \cdots \beta_{n+1}}^{\alpha_1 \cdots \alpha_m}.
\end{align}

* Let $\SigmatTan$ be a $\Sigma_t$-tangent vectorfield and let $Z$ be a spacetime vectorfield.
Relative to the Cartesian coordinates,
we define (see Remark <ref>) $|Z \SigmatTan|_g \geq 0$
as follows:

\begin{align} \label{E:SIGMATPOINTWISENOROMOFZDERIVATIVEOFSIGMATTANGENTVECTORFIELD}
|Z \SigmatTan|_g^2
& \eqdef g_{ab} (Z \SigmatTan^a) Z \SigmatTan^b,
\end{align}

where as usual $Z \SigmatTan^a = Z^{\alpha} \partial_{\alpha} \SigmatTan^a$.

Note that $|Z \SigmatTan|_g$ is the $|\cdot|_g$ norm of the $\Sigma_t$-tangent vectorfield with the Cartesian spatial components
$Z \SigmatTan^i$, $i=1,2,3$.

§.§.§ The $\hfour$-norm of tensorfields

Recall that in Lemma <ref>, we showed that $\hfour$ is
a Riemannian metric on spacetime.
If $\upxi$ is a type $\binom{m}{n}$ spacetime tensorfield,
then we define $|\upxi|_{\hfour}\geq 0$ by:

\begin{align} \label{E:SQUAREPOINTWISENORMWITHRESPECTTORIEMANNIANACOUSTICALMETRIC}
\eqdef
\hfour_{\alpha_1 \widetilde{\alpha}_1}
\cdots
\hfour_{\alpha_m \widetilde{\alpha}_m}
(\hfour^{-1})^{\beta_1 \widetilde{\beta}_1}
\cdots
(\hfour^{-1})^{\beta_n \widetilde{\beta}_n}
\upxi_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_n}
\upxi_{\widetilde{\beta}_1 \cdots \widetilde{\beta}_n}^{\widetilde{\alpha}_1 \cdots \widetilde{\alpha}_m}.
\end{align}

§.§.§ Pointwise comparison results and the $\hfour$-size of
$\Nullhypersurfaceproject$, $\Nullhypersurfacemetric$, and $\Nullhypersurfaceinversemetric$

For any type $\binom{m}{n}$ spacetime tensorfield
$\upxi^{\alpha_1 \cdots \alpha_m}_{\beta_1 \cdots \beta_n}$,
the following comparison estimates
hold relative to the Cartesian coordinates on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$:

\begin{align} \label{E:HRIEMANNIANMETRICNORMCOMPARABLETOEUCLIDEANNORM}
& \approx
\sum_{\substack{0 \leq \alpha_1, \cdots, \alpha_m \leq 3 \\ 0 \leq \beta_1, \cdots, \beta_n \leq 3}}
\left|\upxi_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_m} \right|,
\\
|\pmb{\partial} \upxi|_{\hfour}
& \approx
\sum_{\substack{0 \leq \alpha_1, \cdots, \alpha_m \leq 3 \\ 0 \leq \beta_1, \cdots, \beta_n \leq 3 \\ 0 \leq \gamma \leq 3}}
\left|\partial_{\gamma}
\upxi_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_m} \right|.
\label{E:CARTERSIANDERIVATIVESHRIEMANNIANMETRICNORMCOMPARABLETOEUCLIDEANNORM}
\end{align}

Moreover, if $\SigmatTan$ is a $\Sigma_t$-tangent vectorfield and $Z$ is a spacetime vectorfield,
then the following comparison estimates
hold relative to the Cartesian coordinates
on $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$:

\begin{align}
& \approx
\sum_{a=1,2,3}
\left| \SigmatTan^a \right|,
\label{E:SIGMATTANGENTVECTORFIELDGNORMAPPROXIMATEDBYCARETSIANCOMPONENTNORMS}
\\
|Z \SigmatTan|_g
& \approx
\sum_{a=1,2,3}
\left| Z \SigmatTan^a \right|,
\label{E:ZDERIVATIVEOFSIGMATTANGENTVECTORFIELDGNORMAPPROXIMATEDBYCARETSIANCOMPONENTNORMS}
\\
|\pmb{\partial} \SigmatTan|_{\hfour}
& \approx
\sum_{\substack{\alpha = 0,1,2,3 \\ a=1,2,3}}
\left|\partial_{\alpha} \SigmatTan^a \right|
\approx
\sum_{\alpha = 0,1,2,3}
|\partial_{\alpha} \SigmatTan|_g
\approx
|\Transport \SigmatTan|_g
\sum_{a=1,2,3}
|\partial_a \SigmatTan|_g.
\label{E:TRANSPORTANDCARTERSIANDERIVATIVESHRIEMANNIANMETRICNORMCOMPARABLETOEUCLIDEANNORM}
\end{align}

