is the specific vorticity
\GradEnt^i = \partial_i \Ent
is the entropy gradient vectorfield.
The modified fluid variables
$\VortVort$ and $\DivGradEnt$
are defined in Def. <ref>
and satisfy
\VortVort
\sim
\Flatcurl \vortrenormalized
\DivGradEnt
\sim
\Flatdiv \GradEnt
where “$\sim$” denotes equality up to lower-order (in the sense of regularity) factors and terms.
\square_{\gfour(\wavearray)} f
is the covariant wave operator of $\gfour(\wavearray)$ acting on the scalar function $f$
(see Def. <ref>),
$\pmb{\partial} f = (\partial_t f, \partial_1, \partial_2 f, \partial_3 f)$
is the array of Cartesian coordinate spacetime partial derivatives of $f$,
$\partial f = (\partial_1, \partial_2 f, \partial_3 f)$
is the array of Cartesian spatial partial derivatives of $f$.
The “$\nullform$” are null forms relative to $\gfour$ (see Def. <ref>);
see Sect. <ref> for a discussion of the crucial role they play in the proof.

We highlight that	when deriving
top-order $L^2$ estimates for the specific vorticity and entropy gradient,
we crucially rely on the transport-div-curl equations
to propagate sufficient regularity for the source terms $(\VortVort, \DivGradEnt)$
on RHS (<ref>); see Sect. <ref> for
the details.
That is, even though these terms formally satisfy
\VortVort
\sim
\Flatcurl \vortrenormalized
\DivGradEnt
\sim
\Flatdiv \GradEnt
we cannot control $(\VortVort, \DivGradEnt)$
using only the transport equation
The reason is that (<ref>)
has a source term depending on
$\pmb{\partial} \wavearray$,
and thus, by estimating only the transport equation,
we could prove only that
$\Flatcurl \vortrenormalized, \, \Flatdiv \GradEnt$
are as regular as
$\pmb{\partial}^2 \wavearray$,
which would be insufficient regularity for treating
$\Flatcurl \vortrenormalized$
$\Flatdiv \GradEnt$
as source terms in the wave equation (<ref>).

§.§.§ Three kinds of coordinate systems

The proof of our main results is based on understanding the behavior of the solution
relative to three coordinate systems as well as understanding
the degenerate (near the shock) transformation properties of the coordinate systems.

* (The Cartesian coordinates $(t,x^1,x^2,x^3)$).
These are the fundamental coordinates relative to which the compressible
Euler equations (<ref>)–(<ref>) are posed.
For the solution regime under study, the singularity coincides
with the blowup of $|\partial_1 \RRiemann|$, where $\RRiemann$ is the “almost Riemann invariant” introduced above.

* (The geometric coordinates $(t,u,x^2,x^3)$). As in the other works
cited in Sect. <ref>,
$u$ is an eikonal function, that is, a solution to the
eikonal equation $(\gfour^{-1})^{\alpha \beta}(\wavearray) \partial_{\alpha} u \partial_{\beta} u = 0$,
where $\gfour$ is the acoustical metric. Its level sets $\nullhyparg{u}$ are characteristic for the
wave operator $\square_{\gfour(\wavearray)}$ on LHS (<ref>).
Our analytic framework is based on the fact that
the solution remains rather smooth relative to the geometric coordinates,
except that the high order geometric energies can blow up
as the shock forms, which introduces severe technical
difficulties into the analysis; see Sects. <ref>
and <ref> for discussion
of our geometric $L^2$ estimates.

* (The rough adapted coordinates $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$). These are a new ingredient,
fundamental for our main results. The $\timefunctionarg{\muxmulevelsetvalue}$ are a one-parameter family of
coordinate systems indexed by $\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]$, where $\muxmulevelsetvalue_0 > 0$ is a real number
depending on the solution regime under study.
They are constructed (see Sect. <ref>)
so that their range is $[\timefunction_0,0]$ for some constant $\timefunction_0 < 0$,
such that the initial data are given on the level set $\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction_0 \rbrace$
(see Sect. <ref>),
and such that the level set $\lbrace \timefunctionarg{\muxmulevelsetvalue} = 0 \rbrace$
intersects the singular boundary precisely in one embedded, spacelike, two-dimensional torus.
The union of the $\muxmulevelsetvalue$-indexed tori foliates the singular boundary;
see Fig. <ref>, where the tori on the singular
boundary are denoted by $\twoargmumuxtorus{0}{-\muxmulevelsetvalue}$.

* (The rough adapted coordinates $(\newtimefunction,u,x^2,x^3)$).
With the help of the family of coordinate systems
$\lbrace (\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3) \rbrace_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]}$,
we construct another time function, denoted by $\newtimefunction$,
that foliates the entire region $\MInteresting$. $\newtimefunction$ is a $C^{1,1}$ function
– and this is its optimal regularity regardless of the smoothness of the data
(see Remark <ref>) –
of the geometric coordinates $(t,u,x^2,x^3)$.
In particular, the singular boundary portion under study is a submanifold-with-boundary
that is contained in the level set $\lbrace \newtimefunction =  0 \rbrace$.
See Fig. <ref>, which displays two level sets of
$\newtimefunction$, denoted by $\inthyp{0}{[- \rightu,\leftu]}$
$\inthyp{\timefunction_0}{[- \rightu,\leftu]}$.

§.§.§ The vanishing of $\upmu$ signifies the singularity formation

As in <cit.>, the formation of the shock singularity is precisely characterized by
the vanishing of the inverse foliation density $\upmu$, defined in (<ref>),
where in the present context, $\gfour$ is the acoustical metric.
We recall that the vanishing of $\upmu$ signifies the infinite density of the characteristics;
see Fig. <ref>, where $\upmu$
vanishes along $\mathcal{B}$ and the characteristics pile up there.
The blowup of some Cartesian partial derivative of the almost Riemann invariant $\RRiemann$
then follows from proving a bound of the schematic form
$\upmu |X \RRiemann| \gtrsim 1$ in a neighborhood of the points where
$\upmu$ vanishes, where $X$ is a vectorfield that is transversal to the $\nullhyparg{u}$,
is $L^{\infty}$ close to the Cartesian partial derivative vectorfield $\partial_1$,
and has Euclidean length approximately equal to $1$.
This is essentially the same blowup-mechanism as in the irrotational and isentropic case,
as we described in Sect. <ref>.
Here we highlight that the vectorfield
$\muX \eqdef \upmu X$, which is a geometric replacement for the partial derivative vectorfield
$\geop{u}$ (in the geometric coordinate system), can be used to derive regular estimates for the solution's transversal derivatives:
all fluid quantities and geometric tensors $Q$
that we use to study the solution satisfy $|\muX Q| \lesssim 1$ up to the singularity;
the $\upmu$-weight in the definition of $\muX$ precisely cancels out the singular
behavior $|X Q|$ as $\upmu \downarrow 0$;
see Sect. <ref> for further discussion.

§.§.§ The causal structure of the singular boundary

The discussion in Sect. <ref> suggests
that the singular boundary should be the entire hypersurface $\lbrace \upmu = 0 \rbrace$.
However, as we already discussed in Sect. <ref>,
only a subset of $\lbrace \upmu = 0 \rbrace$ is relevant for the maximal development.
One reason is that formally, for the solutions under study,
$\lbrace \upmu = 0 \rbrace$ is the limit of hypersurfaces
$\lbrace \upmu = \mulevelsetvalue \rbrace$ as $\mulevelsetvalue \downarrow 0$,
and for $\mulevelsetvalue > 0$, these hypersurfaces have a spacelike part, a null part, and timelike part.
For $\mulevelsetvalue > 0$, our analysis yields access to the portions of
$\lbrace \upmu = \mulevelsetvalue \rbrace$ that are $\gfour$-spacelike or $\gfour$-null, but not
necessarily the timelike portion. Hence, the portion of $\lbrace \upmu = 0 \rbrace$ that makes up the singular boundary
can be viewed as the limit as $\mulevelsetvalue \downarrow 0$, of the portion of
$\lbrace \upmu = \mulevelsetvalue \rbrace$ that is $\gfour$-spacelike or $\gfour$-null.
From a careful analysis of the causal structure of the hypersurfaces
$\lbrace \upmu = \mulevelsetvalue \rbrace$, which we provide in Lemma <ref>,
we find that the accessible portion of
$\lbrace \upmu = 0 \rbrace$ that arises in the limit $\mulevelsetvalue \downarrow 0$
is $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu \leq 0 \rbrace$,
where the vectorfield $\muX$ is the same one described in Sect. <ref>.
More precisely, our results are local in spacetime, and our main theorem yields the structure of the singular
boundary portion
$\lbrace \upmu = 0 \rbrace \cap \lbrace - \muxmulevelsetvalue_0 \leq \muX \upmu \leq 0 \rbrace$,
which we denote by $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$ in Fig. <ref>.
We highlight the following key point: even if one were to construct the complementary surface portion
$\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$ in geometric coordinates,
the map from geometric to Cartesian coordinates would fail to be one-to-one
on the full surface $\lbrace \upmu = 0 \rbrace$, though it is one-to-one
on the subset $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$; see Theorem <ref>.
From a different perspective, in the context of Fig. <ref>,
the surface portion $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$
is “cut off” by the Cauchy horizon $\underline{C}$, i.e., in the classical maximal globally hyperbolic development,
$\underline{C}$ develops “before”
$\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$ has a chance to form.

In view of the above discussion, in our main theorem, we
construct the singular boundary portion
\mathcal{B}^{[0,\muxmulevelsetvalue_0]}
\eqdef \bigcup_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]}
\lbrace \upmu = 0 \rbrace
\cap
\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace
see Fig. <ref>.
Our aforementioned transversal convexity assumption on the initial data,
which we are able to propagate throughout the evolution
(see (<ref>)),
ensures that
the hypersurfaces
$\lbrace \upmu = 0 \rbrace$
$\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace$
intersect transversally in embedded, two-dimensional $\gfour$-spacelike tori that we denote by
$\twoargmumuxtorus{0}{-\muxmulevelsetvalue}$. In particular,
we show that $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$ is an embedded three-dimensional
submanifold-with-boundary that is foliated by the tori
Its past boundary, denoted by $\partial_-\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$,
is the torus $\twoargmumuxtorus{0}{0}$, a set that we have also been referring to as the crease.

We next highlight that the crease plays a distinguished role in the
shock development problem described in Sect. <ref>;
once that problem is solved,
the hypersurface of discontinuity $\mathcal{K}$ will emanate
from the crease; see Fig. <ref>
In particular, the crease is a crucial component of the “data” for the
shock development problem.

§.§.§ The one-parameter family of rough time functions $\lbrace \timefunctionarg{\muxmulevelsetvalue} \rbrace_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]}$

We follow the solution up to the singular boundary portion
$\mathcal{B}^{[0,\muxmulevelsetvalue_0]} = \bigcup_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]} \twoargmumuxtorus{0}{-\muxmulevelsetvalue}$
by constructing a one-parameter family of time functions
$\lbrace \timefunctionarg{\muxmulevelsetvalue} \rbrace_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]}$
and controlling the solution on the level sets of each time function.
The $\timefunctionarg{\muxmulevelsetvalue}$ are constructed such that their range is $[\timefunction_0,0]$
for some small parameter $\timefunction_0 < 0$
and such that $\twoargmumuxtorus{0}{-\muxmulevelsetvalue} \subset \lbrace \timefunctionarg{\muxmulevelsetvalue} = 0 \rbrace$.
We construct $\timefunctionarg{\muxmulevelsetvalue}$ by setting it equal to $\upmu$ on the hypersurface
$\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace$
and then transporting it along the flow of a
well-constructed vectorfield that is transversal to $\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace$,
which ensures that the level sets of $\timefunctionarg{\muxmulevelsetvalue}$ are $\gfour$-spacelike
in the region of classical existence.
In Fig. <ref>, we denote the hypersurface
$\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace$ by
and we denote the transversal vectorfield by $\Wtransarg{\muxmulevelsetvalue}$.
See Sect. <ref>
for the details on the construction of $\timefunctionarg{\muxmulevelsetvalue}$.