Finally, the following identities hold:

\begin{align}  \label{E:HRIEMANNIANNORMOFCHARACTERISTICTENSORFIELDSISROOT3}
\sqrt{3}.
\end{align}

We prove (<ref>) when $\upxi$ is a type $\binom{1}{0}$ tensorfield;
the case of general type $\binom{m}{n}$ tensorfields can be handled through similar arguments.
To proceed, we first use
and (<ref>)
to derive the following identity relative to the Cartesian coordinates:

\begin{align} \label{E:HNORMSQUAREDOFVECTORFIELDEXPANDEDOUT}
\hfour_{\alpha \beta} \upxi^{\alpha} \upxi^{\beta}
\Speed^{-2} \sum_{a=1,2,3} (\upxi^a)^2
\left\lbrace
\Speed^{-2}
\sum_{a=1,2,3} (v^a)^2
\right\rbrace
2 \Speed^{-2}
\sum_{a=1,2,3}
v^a \upxi^a \upxi^0.
\end{align}

From the bootstrap assumptions and (<ref>),
we deduce that
$|v^a| \lesssim 1$ and $\Speed \approx 1$. From these estimates and (<ref>),
we conclude that $|\upxi|_{\hfour} \lesssim \sum_{\alpha = 0,1,2,3} |\upxi^{\alpha}|$.
To prove the reverse inequality, we note that
the Cauchy–Schwarz inequality and
Young's inequality imply that the cross term
2 \Speed^{-2}
v^a \upxi^a \upxi^0$ on RHS (<ref>)
is bounded in magnitude by
\leq
\Speed^{-2} \sum_{a=1,2,3} (\upxi^a)^2
\Speed^{-2}
\sum_{a=1,2,3} (v^a)^2
From this bound, (<ref>),
and the estimate $\Speed \approx 1$,
it follows that:

\begin{align} \label{E:HNORMCONTROLSCARTESIAN0COMPONENTS}
|\upxi|_{\hfour}^2 \geq (\upxi^0)^2.
\end{align}

Moreover, the Cauchy–Schwarz inequality and
Young's inequality imply that the cross term
2 \Speed^{-2}
v^a \upxi^a \upxi^0$
is also bounded in magnitude by:

\[
\leq
\left\lbrace
\frac{\Speed^{-2} \sum_{d=1,2,3} (v^d)^2}{1 + \Speed^{-2} \sum_{b=1,2,3} (v^b)^2}
\right\rbrace
\times
\Speed^{-2} \sum_{a=1,2,3} (\upxi^a)^2
\left\lbrace
\Speed^{-2}
\sum_{a=1,2,3} (v^a)^2
\right\rbrace
\]

From this bound, (<ref>),
and the aforementioned estimates $|v^a| \lesssim 1$ and $\Speed \approx 1$,
it follows that:

\begin{align} \label{E:HNORMCONTROLSCARTESIANSPATIALCOMPONENTS}
\geq
\left\lbrace
\frac{1}{1 + \Speed^{-2} \sum_{b=1,2,3} (v^b)^2}
\right\rbrace
\times
\Speed^{-2}
\sum_{a=1,2,3} (\upxi^a)^2
\gtrsim
\sum_{a=1,2,3} (\upxi^a)^2.
\end{align}

Combining (<ref>) and (<ref>),
we see that
$\sum_{\alpha = 0,1,2,3} |\upxi^{\alpha}| \lesssim |\upxi|_{\hfour}$,
which completes the proof of (<ref>).
The comparison estimates
follow from similar arguments, and we omit the straightforward details.

follows as a special case of (<ref>)
with $\pmb{\partial} \upxi$ in the role of $\upxi$.