A key feature of our construction is that for $\timefunction \in [\timefunction_0,0]$, we have:

\begin{align} \label{E:INTROMUMIN}
\min_{\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace} \upmu
& = - \timefunction,
\end{align}

and the minimum value is achieved precisely along the set
\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace
\cap
\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace
Thus, for $\mulevelsetvalue \in [0,-\timefunction_0]$, we have
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
\lbrace \upmu = \mulevelsetvalue \rbrace
\cap
\lbrace \muX \upmu = - \muxmulevelsetvalue \rbrace
Our high order energy estimates feature crucial factors of $1/\upmu$,
and by (<ref>),
in rough adapted coordinates $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$,
such factors are bounded in magnitude on
$\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace$
by $\leq \frac{1}{|\timefunction|}$.
Thus, our energy estimates on the level sets of the rough time function
feature difficult factors of $\frac{1}{|\timefunction|}$,
which leads to singular estimates at the high derivative levels.
While related difficulties were present in all the works that we cited above
on shock formation without symmetry assumptions,
the estimate (<ref>) allows for a simplified approach
to handling the degeneracy; it directly connects the degeneracy to a coordinate function.
We refer to Sect. <ref> for our detailed
analysis of the factors of $\frac{1}{|\timefunction|}$ and the way they affect
our energy estimates, and to Sect. <ref> for an overview of these

We emphasize the following key point:

The rough time functions $\timefunctionarg{\muxmulevelsetvalue}$
are precisely adapted to the structure of the singularity and,
in particular, are not more regular than $\upmu$.
Since it turns out that $\upmu$ is one degree less differentiable[In particular, in view of the transport
equation (<ref>), we see that $\upmu$ cannot be more regular than the source term
$\muX \wavearray$.
This is optimal, as can already be seen in plane-symmetry
via the explicit formula (<ref>).]
than the fluid variables $\wavearray$,
we cannot use commutation vectorfields adapted to the level sets of
$\timefunctionarg{\muxmulevelsetvalue}$ to control the solution up to top-order;
that approach would lead to derivative loss.
For this reason, as in other works on shock formation,
we use the eikonal function $u$
to construct[It does not seem possible to adequately control the solution
by commuting the equations with the standard Cartesian partial derivatives $\partial_{\alpha}$;
the Cartesian vectorfields are not adapted to the singularity, and
that approach would lead to singular error terms that we have no obvious means to control.]
commutation vectorfields that allow us to control the solution.

Finally, we point out that it is not too difficult to construct the $\timefunctionarg{\muxmulevelsetvalue}$;
the difficult part of the analysis is controlling the fluid solution and the acoustical geometry
all the way up the hypersurface $\lbrace \timefunctionarg{\muxmulevelsetvalue} = 0 \rbrace$.

§.§.§ The bootstrap assumptions

To derive estimates, we fix $\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]$ and make an elaborate set of bootstrap assumptions
on an “open-at-the-top” slab $[\timefunction_0,\timefunctionboot) \times [- \rightu,\leftu] \times \mathbb{T}^2$
corresponding to the rough adapted coordinates $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$
(the corresponding region in geometric coordinate space is denoted by
$\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,\leftu]}{\muxmulevelsetvalue}$ in the bulk of the paper).
More precisely, on the slab, our bootstrap assumptions describe the following:

* The behavior of $\upmu$, including the
key property of transversal convexity;
see Sect. <ref>.

* The rough time function $\timefunctionarg{\muxmulevelsetvalue}$;
see Sect. <ref>.

* The properties of various change of variables maps;
see Sect. <ref>.

* The structure and locations of various embedded submanifolds;
see Sect. <ref>.

* The size of the Cartesian coordinates $t$ and $x^1$ on the slab;
see Sect. <ref>.

* The “soft” regularity properties of various quantities on the closure of the slab;
see Sect. <ref>.

* The $L^{\infty}$-size of the fluid variables and their $\nullhyparg{u}$-tangential derivatives up to mid-order;
see Sect. <ref>.

* The $L^{\infty}$-size of the transversal and mixed tangential-transversal derivatives of the fluid variables
and the acoustic geometry;
see Sect. <ref>.

* The size of the energies and null-fluxes for the wave variables $\wavearray$ up to top-order;
see Sect. <ref>.

Our primary analytic tasks in the paper are to derive strict improvements of the bootstrap assumptions
by making suitable assumptions on the data.
Once this has been accomplished,
a standard continuity argument,
provided by Prop. <ref> and
the proof of Theorem <ref>,
implies that the solution exists classically on the maximal slab
$[\timefunction_0,0) \times [- \rightu,\leftu] \times \mathbb{T}^2$
and satisfies the bootstrap assumptions there. Hence, Theorem <ref>
yields the main results at fixed $\muxmulevelsetvalue$,
which in turn form the main ingredients in the proof of the central theorem on the singular boundary,
Theorem <ref>.
In Sect. <ref>,
we provide a road map, indicating the spots in the article where we derive improvements
of the bootstrap assumptions.

§.§.§ Regular estimates with respect to the geometric coordinates on the rough foliations

A key idea of the proof, going back to <cit.>,
is to prove that the solution remains rather smooth relative to the geometric
coordinates $(t,u,x^2,x^3)$. We implement this by following the approach of
and using the eikonal function to construct a set of
commutation vectorfields:

\begin{align} \label{E:INTROCOMMUTATORS}
\Fullset
& \eqdef \lbrace \Lunit, \muX, \Yvf{2}, \Yvf{3} \rbrace
\end{align}

that are adapted to the characteristics; see Fig. <ref> for a picture of these
vectorfields (with one spatial dimension suppressed),
and see Sect. <ref> for our detailed construction
of the elements of $\Fullset$.
In (<ref>) and throughout,
$\Lunit$ is a $\gfour$-null vectorfield (i.e., $\gfour(\Lunit,\Lunit) = 0$)
that is proportional to the (geodesic) gradient vectorfield $(\gfour^{-1})^{\alpha \beta} \partial_{\alpha} u \partial_{\beta}$,
normalized by $\Lunit t = 1$, and tangent to the characteristics $\nullhyparg{u}$.
$\Yvf{2}, \Yvf{3}$ are geometric replacements for $\partial_2$ and $\partial_3$
that span the tangent space of the smooth tori $\ell_{t,u} \eqdef \nullhyparg{u} \cap \Sigma_t$.
The vectorfields $\lbrace \Lunit, \Yvf{2}, \Yvf{3} \rbrace$ span the tangent space of $\nullhyparg{u}$
at each of its points.
$\muX$ is tangent to the flat Cartesian hypersurface $\Sigma_t$,
$\gfour$-orthogonal to $\ell_{t,u}$, and normalized[In this paper, this
normalization condition is equivalent to $\gfour(\muX,\muX) = \upmu^2$.]
by $\muX u = 1$. Note that $\muX$ is transversal to the characteristics. This is important because
all the degeneracies in the problem of shock formation are tied to the behavior of transversal derivatives.
We also note that, in accordance with the discussion in Sect.,<ref>,
we have $\muX = \upmu X$, where $\gfour(X,X) = 1$.

[scale=.6, grid = false, tics=5, trim=-.5cm -1cm -1cm -.5cm, clip]commutatorVFincartesiancoordinates.pdf
(50,15) $\Sigma_0$
(13,35) $(x^2,x^3) \in \mathbb{T}^2$
(4,26)	$x^1 \in \R$
(3,38) $t$
(21,50) $\muX$
(30,57) $\Yvf{A}$
(38,32) $L$
(66,44) $\muX$
(67,50) $L$
(52,69) $\Yvf{A}$

The commutation vectorfields in Cartesian coordinates

As in <cit.>, we use the full set of commutators $\mathcal{Z}$ to derive
$L^{\infty}$ and Hölder estimates for solutions,
while for deriving energy estimates,
we only need to use the following $\nullhyparg{u}$-tangent subset:

\begin{align} \label{E:INTROTANGENTCOMMUTATORS}
\Tanset
& \eqdef \lbrace \Lunit, \Yvf{2}, \Yvf{3} \rbrace.
\end{align}

In particular, our results show that, based on our assumptions on the data,
the up-to-fourth order derivatives
$\wavearray$, $\vortrenormalized$, and $\GradEnt$
with respect to the elements of $\mathcal{Z}$ are bounded in $L^{\infty}$.
This is equivalent to the $L^{\infty}$-boundedness of the up-to-fourth order derivatives of these quantities
with respect to the geometric coordinate partial derivatives.
Similar results hold for the up-to-third order
derivatives of $\VortVort$ and $\DivGradEnt$,
and due to our assumptions on the data,
all fluid variables enjoy additional regularity in directions tangent to
the characteristics $\nullhyparg{u}$.
Similar results also hold for the rough time functions
and various embeddings of various submanifolds into geometric or Cartesian coordinate space,
such as the hypersurfaces
\datahypfortimefunctionarg{-\muxmulevelsetvalue}
\eqdef
\lbrace
\muX \upmu = - \muxmulevelsetvalue
\rbrace
and the $\upmu$-adapted tori
\twoargmumuxtorus{\mulevelsetvalue}{-\muxmulevelsetvalue}
\eqdef
\lbrace
\upmu = \mulevelsetvalue
\rbrace
\cap
\lbrace
\muX \upmu = - \muxmulevelsetvalue
\rbrace

To derive the $L^{\infty}$ and Hölder estimates,
we make “fundamental” $L^{\infty}$ bootstrap assumptions for the fluid
wave variables and their $\Tanset$ derivatives up to mid-order,
as we described in Sect. <ref>.
By combining our fundamental bootstrap assumptions with some auxiliary ones,
we commute all relevant equations with the elements of
$\Fullset$ and treat them as transport equations with
derivative-losing source terms. This allows us to derive
$L^{\infty}$ and Hölder estimates for the solution's transversal
and mixed transversal-tangential derivatives
and to control the embeddings mentioned in the previous paragraph.
We carry out this analysis in
In Sect. <ref>,
we derive related, but much sharper, pointwise estimates for $\upmu$
at the low derivative levels, which in particular yield the crucial estimate
Near the end of the paper, we use our energy estimates and Sobolev embedding
to improve the fundamental bootstrap assumptions; see
Sect. <ref>.

Finally, we will briefly discuss the regularity of the Cartesian components of the elements of $\Fullset$.
They have regularity at the schematic level $\pmb{\partial} u$,
and it turns out that by using renormalizations (which we refer to a “modified quantities”)
and elliptic estimates on co-dimension $2$ hypersurfaces
(specifically, the rough tori
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
\eqdef
\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace
\cap
\nullhyparg{u}
which are techniques that originated in dCsK1993,sKiR2003,dC2007,
we can show that
the elements of $\Fullset$ have just enough regularity
to allow us to derive energy estimates up to top order.
We construct the modified quantities in Sect. <ref>,
we derive the elliptic estimates in
Sect. <ref>,
and we derive the “final” $L^2$ estimates involving these quantities
Sect. <ref>.