We now prove (<ref>).
First, using
and (<ref>),
we compute that
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfacemetric_{\beta \alpha}
\updelta_{\alpha}^{\alpha}
\frac{1}{2} \uLunit_{\alpha} \Lunit^{\alpha}
4 - 1
= 3$,
as is desired.
Next, we use
Lemma <ref>
to compute that:

\begin{align}
\begin{split} \label{E:INTERMEDIATESTEPINPROOFOFHRIEMANNIANNORMOFCHARACTERISTICTENSORFIELDS}
(\hfour^{-1})^{\alpha \beta}
(\hfour^{-1})^{\gamma \delta}
\Nullhypersurfacemetric_{\alpha \gamma}
\Nullhypersurfacemetric_{\beta \delta}
\Nullhypersurfaceproject_{\gamma}^{\ \beta}
\Nullhypersurfaceproject_{\beta}^{\ \gamma}
\left\lbrace
\updelta_{\gamma}^{\beta}
\frac{1}{2} \Lunit^{\beta} \uLunit_{\gamma}
\right\rbrace
\left\lbrace
\updelta_{\beta}^{\gamma}
\frac{1}{2} \Lunit^{\gamma} \uLunit_{\beta}
\right\rbrace
\\
= 4
\Lunit^{\alpha} \uLunit_{\alpha}
\frac{1}{4} (\Lunit^{\alpha} \uLunit_{\alpha})^2
= 3.
\end{split}
\end{align}

A similar calculation based on (<ref>)
yields that $|\Nullhypersurfaceinversemetric|_{\hfour}^2 = 3$.
We have therefore proved (<ref>).

§.§ The coercive elliptic-hyperbolic quadratic form and its coerciveness

§.§.§ The coercive elliptic-hyperbolic quadratic form

In the next definition, we introduce the solution-adapted quadratic form $\ellipticCoerciveQuadratic$
that we will use to control the top-order derivatives of the specific vorticity and entropy gradient.

Let $\smoothtorusproject_b^{\ a}$ denote the $\Sigma_t$-components of the $\ell_{t,u}$ projection tensorfields defined in
and let
be the tensorfields defined in
Let $\SigmatTan$ be a $\Sigma_t$-tangent vectorfield. We define
$\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]$
to be the following quadratic form associated to $\SigmatTan$:

\begin{align}
\begin{split} \label{E:NULLHYPERSURFACEADAPTEDCOERCIVEQUADRATICFORM}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\eqdef
\left(
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}
\right)
\left(
\Nullhypersurfacemetric_{\gamma \delta}
\Transport_{\gamma} \Transport_{\delta}
\right)
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}
\\
& \ \
\frac{1}{16}
\left\lbrace
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2.
\end{split}
\end{align}

§.§.§ The coerciveness of the elliptic-hyperbolic quadratic form

In the next lemma, we exhibit the coerciveness of
$\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]$.

On $\twoargMrough{[\timefunction_0,\timefunctionboot),[- \rightu,\leftu]}{\muxmulevelsetvalue}$,
the quadratic form
$\ellipticCoerciveQuadratic$ from Def. <ref>
is quantitatively positive definite
on the space of Cartesian gradients of $\Sigma_t$-tangent vectorfields $\SigmatTan$
in the following sense,
|\pmb{\partial} \SigmatTan|_{\hfour}^2
\eqdef (\hfour^{-1})^{\alpha \beta} \hfour_{\gamma \delta} (\partial_{\alpha} \SigmatTan^{\gamma}) \partial_{\beta} \SigmatTan^{\delta}

\begin{align} \label{E:COERCIVENESSOFELLIPTICHYPERBOLICQUADRATICFORM}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& \approx
|\pmb{\partial} \SigmatTan|_{\hfour}^2
\approx
\sum_{\alpha = 0}^3 |\partial_{\alpha} \SigmatTan|_g^2.
\end{align}

Throughout the proof, we will use the observations made in Remark <ref>.
To start, we note that Young's inequality implies
that the terms
\frac{1}{16}
\left\lbrace
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2
on RHS (<ref>)
are bounded in magnitude

\begin{align}
\begin{split} \label{E:PROOFSTEP1COERCIVENESSOFELLIPTICHYPERBOLICQUADRATICFORM}
\leq
\frac{4}{16}
[3 (\Transport \SigmatTan^{\alpha}) \uLunit_{\alpha}]^2
\frac{4}{3}
\times
\frac{1}{16}
\left\lbrace
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2
\\
\leq
\frac{9}{4}
[(\Transport \SigmatTan^{\alpha}) \uLunit_{\alpha}]^2
\frac{1}{6}
[(\Lunit \SigmatTan^{\alpha}) \uLunit_{\alpha}]^2
\frac{1}{6}
(\smoothtorusproject_b^{\ a} \partial_a \SigmatTan^b)^2.
\end{split}
\end{align}

Moreover, the Cauchy–Schwarz inequality and the fact that $|\smoothtorusproject|_{\gtorus}^2 = \mytr_{\gtorus} \gtorus = 2$
together imply that:

\begin{align} \label{E:PROOFSTEP2COERCIVENESSOFELLIPTICHYPERBOLICQUADRATICFORM}
\frac{1}{6} (\smoothtorusproject_b^{\ a} \partial_a \SigmatTan^b)^2
& \leq \frac{1}{3} (\gtorus^{-1})^{ab} \gtorus_{cd} (\partial_a \SigmatTan^c) \partial_b \SigmatTan^d
\eqdef \frac{1}{3} |\partial \SigmatTan|_{\gtorus}^2.
\end{align}

we compute that:

\begin{align} \label{E:PROOFSTEP3COERCIVENESSOFELLIPTICHYPERBOLICQUADRATICFORM}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\approx
\left(
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}
\right)
\left(
\Nullhypersurfacemetric_{\gamma \delta}
\Transport_{\gamma} \Transport_{\delta}
\right)
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}.
\end{align}

From (<ref>),
the fact that (in Cartesian coordinates)
\Transport_{\gamma} \partial_{\alpha} \SigmatTan^{\gamma}
\partial_{\alpha} \SigmatTan^0
= 0$,
the identities $\Lunit = \Transport - X$ and $\uLunit = \Transport + X$,
the identities $g_{ab} = \gtorus_{ab} + X_a X_b$ and $(g^{-1})^{ab} = (\gtorus^{-1})^{ab} + X^a X^b$
proved in
and the identities
it follows that:

\begin{align}
\begin{split} \label{E:PROOFSTEP4COERCIVENESSOFELLIPTICHYPERBOLICQUADRATICFORM}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\approx
|\Transport \SigmatTan|_g^2
|\Lunit \SigmatTan|_g^2
(\gtorus^{-1})^{ab} g_{cd} (\partial_a \SigmatTan^c) \partial_b \SigmatTan^d
\approx
|\Transport \SigmatTan|_g^2
|X \SigmatTan|_g^2
(\gtorus^{-1})^{ab} g_{cd} (\partial_a \SigmatTan^c) \partial_b \SigmatTan^d
\\
\approx
|\Transport \SigmatTan|_g^2
(g^{-1})^{ab} g_{cd} (\partial_a \SigmatTan^c) \partial_b \SigmatTan^d
\approx
|\pmb{\partial} \SigmatTan|_{\hfour}^2.
\end{split}
\end{align}

(<ref>) implies
the first “$\approx$” in (<ref>).
The second “$\approx$” in (<ref>)
then follows from Lemma <ref>.

§.§ The characteristic currents

The $\nullhyparg{u}$-tangent vectorfields $\ehcurrent[\SigmatTan,\pmb{\p}\SigmatTan]$ in the next definition play a key role in our analysis.
In our proof of Prop. <ref>,
use them for bookkeeping when integrating by parts.
We sometimes refer to the $\ehcurrent[\SigmatTan,\pmb{\p}\SigmatTan]$
as “characteristic currents” since they are tangent to $\nullhyparg{u}$,
or “elliptic-hyperbolic currents” since they are the basic ingredient for the elliptic-hyperbolic integral identities.
In our prior work <cit.>, we used related – but distinct –
$\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$-tangent currents
to derive elliptic-hyperbolic integral identities.
Compared to the currents in <cit.>,
the ones featured in the next definition are better adapted to the structure of the singularity in the sense
that they do not generate any critical-strength error terms in our top-order $L^2$ estimates.
Moreover, when we integrate by parts over the spacetime region $\twoargMrough{[\timefunction_1,\timefunction_2),[u_1,u_2]}{\muxmulevelsetvalue}$,
the $\nullhyparg{u}$-tangency of the $\ehcurrent[\SigmatTan,\pmb{\p}\SigmatTan]$
allows us to avoid boundary integrals along $\nullhyparg{u}$. This is important because
some of the acoustic geometry error terms (such as the top-order derivatives of $\upmu$)
do not have sufficiently regularity to be controlled in $L^2$ along $\nullhyparg{u}$.

§.§.§ The definition of the $\nullhyparg{u}$-tangent characteristic current

Let $\SigmatTan$ be a $\Sigma_t$-tangent vectorfield.
We define the characteristic current
to be the vectorfield $\ehcurrent = \ehcurrent[\SigmatTan,\pmb{\partial} \SigmatTan]$
with the following components, $(\alpha = 0,1,2,3)$:

\begin{align} \label{E:PUTANGENTELLIPTICHYPERBOLICCURRENT}
\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\SigmatTan^{\gamma}
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha} \partial_{\lambda} \SigmatTan^{\kappa}
\SigmatTan^{\gamma} \Nullhypersurfaceproject_{\gamma}^{\ \alpha}
\Nullhypersurfaceproject_{\lambda}^{\ \kappa}
\partial_{\kappa} \SigmatTan^{\lambda}.
\end{align}

Since for any vectorfield $Z$,
the vectorfield $\Nullhypersurfaceproject_{\beta}^{\ \alpha} Z^{\beta}$
is $\nullhyparg{u}$-tangent,
it follows from (<ref>)
that indeed, $\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]$
is $\nullhyparg{u}$-tangent.