§.§.§ Smooth geometry versus rough geometry and elliptic estimates for $\upchi$

Although regularity considerations force us to commute the equations with
elements of $\Fullset$ and $\Tanset$ (e.g., the rough adapted coordinate partial derivative vectorfields
do not have sufficient regularity for commuting the equations up to top-order),
in order to detect the precise structure of the singular boundary,
we must derive estimates on the rough hypersurfaces
\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace
the characteristics $\nullhyparg{u}$,
and the rough tori
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
\eqdef
\lbrace \timefunctionarg{\muxmulevelsetvalue} = \timefunction \rbrace
\cap
\nullhyparg{u}
For this reason, throughout the analysis,
we have to control the geometry of these surfaces
(e.g., the Gauss curvature of $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$)
by quantitatively relating it to the elements of $\Fullset$ and $\Tanset$
and their derivatives.
A particularly noteworthy manifestation of this issue is the following:
we must control the top-order derivatives of the
null second fundamental form
$\upchi = \frac{1}{2} \angLie_{\Lunit} \gtorus$,
where $\gtorus$ is the Riemannian metric induced by $\gfour$ on $\ell_{t,u} = \Sigma_t \cap \nullhyparg{u}$.
Here, $\angLie_{\Lunit} \gtorus$ denotes Lie differentiation with respect to $\Lunit$ followed by
$\gfour$-orthogonal projection onto $\ell_{t,u}$.
Note that the  “smooth torus” $\ell_{t,u}$ is not adapted (or necessarily fully contained in) the rough spacetime
regions under study. Even though $\upchi$ is $\ell_{t,u}$-tangent,
to avoid derivative loss in the top-order estimates on the rough spacetime regions,
we have to derive elliptic estimates for $\upchi$
on the rough tori
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
To close these elliptic estimates,
we must suitably relate $\upchi$ to a tensor on
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
and control the Gauss curvature of
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
(viewed as a subset of spacetime equipped with the metric $\gfour$),
all while being mindful about crucial factors of $\upmu$,
whose vanishing signifies the shock singularity.
This delicate analysis is located in Sect. <ref>,
where we construct two distinct frames on
one adapted to the smooth tori $\ell_{t,u}$ and one adapted to the rough tori
\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}
and we control the relationship between the two frames
as a key step in obtaining the elliptic estimates.

§.§.§ The role of the good null structure

As in many of the aforementioned works on shock formation, in the present paper,
we crucially rely on the fact that all the derivative-quadratic inhomogeneous terms in the system
are null forms relative to $\gfour$.
From the point of view of analysis,
the crucial point is that when expanded relative to (say) the commutator frame (<ref>),
the $\upmu$-weighted[In our analysis, we introduce a $\upmu$ weight into various equations and estimates.]
null forms satisfy, schematically,
\upmu \nullform(\pmb{\partial} \wavearray, \pmb{\partial} \wavearray)
= \muX \wavearray \cdot P \wavearray
\upmu P \wavearray \cdot P \wavearray
where $P$ schematically denotes elements of the $\nullhyparg{u}$-tangent subset (<ref>);
see Lemma <ref> for a more detailed statement concerning the precise null forms
we encounter in our analysis.
In particular, in the expansion of $\upmu \nullform(\pmb{\partial} \wavearray, \pmb{\partial} \wavearray)$,
terms proportional to $\muX \wavearray \cdot \muX \wavearray$
are completely absent. This is crucial because, if present, signature considerations
would imply that such terms would have to be accompanied by a factor of $\upmu^{-1}$
(i.e., the term would be proportional to $\upmu^{-1} \muX \wavearray \cdot \muX \wavearray$),
and factor of $\upmu^{-1}$ (which blows up as $\upmu \to 0$) would obstruct our philosophy of
deriving regular estimates for the solution's $\mathcal{Z}$-derivatives.
Similar remarks apply to all the other null forms in (<ref>)–(<ref>).

The upshot is the following: all of the derivative-quadratic terms in the system
that drive the formation of the shock are “hidden” in the definition of the covariant wave operator $\square_{\gfour(\wavearray)}$
on LHS (<ref>) and become visible when one expands
$\square_{\gfour(\wavearray)} \Psi$ relative to the Cartesian coordinates (say, via the formula (<ref>)).
The virtue of $\square_{\gfour(\wavearray)}$ is that there is an advanced machinery
for deriving geometric commutator and multiplier energy estimates for such operators,
and we exploit this machinery throughout the paper.

§.§.§ Geometric wave and transport energy estimates on the rough foliations, with singular high order behavior

To derive $L^2$ estimates up to top order, we commute equations[More precisely,
to avoid uncontrollable error terms,
we weight these equations by a factor of $\upmu$ before commuting.]
with the elements of $\Tanset$ (see (<ref>))
and derive energy and elliptic estimates on
regions of the form $\twoargMrough{[\timefunction_0,\timefunction],[- \rightu,u]}{\muxmulevelsetvalue}$,
which are bounded by rough hypersurfaces on the top and bottom and null hypersurfaces on the sides;
see Fig <ref>.

[scale=.4, grid = false, tics=5]Integrationregiongeometric.pdf
(48,99) $\twoargmumuxtorus{0}{-\muxmulevelsetvalue}$
(37,97) $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$
(40,17) $\Sigma_0$
(37,-4) $(x^2,x^3) \in \mathbb{T}^2$
(42,7) $t$
(30.5,3.5) $u \in\mathbb{R}$
(17,75) $\hypthreearg{0}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$
(33,59) $\hypthreearg{\timefunction}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$
(33,43) $\hypthreearg{\timefunction_0}{[- \rightu,\leftu]}{\muxmulevelsetvalue}$
(3,64) $\twoargroughtori{\timefunction,\leftu}{\muxmulevelsetvalue}$
(18,65) $\twoargroughtori{\timefunction,\interestingu}{\muxmulevelsetvalue}$
(56,77) $\twoargroughtori{\timefunction,-\interestingu}{\muxmulevelsetvalue}$
(70,77) $\twoargroughtori{\timefunction,-\rightu}{\muxmulevelsetvalue}$
(3,84) $\twoargroughtori{0,\leftu}{\muxmulevelsetvalue}$
(15,88) $\twoargroughtori{0,\interestingu}{\muxmulevelsetvalue}$
(58,101) $\twoargroughtori{0,-\interestingu}{\muxmulevelsetvalue}$
(74,100) $\twoargroughtori{0,-\rightu}{\muxmulevelsetvalue}$
(30,47) 30$\twoargMrough{[- \rightu,\leftu],[\timefunction_0,\timefunction]}{\muxmulevelsetvalue}$
(-4,43) $\nullhypthreearg{\muxmulevelsetvalue}{\leftu}{[\timefunction_0,\timefunction]}$
(10,44) $\nullhypthreearg{\muxmulevelsetvalue}{\interestingu}{[\timefunction_0,\timefunction]}$
(57,57) $\nullhypthreearg{\muxmulevelsetvalue}{-\interestingu}{[\timefunction_0,\timefunction]}$
(72,59) $\nullhypthreearg{\muxmulevelsetvalue}{-\rightu}{[\timefunction_0,\timefunction]}$

Regions in geometric coordinate space on which we derive estimates

Our elliptic estimates, which are localized in spacetime, come in two flavors: those for the acoustic geometry,
as we mentioned in Sect. <ref>, and elliptic estimates
for some of the fluid variables.
The localized elliptic estimates for the fluid variables
are a difficult new feature of the present work,
and we discuss them in Sect. <ref>;
here we discuss the energy estimates for the wave equations
(<ref>), which in reality are coupled to the elliptic estimates.
More precisely, for solutions $\Psi$ to (<ref>),
we construct coercive energies $\mathbb{E}[\Psi](\timefunction,u)$
on rough hypersurface portions $\hypthreearg{\timefunction}{[- \rightu,u]}{\muxmulevelsetvalue}$
and null fluxes $\mathbb{F}[\Psi](\timefunction,u)$ on null hypersurface portions
(see Def. <ref> for definitions of these portions),
and we derive suitable energy identities by using
a well-known framework based on the energy-momentum tensor for wave equations
and the well-chosen multiplier vectorfield
$\multipliervectorfield = (1 + 2 \upmu) \Lunit
2 \muX$; see Prop. <ref>.
We note in passing that we derive similar energy estimates
for the transport equations
and we will not discuss them in this introduction,
aside from mentioning that the $\gfour$-timelike
property $\gfour(\Transport,\Transport) = - 1$
(see Lemma <ref>) is crucial for those estimates,
as it ensures that $\Transport$ is transversal to the characteristics.

As we mentioned in Sect. <ref>,
to control the top-order derivatives of
the acoustic geometry without derivative loss,
we must rely on renormalizations, which we implement by constructing
appropriate “modified quantities,”
originating in dCsK1993,sKiR2003,dC2007,jS2016b;
see Sect. <ref>.
As in other works on shock waves without symmetry, this renormalization
leads to top-order energy identities for the wave variables $\Psi$ involving
singular terms. Because of the special properties of our rough foliations,
the singular terms are in fact factors of $\frac{1}{\timefunction}$,
and the resulting energy-flux inequalities can be caricatured as
follows, where the LHS is the sum of the energy, the null flux,
and the following coercive spacetime integral:

\begin{align} \label{E:INTROCOERCIVESPACETIME}
\mathbb{K}[\tander^{\Ntop} \Psi](\timefunction,u)
& \eqdef
\int_{(\timefunction',u',x^2,x^3) \in [\timefunction_0,\timefunction) \times [-\rightu,u] \times \mathbb{T}^2}
\mathbf{1}_{\lbrace [-\interestingu,\interestingu] \rbrace}(u')
(\argLrough{\muxmulevelsetvalue} \upmu)
|\angrmD \tander^{\Ntop} \Psi|_{\gtorus}^2
\, \mathrm{d} x^2 \mathrm{d} x^3 \mathrm{d} u' \mathrm{d} \timefunction'
\end{align}

of the top-order derivatives of the wave variable $\Psi$
(see Def. <ref> for the precise definitions):

\begin{align}  \label{E:INTROSCHEMATICTOPORDERENERGYID}
\begin{split}
\mathbb{E}[\tander^{\Ntop} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop} \Psi](\timefunction,u)
\\
\leq
C \initialsmall^2
\boxed{A}
\int_{\timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|}
\mathbb{E}[\tander^{\Ntop} \Psi](\timefunction',u)
\, \mathrm{d} \timefunction'
\cdots.
\end{split}
\end{align}

In (<ref>),
$\initialsmall \geq 0$ is the small size of the perturbation
of the initial data away from a background plane-symmetric solution,
$\timefunction_0 < 0$ denotes the initial rough time (at which the data perturbation is assumed to be small),
and $\timefunction \in [\timefunction_0,0]$.
Moreover, $A > 0$
is a universal constant
(which we have placed in a box to highlight its importance)
that is independent of the equation of state,
and $\cdots$ denotes similar[In reality, in the energy estimates, there are other difficult error terms
that have to be handled with integration by parts with respect to $\frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit$, and these lead to
difficult singular boundary terms that we control in Lemma <ref>.]
or easier error terms or terms that can be handled with the
help of the elliptic estimates described in Sect. <ref>.
In the spacetime integral $\mathbb{K}[\tander^{\Ntop} \Psi](\timefunction,u)$ on LHS (<ref>),
$|\angrmD \tander^{\Ntop} \Psi|_{\gtorus}^2$ denotes the square norm of the derivatives of
$\tander^{\Ntop} \Psi$ in directions tangent to the smooth tori
$\ell_{t',u'} = \Sigma_{t'} \cap \nullhyparg{u'}$,
$\mathbf{1}_{\lbrace [-\interestingu,\interestingu] \rbrace}(u')$
is the characteristic function of a small interval $[-\interestingu,\interestingu]$ of $u$-values
near the crease, and the factor $\argLrough{\muxmulevelsetvalue} \upmu$
is a null derivative of $\upmu$ that is quantitatively negative in this interval.
Hence, in view of the minus sign in (<ref>), we see that
the overall sign of the spacetime integral on LHS (<ref>) is positive,
and this integral is crucially used in the proof to absorb some of the error terms in  “$\cdots$”
on RHS (<ref>).
We also note in passing that some of the error terms in “$\cdots$” are non-singular terms that can be handled
with the help of the null fluxes $\mathbb{F}[\tander^{\Ntop} \Psi](\timefunction,u)$,
and that some of these terms, when treated with Grönwall's inequality in $u$, allow for the possibility
of exponential growth in $u$, which is permissible within the scope of our approach ($u$ is confined to a compact set).
The key point is that by applying Grönwall's inequality to (<ref>),
we deduce (ignoring the “$\cdots$” terms here) the following singular (as $\timefunction \uparrow 0$)
top-order energy estimate:

\begin{align} \label{E:INTROTOPORDERENERGYESTIMATE}
\mathbb{E}[\tander^{\Ntop} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop} \Psi](\timefunction,u)
& \lesssim \initialsmall^2 |\timefunction|^{-A}.
\end{align}

Thus, even though we have used geometrically defined commutators to try to turn
the problem of shock formation into a “regular” problem, a remnant of the singularity
survives[These singular top-order estimates are completely tied to the
energy estimates, and in particular, this difficulty does not arise in $1D$, where one can use transport estimates to close the problem.] at the top-derivative level, i.e, our “unfolding” of the characteristics
does not completely “hide” the singularity.
This calls into question the basic philosophy of the approach.

However, two crucial structural features of the problem rescue the situation.
First, the universal constant $A$ – and hence the singular factor $|\timefunction|^{-A}$ on RHS (<ref>) –
is independent of $\Ntop$. This means that we can choose $\Ntop$ to be large without increasing the
singularity strength. Second, below top-order, one can avoid using the renormalization procedure
to control the acoustic geometry. This leads to the loss of one derivative in the energy estimate hierarchy,
but the loss of one derivative is permissible below top-order. The price one pays is that
this procedure couples the below-top-order estimates to the singular top-order ones.
At one derivative below the top-order, the corresponding energy-null flux inequality
can be caricatured[In reality, the top-order and below-top-order energy estimates are highly coupled
and have to be derived simultaneously through an intricate version of Grönwall's lemma,
which we provide in Sect. <ref>.] as follows:

\begin{align}  \label{E:INTROSCHEMATICBELOWTOPORDERENERGYID}
\begin{split}
\mathbb{E}[\tander^{\Ntop-1} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop-1} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop-1} \Psi](\timefunction,u)
\\
& \leq
C \initialsmall^2
\int_{\timefunction' = \timefunction_0}^{\timefunction}
\frac{1}{|\timefunction'|^{1/2}} \mathbb{E}^{1/2}[\tander^{\Ntop-1} \Psi](\timefunction',u)
\int_{\timefunction'' = \timefunction_0}^{\timefunction'}
\frac{1}{|\timefunction''|^{1/2}}
\mathbb{E}^{1/2}[\tander^{\Ntop} \Psi](\timefunction'',u)
\, \mathrm{d} \timefunction''
\mathrm{d} \timefunction'
\cdots.
\end{split}
\end{align}

By applying Grönwall's inequality to (<ref>)
and accounting for the singular top-order behavior stated in
(<ref>), we deduce that:

\begin{align} \label{E:INTROBELOWTOPORDERENERGYESTIMATE}
\mathbb{E}[\tander^{\Ntop-1} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop-1} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop-1} \Psi](\timefunction,u)
& \lesssim
\initialsmall^2 |\timefunction|^{- (A-2)}.
\end{align}

(<ref>) shows that if we descend one derivative level below top-order,
then the energy becomes less singular by a factor of $|\timefunction|^2$.
As in dC2007,jS2016b,jLjS2018,jLjS2021,
one can continue the descent, proving that:

\begin{align} \label{E:INTROENERGYESTIMATEDESCENTTWOBELOWTOP}
\mathbb{E}[\tander^{\Ntop-2} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop-2} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop-2} \Psi](\timefunction,u)
\lesssim
\initialsmall^2 |\timefunction|^{- (A-4)},
\end{align}

and finally arriving at the non-singular estimate:

\begin{align} \label{E:INTRONONSINGULARENERGYESTIMATE}
\mathbb{E}[\tander^{\Ntop-\frac{A}{2}} \Psi](\timefunction,u)
\mathbb{F}[\tander^{\Ntop-\frac{A}{2}} \Psi](\timefunction,u)
\mathbb{K}[\tander^{\Ntop-\frac{A}{2}} \Psi](\timefunction,u)
\lesssim
\initialsmall^2.
\end{align}

The crucial non-singular energy estimate (<ref>) is what
allows us to show, via Sobolev embedding,
that the solution is bounded with respect to the geometric coordinates $(t,u,x^2,x^3)$
at derivative levels $\approx \Ntop-\frac{A}{2}$ and below.

Finally, we note that a related energy estimate hierarchy holds for the other fluid variables
(including $\vortrenormalized$, $\GradEnt$, $\VortVort$, $\DivGradEnt$)
and the acoustic geometry (including quantities such as $\upmu$ and $\upchi$),
and that in practice, all these estimates are coupled
(though in certain spots in our bootstrap argument, we exploit weak coupling, which allows us to derive
some estimates before deriving others).
We refer to Sect. <ref> for
detailed statements of the full hierarchy of energy estimates.

§.§.§ Localized elliptic fluid variable estimates via the characteristic current

As we mentioned earlier, to close the top-order energy estimates
for the vorticity and entropy
on the rough domains
\twoargMrough{[\timefunction_0,\timefunction],[- \rightu,u']}{\muxmulevelsetvalue}
\lbrace \timefunction_0 \leq \timefunctionarg{\muxmulevelsetvalue} \leq \timefunction \rbrace
\cap
\lbrace - \rightu \leq u \leq u' \rbrace
we cannot rely exclusively on the transport equation (<ref>); that approach
would lead to the loss of a derivative in our scheme.
Instead, we use the full structure of the transport-div-curl system
We also have to show that our elliptic estimates, which are singular,
are compatible with the
blowup-rates of the high order wave and transport energies, described in
Sect. <ref>.
We used a version of this approach in <cit.>, where we followed the solution precisely to the constant-Cartesian-time
hypersurface of first blowup. The elliptic estimates in <cit.>
were much simpler because we derived them only on the flat hypersurfaces $\Sigma_t$, whose geometry is trivial,
and because we did not try to derive the localized structure of the singular boundary; this allowed us to close the proof
by deriving the elliptic estimates only across all of space, thereby
(also exploiting the assumption of compactly supported data in <cit.>) avoiding the
difficult spatial boundary terms that we encounter in the present work.

To derive the desired estimates, in particular to handle the difficult boundary terms,
we use the technology of <cit.>,
which allows one to combine the Euclidean div-curl system
with the transport equations (<ref>)
to derive “elliptic-hyperbolic identities” that yield
a sufficient amount of Sobolev regularity on any region
– including $\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u']}{\muxmulevelsetvalue}$ –
that is globally hyperbolic with respect to $\gfour$.
See Prop. <ref>
for the precise elliptic-hyperbolic identity that we use to close the top-order estimates.

There are three new ingredients in our elliptic estimates here compared to <cit.>:

* The elliptic-hyperbolic identities depend, roughly, on certain
curvature components of the boundaries of the domain.
Some of these components become very singular as $\upmu \to 0$,
and we need to ensure that the singularity strength
is compatible with the blowup-rates of the high order wave and transport energies.
In Prop. <ref>,
we provide pointwise estimates guaranteeing that indeed,
all of the error terms in the identities are controllable within the scope of our approach.

* We derive a new version of the identities from <cit.>
based on applying the divergence theorem to a well-constructed
characteristic current $\ehcurrent^{\alpha}$,
which is tangent to the characteristic hypersurfaces $\nullhyparg{u}$;
see Def. <ref>.
Because of the $\nullhyparg{u}$-tangency, when we apply
the divergence theorem to $\ehcurrent^{\alpha}$ on $\twoargMrough{[\timefunction_0,\timefunction],[- \rightu,u']}{\muxmulevelsetvalue}$,
there are no boundary integrals along the lateral boundaries $\nullhyparg{u}$.
This allows us to completely avoid
error integrals on $\nullhyparg{u}$ that feature the top-order derivatives of $\upmu$;
the point is that it is not possible to derive top-order $L^2$ estimates for $\upmu$ on $\nullhyparg{u}$
because it satisfies a transport equation $\Lunit \upmu = \cdots$, where $\Lunit$ is tangent to the $\nullhyparg{u}$.
We also highlight that, unlike the currents in <cit.>,
our characteristic current here does not involve the future-directed normal $\hypunitnormalarg{\muxmulevelsetvalue}$
to the constant-rough-time hypersurfaces $\hypthreearg{\timefunction}{[- \rightu,u']}{\muxmulevelsetvalue}$.
Avoiding $\hypunitnormalarg{\muxmulevelsetvalue}$-dependent terms
is advantageous because the derivatives of
$\hypunitnormalarg{\muxmulevelsetvalue}$ become very singular as $\upmu \to 0$,
and it is not at all clear that such terms would have been compatible
the blowup-rates of the high order wave and transport energies.

* Upon applying the divergence theorem to $\ehcurrent^{\alpha}$ on
$\twoargMrough{[\timefunction_0,\timefunction),[- \rightu,u']}{\muxmulevelsetvalue}$,
we generate boundary integrals along the top boundary portion
$\hypthreearg{\timefunction}{[- \rightu,u']}{\muxmulevelsetvalue}$; this is a key difference from <cit.>,
where we integrated across all of space.
To control some of these boundary
integrals, we need to identify and integrate some perfect $\hypthreearg{\timefunction}{[- \rightu,u']}{\muxmulevelsetvalue}$-divergences,
which leads to further co-dimension-two boundary integrals on the rough tori
$\twoargroughtori{\timefunction,u'}{\muxmulevelsetvalue}$ and $\twoargroughtori{\timefunction,- \rightu}{\muxmulevelsetvalue}$.
The key point is that, as in <cit.>, the rough tori boundary integrals either are controlled by the data or
enter with a good sign,
which is crucial for closing the elliptic estimates;
see the tori integrals in the identity (<ref>).
The analysis tied to these issues is very technical and is captured in divergence-form
in Lemma <ref>.

§.§ Outline of the remainder of the paper

* In Sect. <ref>, we define the fluid variables that we use in our analysis and
recall the geometric formulation of the flow derived in <cit.>.

* In Sects. <ref>–<ref>,
we derive basic properties and identities (not yet estimates)
tied to the eikonal function $u$, the rough time function $\timefunctionarg{\muxmulevelsetvalue}$,
the corresponding geometries,
and changes of variables between various coordinate systems and vectorfields.