§.§.§ The covariant divergence identity satisfied by the elliptic-hyperbolic current

In the next lemma, we provide the main covariant divergence identity
satisfied by the current $\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]$
from Def. <ref>.
The identity forms the starting point for the divergence-theorem based proof of Prop. <ref>.

$\SigmatTan$ be a $\Sigma_t$-tangent vectorfield,
let $\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]$
be the quadratic form defined by (<ref>),
and let $\weight$ be a “weight function.”
Then the following identity holds relative to the Cartesian coordinates,
where $\Dfour$ is the Levi-Civita connection of $\gfour$:

\begin{align}
\begin{split} \label{E:COVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\weight \ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& = \Dfour_{\alpha} (\weight \ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan])
\weight
\mathfrak{J}_{(\textnormal{Antisymmetric})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\weight
\mathfrak{J}_{(\textnormal{Div})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\\
& \ \
\mathfrak{J}_{(\pmb{\partial} \weight)}[\SigmatTan,\pmb{\partial} \SigmatTan]
\weight
\mathfrak{J}_{(\textnormal{Absorb-1})}[\SigmatTan,\pmb{\partial} \SigmatTan]
\weight
\mathfrak{J}_{(\textnormal{Absorb-2})}[\SigmatTan,\pmb{\partial} \SigmatTan]
\\
& \ \
\weight
\mathfrak{J}_{(\textnormal{Material})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
\weight
\mathfrak{J}_{(\textnormal{Null Geometry})}[\SigmatTan,\pmb{\partial} \SigmatTan],
\end{split}
\end{align}

where with
$(\mathrm{d} \SigmatTan_{\flat})_{\alpha \beta}
\eqdef \partial_{\alpha} \SigmatTan_{\beta}
\partial_{\beta} \SigmatTan_{\alpha}
we have:

\begin{align} \label{E:ANTISYMMETRICNULLCURRENTSPACETIMERRORTERM}
\mathfrak{J}_{(\textnormal{Antisymmetric})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\frac{1}{2}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\mathrm{d} \SigmatTan_{\flat})_{\alpha \gamma}
(\mathrm{d} \SigmatTan_{\flat})_{\beta \delta},
\\
\mathfrak{J}_{(\textnormal{Div})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\frac{9}{16}
(\partial_a \SigmatTan^a)^2,
\label{E:DIVERGENCENULLCURRENTSPACETIMERRORTERM}
\\
\mathfrak{J}_{(\pmb{\partial} \weight)}[\SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan] \partial_{\alpha} \weight,
\label{E:DERIVATIVEOFWEIGHTNULLCURRENTSPACETIMERRORTERM}
\\
\begin{split} 		\label{E:ABSORBABLEANTISYMMETRICANDDIVERGENCENULLCURRENTSPACETIMERRORTERM}
\mathfrak{J}_{(\textnormal{Absorb-1})}[\SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
\SigmatTan^{\kappa}
(\partial_{\alpha} \gfour_{\delta \kappa})
(\mathrm{d} \SigmatTan_{\flat})_{\beta \gamma}
\\
& \ \
\frac{3}{8}
(\partial_a \SigmatTan^a)
\left\lbrace
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace,
\end{split}
\\
\mathfrak{J}_{(\textnormal{Absorb-2})}[\SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\Chfour_{\alpha \ \beta}^{\ \alpha}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
(\partial_{\beta} \gfour_{\delta \kappa})
\SigmatTan^{\kappa}
\partial_{\alpha} \SigmatTan^{\gamma}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
\SigmatTan^{\kappa}
(\partial_{\delta} \gfour_{\beta \kappa})
\partial_{\alpha} \SigmatTan^{\gamma},
\label{E:EASYABSORBABLENULLCURRENTSPACETIMERRORTERM}
\\
\mathfrak{J}_{(\textnormal{Material})}[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\Nullhypersurfacemetric_{\alpha \beta}
(\Transport \SigmatTan^{\alpha})
\Transport \SigmatTan^{\beta},
\label{E:QUADRATICINMATERIALDERIVATIVESNULLCURRENTSPACETIMERRORTERM}
\\
\mathfrak{J}_{(\textnormal{Null Geometry})}[\SigmatTan,\pmb{\partial} \SigmatTan]
& \eqdef
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\right)
\right\rbrace
\partial_{\lambda} \SigmatTan^{\kappa}
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \alpha}
\Nullhypersurfaceproject_{\lambda}^{\ \kappa}
\right)
\right\rbrace
\partial_{\kappa}
\SigmatTan^{\lambda},
\label{E:DERIVATIVESOFNULLGEOMETRYNULLCURRENTSPACETIMERRORTERM}
\end{align}