* In Sect. <ref>, we construct
the “ingoing” $\gfour$-null vectorfield $\uLunit$, which is transversal to the characteristics $\nullhyparg{u}$
and complements the $\gfour$-null vectorfield $\Lunit$, which generates the $\nullhyparg{u}$.
We use $\uLunit$ in Sect. <ref>,
when we derive the “elliptic-hyperbolic” identities
that we will use to control the top-order derivatives of
$\vortrenormalized$ and $\GradEnt$.

* In Sect. <ref>, we define various norms, area forms, and volume forms.
We also introduce notation for various strings of commutation vectorfields.

* In Sect. <ref>, we introduce schematic notation and derive
various identities in schematic form. These will be used throughout the remainder of the paper.

* In Sects. <ref>–<ref>,
we list various parameters corresponding to the solutions under study,
state our assumptions on the initial data,

* In Sect. <ref>,
we state all our bootstrap assumptions – except for the energy bootstrap assumptions.

* In Sect. <ref>,
we use the bootstrap assumptions to derive preliminary pointwise, commutator, and differential operator comparison estimates.

* In Sect. <ref>,
we use the bootstrap assumptions to analyze
the data hypersurface $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot)}$
for the rough time function $\timefunctionarg{\muxmulevelsetvalue}$,
which solves the transport $\Wtransarg{\muxmulevelsetvalue} \timefunctionarg{\muxmulevelsetvalue} = 0$
with data prescribed on $\datahypfortimefunctiontwoarg{-\muxmulevelsetvalue}{[\timefunction_0,\timefunctionboot)}$.
We also derive basic properties of the flow map of $\Wtransarg{\muxmulevelsetvalue}$.

* In Sect. <ref>,
we use the bootstrap assumptions to derive estimates for
$\timefunctionarg{\muxmulevelsetvalue}$. We also show that various quantities extend to the
closure of the bootstrap region as elements of various Hölder spaces.
Finally, we study the properties of the map
$\CHOVroughtomumuxmu{\muxmulevelsetvalue}(\timefunction,u,x^2,x^3) = (\upmu,\muX \upmu,x^2,x^3)$,
which is important for understanding the structure of the singular boundary.

* In Sect. <ref>,
we use the bootstrap assumptions to control the flow map of the $\gfour$-null vectorfield
$\argLrough{\muxmulevelsetvalue} = \frac{1}{\Lunit \timefunctionarg{\muxmulevelsetvalue}} \Lunit$,
which is the principal operator in many of the transport equations that we will later study.
construct $\timefunctionarg{\muxmulevelsetvalue}$,
derive estimates for $\timefunctionarg{\muxmulevelsetvalue}$ and its geometry,
and study the diffeomorphism properties of the change of variables map
$(t,u,x^2,x^3) \rightarrow (\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$.

* In Sect. <ref>,
we derive $L^{\infty}$ estimates that yield improvements of many of
our quantitative bootstrap assumptions.

* In Sect. <ref>, we derive
sharp estimates for $\upmu$. These are crucial for the energy estimates and for understanding
the structure of the singular boundary.
We also study the homeomorphism and diffeomorphism properties of the change of variables map
$(t,u,x^2,x^3) \rightarrow (t,x^1,x^2,x^3)$.

* In Sect. <ref>, we construct the modified quantities
that we use to control the acoustic geometry in $L^2$ without derivative loss.

* In Sect.<ref>,
we construct some basic ingredients needed for the hyperbolic energy estimates.
In particular, we define energies and null-fluxes, derive energy–null-flux identities for solutions
to wave equations and transport equations, and exhibit the coerciveness of the energies
and null fluxes.

* In Sect. <ref>,
we derive the “elliptic-hyperbolic” identities
that we will use to control the top-order derivatives of
$\vortrenormalized$ and $\GradEnt$.

* In Sect. <ref>, we commute the wave equations
satisfied by the wave variables  $\wavearray = (\RRiemann,\LRiemann,v^2,v^3,\Ent)$ up to top-order and
derive pointwise estimates for the inhomogeneous terms.
These are a preliminary ingredient
in our $L^2$ analysis of the inhomogeneous terms.

* In Sect. <ref>,
we provide an analog of Sect. <ref> for the transport variables.
That is, we commute the transport equations satisfied by
$\VortVort$, and $\DivGradEnt$
up to top-order and
derive pointwise estimates for the inhomogeneous terms.
We also derive pointwise estimates at the low derivative levels for the rough acoustical
geometry. All the estimates in this section
are preliminary ingredients for our derivation of
energy and elliptic estimates.

* In Sect. <ref>,
we state all of our a priori energy estimates.
We also state bootstrap assumptions for the energies of the
“wave variables” $\wavearray \eqdef (\RRiemann,\LRiemann,v^2,v^3,\Ent)$.
The proof of the energy estimates occupies a substantial portion of the remainder of the paper,
all the way through Sect. <ref>.

* In Sect. <ref>,
we derive preliminary $L^2$ estimates for the below-top-order derivatives of the
eikonal function quantities $\upmu$, $\Lunit^i$, $\upchi$, and $\mytr_{\gtorus}\upchi$.
We also derive preliminary $L^2$ for $\wavearray$ that lose one derivative.

* In Sect. <ref>,
we use the wave energy bootstrap assumptions to derive below-top-order energy estimates for
and $\DivGradEnt$.

* In Sect. <ref>,
we use the wave energy bootstrap assumptions and the below-top-order energy estimates of
Sect. <ref>
to derive the top-order “elliptic-hyperbolic” energy estimates
and $\DivGradEnt$.

* In Sect. <ref>, we derive general elliptic estimates on the rough
tori, which we will later use to control the top-order derivatives of the acoustic geometry
along the rough foliations.

* In Sect. <ref>, we derive up-to-top-order
energy estimates for the wave variables and the acoustic geometry.
This completes the proof of the energy estimates
stated in Sect. <ref>
and in particular yields a strict improvement of our wave energy bootstrap assumptions.

* In Sect. <ref>,
we use the energy estimates and Sobolev embeddings to derive $L^{\infty}$ estimates
that yield strict improvements of the remaining quantitative bootstrap assumptions.
This closes the bootstrap argument and completes our proof of a priori estimates.

* In Sect. <ref>, we use the a priori estimates
and a continuation principle to show that we can extend the solution all the way up to the level
set $\lbrace \timefunctionarg{\muxmulevelsetvalue} = 0 \rbrace$, which contains
the two-dimensional torus $\twoargmumuxtorus{0}{-\muxmulevelsetvalue}$, which in turn is contained in
the singular boundary.
We provide these results as Theorem <ref>,
which is the first main theorem of the paper.
This theorem provides a development of the data
containing the portion of the singular boundary that is “accessible” via the foliation of spacetime
by the level sets of $\timefunctionarg{\muxmulevelsetvalue}$.

* In Sect. <ref>, we study the union
of the developments as $\muxmulevelsetvalue$ varies, and we define an interesting sub-region,
$\MInteresting$, which contains
a portion of the singular boundary, namely $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$, and its past boundary
$\partial_- \mathcal{B}^{[0,\muxmulevelsetvalue_0]}$, the crease.
We also construct a new rough time function
$\newtimefunction$, whose level sets foliate $\MInteresting$.
Finally, we derive various quantitative and qualitative properties of
various geometric objects tied to $\MInteresting$
and $\newtimefunction$.

* In Sect. <ref>, we study
the homeomorphism and diffeomorphism properties of the change of variables map
$\Upsilon(t,u,x^2,x^3) = (t,x^1,x^2,x^3)$ on $\MInteresting$.
We also exhibit the properties of $\Upsilon(\mathcal{B}^{[0,\muxmulevelsetvalue_0]})$,
i.e., the embedding of the singular boundary in Cartesian coordinate space.

* In Sect. <ref>, we state and prove Theorem <ref>,
which is the main result of the paper. The theorem shows that
$\MInteresting$ contains the portion $\mathcal{B}^{[0,\muxmulevelsetvalue_0]}$
of the singular boundary and the crease, and it gives a detailed description of the solution
in the different coordinate systems as well as the change of variables maps.
The theorem is essentially a conglomeration of results derived
earlier in the paper.

§ BASIC SETUP, COMPRESSIBLE EULER FLOW, AND ITS GEOMETRIC REFORMULATION

In this section, we first introduce some basic notational conventions and definitions.
We then provide a standard first-order quasilinear hyperbolic formulation of compressible Euler flow.
Next, we define a series of additional fluid variables and geometric tensors associated to the flow.
Finally, we recall the new formulation of the flow derived in <cit.>. More precisely,
we use a slightly modified version of the formulation in <cit.> that is adapted to the nearly
plane-symmetric solutions featured in our main results. The only difference with <cit.>
is that here, we replace the density and
velocity component $v^1$ with our “almost Riemann invariants”
$\RRiemann$ and $\LRiemann$, which are useful
for capturing smallness in the regime under study.
In total, the new formulation comprises covariant wave equations, which govern the propagation of sound waves,
coupled to systems of transport-div-curl systems, which drive the evolution of the vorticity and entropy.

§.§ Basic notation and conventions

The precise definitions of some of the concepts referred to
here are provided later in the article.

* (Cartesian coordinates)
Our analysis takes place on subsets of the spacetime manifolds
$\mathbb{R} \times \Sigma$,
where $\Sigma \eqdef \mathbb{R} \times \mathbb{T}^2$ is the spatial manifold.
We fix a standard Cartesian coordinate system $\lbrace x^{\alpha} \rbrace_{\alpha = 0,1,2,3}$
on $\mathbb{R} \times \Sigma$, where $t \eqdef x^0 \in \mathbb{R}$
is the time coordinate and $(x^1,x^2,x^3) \in \Sigma$
are the spatial coordinates (where $(x^2,x^3)$ are standard coordinates[While the coordinates $x^2,x^3$ on $\mathbb{T}^2$ are only locally defined, the corresponding partial derivative vectorfields $\partial_2,\partial_3$ can be extended
so as to form a global smooth frame on $\mathbb{T}^2$. Similar remarks apply to the one-forms $dx^2,dx^3$
These simple observations are relevant for this paper because when we derive estimates, the coordinate functions $x^2,x^3$
themselves are never directly relevant; what matters are estimates for the components of various tensorfields
with respect to the frame $\lbrace \partial_t, \partial_1, \partial_2, \partial_3 \rbrace$
and the basis dual co-frame $\lbrace dt, dx^1, dx^2, dx^3 \rbrace$, which are everywhere smooth.
on $\mathbb{T}^2$). By a plane-symmetric solution,
we mean one whose fluid variables are independent of $(x^2,x^3)$ in this coordinate system.
We sometimes refer to $t$ as the Cartesian time function.”
$\Sigma_{t'} \eqdef \lbrace (t,x^1,x^2,x^3) \in \R \times \R \times \T^2 | \ t \equiv t' \rbrace$
denotes the standard flat hypersurface of constant Cartesian time.

* (Cartesian coordinate partial derivatives)
We use the notation $\lbrace \partial_{\alpha} \rbrace_{\alpha = 0,1,2,3}$
(or $\partial_t \eqdef \partial_0$) to denote the Cartesian coordinate partial derivative

* (Lowercase Greek index conventions) Lowercase Greek spacetime indices
$\alpha$, $\beta$, etc. correspond to the Cartesian coordinate spacetime coordinates
and vary over $0,1,2,3$. All lowercase Greek indices are lowered and raised with the acoustical metric
$\gfour$ (see definition (<ref>))
and its inverse $\gfour^{-1}$, and not with the Minkowski metric.
Throughout the article, if $\upxi$ is a type $\binom{m}{n}$ spacetime tensorfield,
then unless we indicate otherwise, in our identities and estimates,
$\left\lbrace \upxi_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_m} \right\rbrace_{
\alpha_1, \cdots \alpha_m, \beta_1, \cdots, \beta_n = 0,1,2,3}$ denotes its
components with respect to the Cartesian coordinates.
This is important because some of our identities
and estimates hold only with respect to the Cartesian coordinates.

* (Lowercase Latin index conventions)
Lowercase Latin spatial indices $a$, $b$, etc. correspond to the Cartesian spatial coordinates and vary over $1,2,3$.
Much like in the previous point,
if $\upxi$ is a type $\binom{m}{n}$ $\Sigma_t$-tangent tensorfield
(see Def. <ref>),
then unless we indicate otherwise, in our identities and estimates,
$\left\lbrace \upxi_{b_1 \cdots b_n}^{a_1 \cdots a_m} \right\rbrace_{a_1,\cdots,a_m,b_1,\cdots,b_n = 1,2,3}$
denotes its components with respect to the Cartesian spatial coordinates.

* (Uppercase Latin index conventions)
Uppercase Latin spatial indices $A$, $B$, etc. correspond to the coordinates
$(x^2,x^3)$ on $\mathbb{T}^2$ and vary over $2,3$. In particular, if $V$ is a vectorfield,
and $A \in \lbrace 2,3 \rbrace$, then $V^A = V^{\alpha} \partial_{\alpha} x^A$,
where $(x^2,x^3)$ are the standard Cartesian coordinates on $\mathbb{T}^2$.

* (Einstein summation)
We use Einstein's summation convention in that repeated indices are summed,
e.g., $\Lunit^A X^A \eqdef \Lunit^2 X^2 + \Lunit^3 X^3$.

* (Use of “$\cdot$”)
We sometimes use “$\cdot$” to denote the natural contraction between two tensors.
For example, if $\upxi$ is a spacetime one-form and $V$ is a
spacetime vectorfield,
then $\upxi \cdot V \eqdef \upxi_{\alpha} V^{\alpha}$.
At other times, we use “$\cdot$” to schematically denote products,
e.g., $A_1 \cdot A_2 \cdot A_3$ is a trilinear form in $A_1, A_2, A_3$.

* (Tensor contractions)
If $V$ and $W$ are vectorfields, then $V_W \eqdef V^{\alpha} W_{\alpha} = \gfour_{\alpha \beta} V^{\alpha} W^{\beta}$.
If $\upxi$ is a one-form and $V$ is a vectorfield, then $\upxi_V \eqdef \upxi_{\alpha} V^{\alpha}$.
We use similar notation when contracting higher-order tensorfields against vectorfields.
For example, if $\upxi$ is a type $\binom{0}{2}$ tensorfield and
$V$ and $W$ are vectorfields, then $\upxi_{VW} \eqdef \upxi_{\alpha \beta} V^{\alpha} W^{\beta}$.

* (Commutator of operators)  If $Q_1$ and $Q_2$ are two operators, then $[Q_1,Q_2] \eqdef Q_1 Q_2 - Q_2 Q_1$ denotes their commutator.

* (Constants) We establish conventions for constants in Sect. <ref>.

§.§ Basic differential operators

In our analysis, we will encounter many kinds of differential operators.
Here, we define some basic operators.

If $f$ is a scalar function, then $\rmD f$
denotes the gradient one-form associated to $f$,
e.g., $(\rmD f)_{\alpha} \eqdef \rmD f \cdot \partial_{\alpha} = \partial_{\alpha} f$.

If $V$ is a vectorfield and $f$ is a scalar function, then
$V f \eqdef V^{\alpha} \partial_{\alpha} f = V \cdot \rmD f$
denotes the derivative of $f$ in the direction $V$.

$\Flatdiv$ and $\Flatcurl$ respectively denote the Euclidean spatial divergence and curl operators.
That is, given a $\Sigma_t$-tangent
vectorfield $V = V^a \partial_a$, we define,
relative to the Cartesian spatial coordinates,
$\Flatdiv V$ and $\Flatcurl V$ to be the following
scalar function and $\Sigma_t$-tangent vectorfield:

\begin{align} \label{E:FLATDIVANDCURL}
\Flatdiv V
& \eqdef \partial_a V^a,
(\Flatcurl V)^i
& \eqdef \upepsilon_{iab} \partial_a V^b,
\end{align}

where $\upepsilon_{iab}$ is the fully antisymmetric symbol normalized by $\upepsilon_{123} = 1$.

§.§ A first-order formulation involving the logarithmic density

§.§.§ The logarithmic density, assumptions on the equation of state, and normalizations

We find it convenient to work with the logarithmic density featured in the next
definition, rather than the density.
In the rest of the paper,

\begin{align} \label{E:BACKGROUNDDENSITY}
\overline{\varrho} & > 0
\end{align}

denotes a fixed constant “background density.”

We define the logarithmic density $\LogDensity$ as follows:

\begin{align} \label{E:LOGDENS}
\LogDensity
& \eqdef \ln\left(\frac{\varrho}{\overline\varrho}\right).
\end{align}

In the rest of the paper,
we view the speed of sound $\Speed$ (which is defined in (<ref>))
to be a function of $(\LogDensity,\Ent)$.
Note that by (<ref>) and the chain rule,
we have $\Speed(\LogDensity,\Ent)
= \sqrt{(\overline{\varrho})^{-1} \exp(-\LogDensity) p;_{\LogDensity}}$,
where $p;_{\LogDensity} \eqdef \tfrac{\p p}{\p \LogDensity}$ denotes the partial derivative of the equation of state
with respect to the logarithmic density at fixed $\Ent$.

[Partial differentiation with respect to state-space variables]

In accordance with the above notation,
for any scalar function $f = f(\LogDensity,\Ent)$, we use the notation
$f_{;\Ent} \eqdef \frac{\partial f}{\partial \LogDensity}$ to denote
the partial derivative of $f$
with respect to the logarithmic density at fixed $\Ent$.
Similarly, we denote the partial derivative of $f$ with respect to $\Ent$ at fixed $\LogDensity$
by $f_{;\Ent} \eqdef \frac{\partial f}{\partial \Ent}$.
We also write $ f_{;\LogDensity;\Ent} \eqdef \frac{\partial^2 f}{\partial \Ent \partial \LogDensity}$ and use similar notation for other higher partial derivatives of $f$ with respect to $\LogDensity, \Ent$.

To ensure that shocks occur for solutions near static isentropic fluid states with constant density $\overline{\varrho} > 0$,
we assume the following non-degeneracy condition:

\begin{align}\label{E:NONDEGENCONDITION}
\overline{\Speed}^{-1} \overline{\Speed_{;\uprho}} + 1
& \neq 0,
\end{align}

where LHS (<ref>) is defined to be the constant
obtained by evaluating $\Speed^{-1} \Speed_{;\LogDensity} + 1$ at $\LogDensity = \Ent \equiv 0$.
Equation (<ref>) ensures that the null condition fails to hold for perturbations
of the background solution $\LogDensity = \Ent \equiv 0$; see Sect. <ref>.
Our main results hold for all equations of state except for that of the Chaplygin gas,
namely $p = C_0 - C_1 \exp(- \LogDensity)$, where $C_0 \geq 0$ and $C_1 > 0$ are constants.
This equation of state is degenerate in the following sense: $\Speed^{-1} \Speed_{;\LogDensity} + 1 \equiv 0$.

By rescaling Cartesian time if necessary, we can assume the following convenient normalization condition:

\begin{align} \label{E:BACKGROUNDSOUNDSPEEDISUNITY}
\Speed(\LogDensity = 0, \Ent = 0)
& = 1.
\end{align}

§.§.§ A first-order formulation involving the logarithmic density

From definition (<ref>) and equations (<ref>)–(<ref>),
it follows that
relative to the standard Cartesian coordinates on $\mathbb{R} \times \Sigma$,
the compressible Euler equations can be expressed as the following system
in $\LogDensity$, $v$, and $\Ent$:

\begin{align}
\Transport v^i
& =
- \Speed^2 \updelta^{ia} \partial_a \LogDensity
- \exp(-\LogDensity) \frac{p_{;\Ent}}{\overline{\varrho}} \updelta^{ia} \partial_a \Ent,
\label{E:BVIEVOLUTION}
\\
\label{E:BLOGDENSITYEVOLUTION}
\Transport \LogDensity
& = - \Flatdiv v,
\\
\Transport \Ent
& = 0.
\label{E:BENTROPYEVOLUTION}
\end{align}

§.§ The almost Riemann invariants

To study solutions close to simple isentropic plane-symmetric solutions,
we find it convenient to replace
$\LogDensity$ and $v^1$ with a pair of “almost Riemann invariants,”
denoted by $\RRiemann$ and $\LRiemann$.
In this paper, exact isentropic plane-symmetric simple solutions are such that
$\RRiemann$ is a function of only $(t,x^1)$
$\LRiemann = \Ent = v^2 = v^3 \equiv 0$.

We define the almost Riemann invariants[Compare $\almostRiemann_{(\pm)}$ with the true Riemann invariants for the plane-symmetric solutions given by (<ref>).] away from symmetry $\almostRiemann_{(\pm)}$ as follows:

\begin{align} \label{E:ALMOSTRIEMANNINVARIANTS}
\almostRiemann_{(\pm)}
& \eqdef
\pm
\almostRiemannfunction(\LogDensity,\Ent),
\mbox{where }
\almostRiemannfunction(\LogDensity,\Ent)
\eqdef
\int_0^{\LogDensity}
\Speed(\LogDensity',\Ent)
\, \rmD \LogDensity'.
\end{align}

We have introduced $\almostRiemann_{(\pm)}$ because they are convenient for
studying perturbations of simple isentropic plane-waves (for which only $\RRiemann$ is non-vanishing);
$\almostRiemann_{(\pm)}$ allow us to capture various kinds of smallness of the
It is well-known that for isentropic plane-symmetric solutions,
one can use $\lbrace \RRiemann, \LRiemann \rbrace$ as
the unknowns in place of $\lbrace \LogDensity, v^1 \rbrace$;
see Appendix <ref>.
Away from symmetry, a similar remark also holds for our almost Riemann invariants,
provided we take into account the entropy.
Specifically, from (<ref>) and definition (<ref>),
it follows that $v^1 = \frac{1}{2}(\RRiemann - \LRiemann)$,
and that when $\LogDensity$, $v^1$, and $\Ent$ are sufficiently small
(as is captured by the smallness parameters $\mathring{\upalpha}$ and $\initialsmall$ that we
introduce in Sect. <ref>),
we have (via the implicit function theorem)
$\LogDensity =
(\RRiemann - \LRiemann)
\cdot
\widetilde{F}(\RRiemann - \LRiemann,\Ent)$,
where $\widetilde{F}$ is a smooth function.
This allows us to control $\LogDensity$ and $v^1$ in terms of
$\RRiemann$, $\LRiemann$, and $\Ent$.
Throughout the article, we use this observation without explicitly pointing it out.
In particular, even though many of the equations that we study explicitly involve
$\LogDensity$ and $v^1$, it should be understood that we always
estimate these quantities in terms of the “wave variables”
$\RRiemann$, $\LRiemann$, and $\Ent$,
which are featured in the array (<ref>)
defined below.

§.§ The higher order fluid variables

The “higher order” fluid variables in the next definition appear in Theorem <ref>,
which provides the formulation of compressible Euler flow that we use throughout our analysis.