and on RHS (<ref>),
$\Chfour_{\alpha \ \beta}^{\ \gamma}
\eqdef \frac{1}{2} (\gfour^{-1})^{\gamma \delta}
\left(\partial_{\alpha} \gfour_{\delta \beta}
\partial_{\beta} \gfour_{\alpha \delta}
\partial_{\delta} \gfour_{\alpha \beta}
\right)$
are the Cartesian Christoffel symbols of $\gfour$.

Throughout the proof,
we silently use the simple fact that $\Transport_{\alpha} \SigmatTan^{\alpha} = - \SigmatTan^0 = 0$
and thus $\SigmatTan_{\alpha} \eqdef \gfour_{\alpha \beta} \SigmatTan^{\beta} = \hfour_{\alpha \beta} \SigmatTan^{\beta}$
$(\gfour^{-1})^{\alpha \beta} \SigmatTan_{\beta} = (\hfour^{-1})^{\alpha \beta} \SigmatTan_{\beta}$.
We also silently use the simple identity
$\partial_{\alpha} (\hfour^{-1})^{\beta \gamma}
- (\hfour^{-1})^{\beta \beta'}
(\hfour^{-1})^{\gamma \gamma'}
\partial_{\alpha} \hfour_{\beta' \gamma'}$
and the fact that in Cartesian coordinates,
$\partial_{\alpha} \gfour_{\beta \gamma} = \partial_{\alpha} \hfour_{\beta \gamma}$
(this follows easily from (<ref>) and (<ref>)).
Moreover, we frequently relabel indices from line to line whenever convenient.
We will also silently use the observations made in Remark <ref>
and the symmetry of $\Nullhypersurfacemetric$
shown in Lemma <ref>.

We start by using (<ref>)
to compute that relative to the Cartesian coordinates, we have:

\begin{align}
\begin{split} \label{E:FIRSTSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Dfour_{\alpha} \ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
& =
\partial_{\alpha} \ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
\Chfour_{\alpha \ \beta}^{\ \alpha}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
\\
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\lambda} \SigmatTan^{\kappa}
(\Nullhypersurfaceproject_{\beta}^{\ \alpha}
\partial_{\alpha} \SigmatTan^{\beta})^2
\\
& \ \
\Chfour_{\alpha \ \beta}^{\ \alpha}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\right)
\right\rbrace
\partial_{\lambda} \SigmatTan^{\kappa}
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \alpha}
\Nullhypersurfaceproject_{\lambda}^{\ \kappa}
\right)
\right\rbrace
\partial_{\kappa}
\SigmatTan^{\lambda},
\end{split}
\end{align}

where we stress that due to cancellations, the second derivatives of $\SigmatTan$ are absent from
RHS (<ref>).
Next, we compute the following identity:

\begin{align}
\begin{split}  \label{E:SECONDSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\partial_{\lambda} \SigmatTan^{\kappa}
& =
\partial_{\lambda}[(\hfour^{-1})^{\kappa \sigma} \SigmatTan_{\sigma}]
(\hfour^{-1})^{\kappa \sigma}
\partial_{\lambda} \SigmatTan_{\sigma}
[\partial_{\lambda}(\hfour^{-1})^{\kappa \sigma}]
\SigmatTan_{\sigma}
\\
& =
(\hfour^{-1})^{\kappa \sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\lambda} \SigmatTan_{\sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
[\partial_{\lambda}(\hfour^{-1})^{\kappa \sigma}]
\SigmatTan_{\sigma}
\\
(\hfour^{-1})^{\kappa \sigma}
\partial_{\sigma}
(\hfour_{\lambda \lambda'} \SigmatTan^{\lambda'})
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\lambda} \SigmatTan_{\sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
(\hfour^{-1})^{\kappa \kappa'}
\SigmatTan^{\sigma}
(\partial_{\lambda} \hfour_{\kappa' \sigma})
\\
& =
(\hfour^{-1})^{\kappa \sigma}
\hfour_{\lambda \lambda'}
(\partial_{\sigma} \SigmatTan^{\lambda'})
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\lambda} \SigmatTan_{\sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\sigma} \hfour_{\lambda \lambda'})
\SigmatTan^{\lambda'}
(\hfour^{-1})^{\kappa \kappa'}
\SigmatTan^{\sigma}
(\partial_{\lambda} \hfour_{\kappa' \sigma}).
\end{split}
\end{align}