* We define the specific vorticity to be the $\Sigma_t$-tangent vectorfield whose Cartesian spatial components are:

\begin{align} \label{E:SPECIFICVORTICITYDEF}
\vortrenormalized^i
& \eqdef
\frac{(\Flatcurl v)^i}{\exp(\LogDensity)}
\frac{\upepsilon_{ijk} \updelta^{jl}\p_lv^k}{\exp(\LogDensity)},
\end{align}

where $\updelta^{jl}$ is the Kronecker delta.

* We define the entropy gradient to be the $\Sigma_t$-tangent vectorfield whose Cartesian spatial components are:

\begin{align} \label{E:GRADENTDEF}
\GradEnt^i
& \eqdef \updelta^{ia}\p_a \Ent = \p_i \Ent.
\end{align}

* We define the modified fluid variables to be the $\Sigma_t$-tangent vectorfield $\VortVort$ and the scalar function
$\DivGradEnt$ whose Cartesian spatial components are:

\begin{align} \label{E:MODIFIEDCURLOFVORTICITY}
\VortVort^i
& \eqdef
\exp(-\LogDensity) (\Flatcurl \vortrenormalized)^i
\exp(-3\LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}} \GradEnt^a \partial_a v^i
\exp(-3\LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}} (\Flatdiv v) \GradEnt^i,
\\
\DivGradEnt
& \eqdef
\exp(-2 \LogDensity) \Flatdiv \GradEnt
\exp(-2 \LogDensity) \GradEnt^a \partial_a \LogDensity.
\label{E:MODIFIEDDIVERGENCEOFENTROPYGRADIENT}
\end{align}

§.§ Arrays of fluid variables and array norm notation

We provide the next definition for notational convenience.

We define the array of wave[These “wave variables” solve wave equations;
see Theorem <ref>.] variables as follows:

\begin{align} \label{E:ARRAYOFWAVEVARIABLES}
\wavearray
& \eqdef (\Psi_0,\Psi_1,\Psi_2,\Psi_3,\Psi_4)
\eqdef (\RRiemann,\LRiemann,v^2,v^3,\Ent).
\end{align}

We define the partial array of wave variables by:

\begin{align} \label{E:PARTIALWAVEARRAY}
\wavearraypartial
& \eqdef (\LRiemann,v^2,v^3,\Ent).
\end{align}

We view $\wavearray$ to be an array of scalar functions $\Psi_{\iota}$, where
$\iota = 0,\cdots,4$. We will not attribute any tensorial structure to the labeling index $\iota$ besides simple contractions, denoted by $\diamond$, corresponding to the chain rule; see Def. <ref>.

In the next definition, we introduce notation for norms of arrays.

* Given the fluid variable array $\wavearray$ from Def. <ref>, we define:

\begin{align} \label{E:ABSOLUTVALUEOFPSIARRAY}
& \eqdef \max_{\iota \in \{0,\cdots,4\}} | \Psi_\iota|.
\end{align}

For any norm $\|\cdot\|$ on scalar functions that appears in the paper, we set:

\begin{align} \label{E:NORMOFPSIARRAY}
\| \wavearray\|
& \eqdef \max_{\iota \in \{0,\cdots,4\}} \| \Psi_\iota \|.
\end{align}

We use a similar convention for $\vortrenormalized$:
$|\vortrenormalized| = \max_{a = 1,2,3} |\vortrenormalized^a|$,
and similarly for $\wavearraypartial$, $\GradEnt$, $\VortVort$,

* We use the following convention when taking norms of more than one variable at a time:

\begin{align} \label{E:NORMSOFMORETHANONEVARIABLE}
\| (\vortrenormalized,\GradEnt)\|
& \eqdef \max\{  \| \vortrenormalized\|,  \| \GradEnt \|\}.
\end{align}

§.§ The acoustical metric and related geometric objects

In the following definition, we introduce the acoustical metric and its inverse.
This Lorentzian[By “Lorentzian,” we mean that viewed as a quadratic form, the symmetric $4 \times 4$ matrix
$(\gfour_{\alpha \beta})_{\alpha,\beta=0,1,2,3}$ has signature $(-,+,+,+)$.]
metric drives the propagation of sound waves and is necessary to reveal the full geometry of the singular boundary.

Relative to the Cartesian coordinates $(t,x^1,x^2,x^3)$,
we define the acoustical metric $\gfour$ and the inverse acoustical metric $\gfour^{-1}$ as
follows, where the material derivative vectorfield $\Transport$ is defined in (<ref>)
and the speed of sound $\Speed$ is defined in (<ref>):

\begin{align}
\gfour & = - \rmD t \otimes \rmD t + \Speed^{-2} \sum_{a=1}^3 (\rmD x^a - v^a \rmD t)\otimes  (\rmD x^a - v^a \rmD t),
\label{E:ACOUSTICALMETRIC} \\
\gfour^{-1} & = - \Transport \otimes \Transport + \Speed^2 \sum_{a=1}^3 \p_a\otimes \p_a.
\label{E:INVERSEACOUSTICALMETRIC}
\end{align}

Straightforward calculations yield that $\gfour_{\alpha \gamma} (\gfour^{-1})^{\gamma \beta} = \updelta_{\alpha}^{\beta}$,
where $\updelta_{\alpha}^{\beta}$ is the Kronecker delta, i.e., $\gfour^{-1}$ is indeed the inverse of $\gfour$.
In the remainder of the article, we silently lower and raise lowercase Greek indices with $\gfour$ and $\gfour^{-1}$,
e.g., $V^{\alpha} = (\gfour^{-1})^{\alpha \beta} V_{\beta}$.

In our forthcoming analysis, the undifferentiated quantities
$v^i$ and $\Speed - 1$ will be small, where
we quantify their smallness via the parameters $\mathring{\upalpha}$ and $\initialsmall$,
which we introduce in Sect. <ref>.
Hence, in view of (<ref>), we find it convenient to introduce the following decomposition:

\begin{align} \label{E:SPLITMETRICINTOMINKOWSKIANDREMAINDERPART}
\gfour_{\alpha \beta}(\wavearray)
& = m_{\alpha \beta} + \gfour_{\alpha \beta}^{(\textnormal{Small})}(\wavearray)
\end{align}

where $m_{\alpha \beta} = \diag(-1,1,1,1)$ is the Minkowski metric and $\gfour_{\alpha \beta}^{(\textnormal{Small})}(\wavearray)$ is a smooth function of $\wavearray$ satisfying:

\begin{align} \label{E:METRICPERTURBATIONVANISHESATTRIVIALPSISOLUTION}
\gfour_{\alpha \beta}^{(\textnormal{Small})}(\wavearray = 0)
& = 0.
\end{align}

The scalar functions $G^{\iota}_{\alpha \beta}$ in the following definition
will appear as coefficients in many of the equations that we study.

Viewing the Cartesian component functions
$\gfour_{\alpha \beta} = \gfour_{\alpha \beta}(\wavearray)$
as functions of the wave variables, for $\alpha,\beta = 0,1,2,3$ and $\iota = 0,1,2,3,4$,
we define:

\begin{align} \label{E:DERIVATIVESOFMETRICWRTFLUID}
G^{\iota}_{\alpha \beta}(\wavearray)
& \eqdef \frac{\p}{\p \Psi_{\iota}} \gfour_{\alpha \beta}(\wavearray),
\\
\vec{G}_{\alpha \beta}
& = \vec{G}_{\alpha \beta}(\wavearray)
\eqdef
\left(G_{\alpha \beta}^0 (\wavearray),
G_{\alpha \beta}^1 (\wavearray),
G_{\alpha \beta}^2 (\wavearray),
G_{\alpha \beta}^3 (\wavearray),
G_{\alpha \beta}^4 (\wavearray)
\right).
\label{E:ARRAYVERSIONDERIVATIVESOFMETRICWRTFLUID}
\end{align}

For each fixed $\iota \in \{0,\cdots,4\}$,
we view $\{G^{\iota}_{\alpha \beta}\}_{\alpha,\beta = 0,\cdots,3}$ to be the Cartesian components of the spacetime tensorfield
Similarly, we view
$\{ \vec{G}_{\alpha \beta}\}_{\alpha,\beta = 0,\cdots,3}$ to be the Cartesian components of
the array-valued spacetime tensorfield $\vec{G}$.

Let $V_1,V_2$ be vectorfields,
and let $D$ be a differential operator. We define:

\begin{align} \label{D:DERIVATIVEOFARRAY}
D \wavearray
& \eqdef (D \Psi_0, D \Psi_1, D \Psi_2, D \Psi_3, D \Psi_4),
\vec{G}_{V_1 V_2} \diamond D \wavearray
& \eqdef \sum_{\iota = 0}^4 G_{\alpha \beta}^\iota V_1^{\alpha}V_2^\beta D \Psi_\iota.
\end{align}

§.§ Covariant wave operator and $\gfour$-null forms

In this section, we provide some definitions that we need
to state Theorem <ref>, which provides
the geometric formulation of compressible Euler flow that we use throughout our analysis.

We start by recalling the standard definition of the covariant wave operator $\square_{\gfour}$.

The covariant wave operator $\square_{\gfour}$
of the acoustical metric $\gfour = \gfour(\wavearray)$
acts on scalar-valued functions $\varphi$ as follows:[The formula (<ref>)
holds relative to arbitrary coordinates.]

\begin{align} \label{E:WAVEOPERATORARBITRARYCOORDINATES}
\square_{\gfour} \varphi
:= \frac{1}{\sqrt{|\mbox{\upshape det} \gfour|}}
\partial_{\alpha}
\left\lbrace
\sqrt{|\mbox{\upshape det} \gfour|} (\gfour^{-1})^{\alpha \beta}
\partial_{\beta} \varphi
\right\rbrace.
\end{align}

We now recall the definition of a standard null form with respect to the acoustical metric
(“$\gfour$-null form” for short).

Let $\varphi$ and $\widetilde{\varphi}$ be scalar functions.
We define
$\nullform^{(g)}(\partial\varphi,\partial\widetilde{\varphi})$ to be the following
derivative-quadratic term:

\begin{align}\label{E:Q0DEF}
\nullform^{(\gfour)}(\partial \varphi,\partial \widetilde{\varphi})
& =
(\gfour^{-1})^{\alpha \beta}
\partial_{\alpha} \varphi \partial_{\beta} \widetilde{\varphi}.
\end{align}

For $0 \leq \alpha < \beta \leq 3$,
we define $\nullform_{\alpha \beta}(\partial \varphi,\partial\widetilde{\varphi})$
to be the following derivative-quadratic term:

\begin{align}\label{E:QIJDEF}
\nullform_{\alpha \beta}(\partial\varphi,\partial\widetilde{\varphi})
= 	\partial_{\alpha} \varphi \partial_{\beta} \widetilde{\varphi}
\partial_{\beta} \varphi \partial_{\alpha} \widetilde{\varphi}.
\end{align}

In the rest of the paper, we use the terminology
null form relative to $\gfour$
$\gfour$-null form
to denote any linear combination
of the standard null forms
with (possibly solution dependent) coefficients that are controllable under the scope of our approach.

§.§ The geometric wave-transport-divergence-curl formulation of the compressible Euler equations

Our main results fundamentally rely on the following formulation of the compressible Euler equations, first derived in <cit.>.