Next, taking into account the definition (<ref>),
we compute that the contribution of the second product
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\lambda} \SigmatTan_{\sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
RHS (<ref>)
to the first product on RHS (<ref>)
is as follows:

\begin{align}
\begin{split} \label{E:THIRDSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
(\partial_{\alpha} \SigmatTan^{\gamma})
(\hfour^{-1})^{\kappa \sigma}
(\partial_{\lambda} \SigmatTan_{\sigma}
\partial_{\sigma} \SigmatTan_{\lambda}
\\
& =
\Nullhypersurfaceinversemetric^{\alpha \sigma}
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
\left\lbrace
\partial_{\alpha} [(\hfour^{-1})^{\gamma \gamma'} \SigmatTan_{\gamma'}]
\right\rbrace
(\mathrm{d} \SigmatTan_{\flat})_{\lambda \sigma}
\\
& =
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\partial_{\alpha} \SigmatTan_{\gamma})
(\mathrm{d} \SigmatTan_{\flat})_{\delta \beta}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
\SigmatTan^{\kappa}
(\partial_{\alpha} \hfour_{\delta \kappa})
(\mathrm{d} \SigmatTan_{\flat})_{\gamma \beta}
\\
& =
\frac{1}{2}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\mathrm{d} \SigmatTan_{\flat})_{\alpha \gamma}
(\mathrm{d} \SigmatTan_{\flat})_{\beta \delta}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
\SigmatTan^{\kappa}
(\partial_{\alpha} \hfour_{\delta \kappa})
(\mathrm{d} \SigmatTan_{\flat})_{\beta \gamma}.
\end{split}
\end{align}

Next, using
and (<ref>),
we rewrite the first product on RHS (<ref>)
as follows:

\begin{align}
\begin{split} \label{E:FOURTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\lambda} \SigmatTan^{\kappa}
& =
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfacemetric_{\gamma \delta}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}
\frac{1}{2}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\mathrm{d} \SigmatTan_{\flat})_{\alpha \gamma}
(\mathrm{d} \SigmatTan_{\flat})_{\beta \delta}
\\
& \ \
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
\SigmatTan^{\kappa}
(\partial_{\alpha} \hfour_{\delta \kappa})
(\mathrm{d} \SigmatTan_{\flat})_{\beta \gamma}
\\
& \ \
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
(\partial_{\beta} \hfour_{\delta \kappa})
\SigmatTan^{\kappa}
\partial_{\alpha} \SigmatTan^{\gamma}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
\SigmatTan^{\kappa}
(\partial_{\delta} \hfour_{\beta \kappa})
\partial_{\alpha} \SigmatTan^{\gamma}.
\end{split}
\end{align}

Next, using (see Lemma <ref>) the identities
$\Transport = \frac{1}{2} (\Lunit + \uLunit)$,
$\Lunit = \Transport - X$,
$\uLunit = \Transport + X$,
$\Transport_{\alpha} \SigmatTan^{\alpha} = 0$,
$\Transport_{\alpha} \partial_{\beta} \SigmatTan^{\alpha} = 0$,
as well as the identity
\partial_a \SigmatTan^a
X_a X \SigmatTan^a
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
(see (<ref>)),
we compute that:

\begin{align} \label{E:FIFTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Lunit_{\alpha}
\uLunit \SigmatTan^{\alpha}
& =
\uLunit_{\alpha}
\uLunit \SigmatTan^{\alpha}
\uLunit_{\alpha}
\Lunit \SigmatTan^{\alpha}
\uLunit_{\alpha}
\Transport \SigmatTan^{\alpha},
\\
\Lunit_{\alpha}
\uLunit \SigmatTan^{\alpha}
& =
\Lunit_{\alpha}
X \SigmatTan^{\alpha}
\Lunit_{\alpha}
\Transport \SigmatTan^{\alpha}
X \SigmatTan^a
\Lunit_{\alpha}
\Transport \SigmatTan^{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\partial_a \SigmatTan^a
\uLunit_{\alpha}
\Transport \SigmatTan^{\alpha}.
\label{E:SIXTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\end{align}

Next, using (<ref>),
we compute that:

\begin{align} \label{E:SEVENTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Nullhypersurfaceproject_{\beta}^{\ \alpha}
\partial_{\alpha} \SigmatTan^{\beta}
& =
\partial_a \SigmatTan^a
\frac{1}{4}
\Lunit_{\alpha}
\uLunit \SigmatTan^{\alpha}
\frac{1}{4}
\Lunit_{\alpha}
\uLunit \SigmatTan^{\alpha}
\\
& =
\frac{3}{4}
\partial_a \SigmatTan^a
\frac{3}{4}
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
\frac{1}{4}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\frac{1}{4}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b.
\label{E:EIGHTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\end{align}

we rewrite the second product on RHS (<ref>)
as follows:

\begin{align} \label{E:NINTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
(\Nullhypersurfaceproject_{\beta}^{\ \alpha}
\partial_{\alpha} \SigmatTan^{\beta})^2
& =
\frac{1}{16}
\left\lbrace
\partial_a \SigmatTan^a
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2.
\end{align}

and (<ref>),
we deduce the following identity:

\begin{align}
\begin{split} \label{E:TENTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfacemetric_{\gamma \delta}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}
& =
\Dfour_{\alpha} \ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
\\
& \ \
\frac{1}{2}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\mathrm{d} \SigmatTan_{\flat})_{\alpha \gamma}
(\mathrm{d} \SigmatTan_{\flat})_{\beta \delta}
\frac{1}{16}
\left\lbrace
\partial_a \SigmatTan^a
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2
\\
& \ \
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
\SigmatTan^{\kappa}
(\partial_{\alpha} \hfour_{\delta \kappa})
(\mathrm{d} \SigmatTan_{\flat})_{\beta \gamma}
\\
& \ \
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \lambda}
\Nullhypersurfaceproject_{\kappa}^{\ \alpha}
\right)
\right\rbrace
\partial_{\lambda} \SigmatTan^{\kappa}
\SigmatTan^{\gamma}
\left\lbrace
\partial_{\alpha}
\left(
\Nullhypersurfaceproject_{\gamma}^{\ \alpha}
\Nullhypersurfaceproject_{\lambda}^{\ \kappa}
\right)
\right\rbrace
\partial_{\kappa}
\SigmatTan^{\lambda}
\\
& \ \
\Chfour_{\alpha \ \beta}^{\ \alpha}
\ehcurrent^{\beta}[\SigmatTan,\pmb{\partial} \SigmatTan]
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
(\partial_{\beta} \hfour_{\delta \kappa})
\SigmatTan^{\kappa}
\partial_{\alpha} \SigmatTan^{\gamma}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceproject_{\gamma}^{\ \delta}
\SigmatTan^{\kappa}
(\partial_{\delta} \hfour_{\beta \kappa})
\partial_{\alpha} \SigmatTan^{\gamma}.
\end{split}
\end{align}

Next, using (<ref>),
we observe that:

\begin{align}
\begin{split} \label{E:ELEVENTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& =
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfacemetric_{\gamma \delta}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}
\Nullhypersurfacemetric_{\alpha \beta}
(\Transport \SigmatTan^{\alpha})
\Transport \SigmatTan^{\beta}
\\
& \ \
\left(
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}
\right)
\Transport_{\gamma} \Transport_{\delta}
(\partial_{\alpha} \SigmatTan^{\gamma})
\partial_{\beta} \SigmatTan^{\delta}
\\
& \ \
\frac{1}{16}
\left\lbrace
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2.
\end{split}
\end{align}

Hence, noting that
the terms on the second line of RHS (<ref>)
vanish, we can add

\Nullhypersurfacemetric_{\alpha \beta}
(\Transport \SigmatTan^{\alpha})
\Transport \SigmatTan^{\beta}
\frac{1}{16}
\left\lbrace
(\Transport \SigmatTan^{\alpha})
\uLunit_{\alpha}
(\Lunit \SigmatTan^{\alpha})
\uLunit_{\alpha}
\smoothtorusproject_b^{\ a}
\partial_a \SigmatTan^b
\right\rbrace^2

to each side of (<ref>)
to obtain the following identity:

\begin{align}
\begin{split} \label{E:TWELFTHSTEPPROOFCOVARIANTDIVERGENCEIDENTITYFORELLIPTICHYPERBOLICCURRENT}
\ellipticCoerciveQuadratic[\pmb{\partial} \SigmatTan,\pmb{\partial} \SigmatTan]
& =
\Dfour_{\alpha} \ehcurrent^{\alpha}[\SigmatTan,\pmb{\partial} \SigmatTan]
\\
& \ \
\frac{1}{2}
\Nullhypersurfaceinversemetric^{\alpha \beta}
\Nullhypersurfaceinversemetric^{\gamma \delta}
(\mathrm{d} \SigmatTan_{\flat})_{\alpha \gamma}