Let $\overline{\varrho} > 0$ be any constant background density,[Recall
that $\LogDensity$ depends on $\overline{\varrho}$; see Def. <ref>.]
and assume that $(\LogDensity,v^1,v^2,v^3,\Ent)$ is
a solution
to the compressible Euler equations
in three spatial dimensions under an arbitrary equation of state $p = p(\varrho,\Ent)$ with positive sound speed
$\Speed$ (see (<ref>)).
Let $\Transport$ be the material derivative vectorfield defined in (<ref>),
let $\gfour$ be the acoustical metric from Def. <ref>,
let $\square_{\gfour}$ be the corresponding covariant wave operator from Def. <ref>,
let $\almostRiemann_{(\pm)}$ be the almost Riemann invariants from Def. <ref>
(see Remark <ref> concerning their significance for this paper),
let $\almostRiemannfunction  = \almostRiemannfunction(\LogDensity,\Ent)$ be the function from
Def. <ref>,
and let
be the higher order variables from Def. <ref>.
Then the scalar-valued functions
$\Flatdiv \vortrenormalized$,
$(\Flatcurl \GradEnt)^i$,
also solve the following equations,
where $\upepsilon_{ijk}$ is the fully antisymmetric symbol normalized by $\upepsilon_{123}=1$,
and the Cartesian component functions $v^i$ are
treated as scalar-valued functions
under covariant differentiation on LHS (<ref>):

Covariant wave equations.

\begin{align}
\square_{\gfour(\wavearray)} v^i
& =
\Speed^2 \exp(2 \LogDensity) \VortVort^i
\nullform_{(v)}^i
\mathfrak{L}_{(v)}^i,
\label{E:VELOCITYWAVEEQUATION}	\\
\square_{\gfour(\wavearray)} \almostRiemann_{(\pm)}
= - \Speed^2 \exp(2 \LogDensity) \VortVort^1
\pm
\left\lbrace
F_{;\Ent} c^2 \exp(2\LogDensity)
\Speed \exp(\LogDensity) \frac{p_{;\Ent}}{\overline{\varrho}}
\right\rbrace
\DivGradEnt
\nullform_{(\pm)}
\mathfrak{L}_{(\pm)}, \label{E:WAVEEQUATIONFORALMOSTRIEMANNINVARIANTS} \\
\square_{\gfour(\wavearray)} \LogDensity
& =
\exp(\LogDensity) \frac{p_{;\Ent}}{\overline{\varrho}} \DivGradEnt
\nullform_{(\LogDensity)}
\mathfrak{L}_{(\LogDensity)},
\label{E:RENORMALIZEDDENSITYWAVEEQUATION}
\\
\square_{\gfour(\wavearray)} \Ent
& =
\Speed^2 \exp(2 \LogDensity)  \DivGradEnt
\mathfrak{L}_{(\Ent)}.
\label{E:ENTROPYWAVEEQUATION}
\end{align}

Transport equations.

\begin{align}	\Transport \vortrenormalized^i
& = \mathfrak{L}_{(\vortrenormalized)}^i,
\label{E:RENORMALIZEDVORTICTITYTRANSPORTEQUATION}
\\
\Transport \Ent
& = 0,
\label{E:ENTROPYTRANSPORTMAINSYSTEM}
\\
\Transport \GradEnt^i
& = \mathfrak{L}_{(\GradEnt)}^i.
\label{E:GRADENTROPYTRANSPORT}
\end{align}

Transport-divergence-curl system for the specific vorticity.

\begin{align} \label{E:FLATDIVOFRENORMALIZEDVORTICITY}
\Flatdiv \vortrenormalized
& =
\mathfrak{L}_{(\Flatdiv \vortrenormalized)},
\\
\Transport \VortVort^i
& =
\mainnullform_{(\VortVort)}^i
\nullform_{(\VortVort)}^i
\mathfrak{L}_{(\VortVort)}^i.
\label{E:EVOLUTIONEQUATIONFLATCURLRENORMALIZEDVORTICITY}
\end{align}

Transport-divergence-curl system for the entropy gradient.

\begin{align}
\Transport \DivGradEnt
& =  \mainnullform_{(\DivGradEnt)} +
\nullform_{(\DivGradEnt)},
\label{E:TRANSPORTFLATDIVGRADENT}
\\
(\Flatcurl \GradEnt)^i & = 0.
\label{E:CURLGRADENTVANISHES}
\end{align}

Above, the main terms $\mainnullform_{(\VortVort)}^i$ and $\mainnullform_{(\DivGradEnt)}$
in the transport equations for the modified fluid variables are
the null forms relative to $\gfour$ (see Def. <ref>)
defined by:[Actually, the last the last term on
RHS (<ref>) is not a null form, but rather
a simpler harmless error term.]

\begin{align}
\begin{split} \label{E:TRANSPORTVORTVORTMAINTERMS}
\mainnullform_{(\VortVort)}^i & \eqdef
2 \updelta_{jk} \upepsilon_{iab} \exp(-\LogDensity) (\partial_a v^j) \partial_b \vortrenormalized^k
\upepsilon_{ajk}
\exp(-\LogDensity)
(\partial_a v^i)
\partial_j \vortrenormalized^k
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}}
\left\lbrace
(\Transport \GradEnt^a) \partial_a v^i
(\Transport v^i) \partial_a \GradEnt^a
\right\rbrace
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}}
\left\lbrace
(\Transport v^a) \partial_a \GradEnt^i
( \Transport \GradEnt^i) \partial_a v^a
\right\rbrace,
\end{split}
\\
\mainnullform_{(\DivGradEnt)} & = 	2 \exp(-2 \LogDensity)
\left\lbrace
(\partial_a v^a) \partial_b \GradEnt^b
(\partial_a \GradEnt^b) \partial_b v^a
\right\rbrace
\exp(-\LogDensity) \updelta_{ab} (\Flatcurl \vortrenormalized)^a \GradEnt^b.
\label{E:TRANSPORTDIVGRADENTMAINTERMS}
\end{align}

are[The term
on RHS (<ref>)
and the term $\mainnullform_{(\DivGradEnt)}$
on RHS (<ref>) are also null forms relative to $\gfour$.
We have isolated these two null forms with different notation because
are more difficult to treat than
$\nullform_{(v)}^i$, $\nullform_{(\pm)}$
to bound the top-order derivatives of the “$\mainnullform$” terms,
we rely on the delicate “elliptic-hyperbolic” identities
that we derive in Sect. <ref>.
the null forms relative to $\gfour$ defined by:

\begin{align}
\nullform_{(v)}^i
& \eqdef 	-
\left\lbrace
\Speed^{-1} \Speed_{;\LogDensity}
\right\rbrace
(\gfour^{-1})^{\alpha \beta} (\partial_{\alpha} \LogDensity) \partial_{\beta} v^i,
\label{E:VELOCITYNULLFORM}
\\
\nullform_{(\pm)}
\eqdef \nullform_{(v)}^1 \mp 2 \Speed_{;\LogDensity}
(\gfour^{-1})^{\alpha \beta} \p_\alpha \LogDensity \p_\beta \LogDensity
\pm
\Speed \left\lbrace (\p_a v^a) (\p_b v^b) - (\p_a v^b) \p_b v^a
\right\rbrace,
\label{E:RIEMANNINVARIANTWAVEEQUATIONNULLFORM}
\\
\nullform_{(\LogDensity)}
& \eqdef
3 \Speed^{-1} \Speed_{;\LogDensity}
(\gfour^{-1})^{\alpha \beta} (\partial_{\alpha} \LogDensity) \partial_{\beta} \LogDensity
\left\lbrace
(\partial_a v^a) \partial_b v^b
(\partial_a v^b) \partial_b v^a
\right\rbrace,
\label{E:DENSITYNULLFORM}
\\
\begin{split} \label{E:RENORMALIZEDVORTICITYCURLNULLFORM}
\nullform_{(\VortVort)}^i
& \eqdef
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}}  \GradEnt^i
\left\lbrace
(\partial_a v^b) \partial_b v^a
(\partial_a v^a) \partial_b v^b
\right\rbrace
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}}
\GradEnt^b
\left\lbrace
(\partial_a v^a) \partial_b v^i
(\partial_a v^i) \partial_b v^a
\right\rbrace
\\
& \ \
2 \exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}}
\GradEnt^a
\left\lbrace
(\partial_a \LogDensity)  \Transport v^i
(\partial_a v^i) \Transport \LogDensity
\right\rbrace
\\
&  \ \
2 \exp(-3 \LogDensity) \Speed^{-3} \Speed_{;\LogDensity} \frac{p_{;\Ent}}{\overline{\varrho}}
\GradEnt^a
\left\lbrace
(\partial_a \LogDensity)  \Transport v^i
(\partial_a v^i) \Transport \LogDensity
\right\rbrace
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent;\LogDensity}}{\overline{\varrho}}
\GradEnt^a
\left\lbrace
(\partial_a v^i)
\Transport \LogDensity
(\partial_a \LogDensity) \Transport v^i
\right\rbrace
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent;\LogDensity}}{\overline{\varrho}} \GradEnt^i
\left\lbrace
(\Transport v^a) \partial_a \LogDensity
(\Transport \LogDensity) \partial_a v^a
\right\rbrace
\\
& \ \
2 \exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}} \GradEnt^i
\left\lbrace
(\Transport \LogDensity) \partial_a v^a
(\Transport v^a) \partial_a \LogDensity
\right\rbrace
\\
& \ \
2 \exp(-3 \LogDensity) \Speed^{-3} \Speed_{;\LogDensity} \frac{p_{;\Ent}}{\overline{\varrho}} \GradEnt^i
\left\lbrace
(\Transport \LogDensity) \partial_a v^a
(\Transport v^a) \partial_a \LogDensity
\right\rbrace,
\end{split}
\\
\label{E:DIVENTROPYGRADIENTNULLFORM}
\nullform_{(\DivGradEnt)}
& \eqdef
2 \exp(-2 \LogDensity)
\GradEnt^a
\left\lbrace
(\partial_a v^b) \partial_b \LogDensity
(\partial_a \LogDensity)
\partial_b v^b
\right\rbrace.
\end{align}

In addition, the terms
$\mathfrak{L}_{(v)}^i$, $\mathfrak{L}_{(\pm)}$,
$\mathfrak{L}_{(\Flatdiv \vortrenormalized)}$,

which are at most linear in the derivatives of the unknowns, are defined as follows:

\begin{align}
\begin{split} \label{E:VELOCITYILINEARORBETTER}
\mathfrak{L}_{(v)}^i
& \eqdef
2 \exp(\LogDensity) \upepsilon_{iab} (\Transport v^a) \vortrenormalized^b
\frac{p_{;\Ent}}{\overline{\varrho}} \upepsilon_{iab} \vortrenormalized^a \GradEnt^b
\\
& \ \
\frac{1}{2} \exp(-\LogDensity) \frac{p_{;\LogDensity;\Ent}}{\overline{\varrho}} \GradEnt^a \partial_a v^i
\\
& \ \
2 \exp(-\LogDensity) \Speed^{-1} \Speed_{;\LogDensity} \frac{p_{;\Ent}}{\overline{\varrho}}
(\Transport \LogDensity) \GradEnt^i
\exp(-\LogDensity) \frac{p_{;\Ent;\LogDensity}}{\overline{\varrho}} (\Transport \LogDensity) \GradEnt^i,
\end{split}
\\
\begin{split} \label{E:RIEMANNLINEARORBETTER}
\mathfrak{L}_{(\pm)}
\eqdef
\mathfrak{L}^1_{(v)} \mp \frac{5}{2} \Speed \exp(-\LogDensity) \frac{p_{;\GradEnt^b\LogDensity}}{\overline{\varrho}} \GradEnt^a \p_a \LogDensity \pm 2 \Speed^2 c_{;\Ent} \GradEnt^a\p_a \LogDensity
\\
& \ \
\mp \Speed \exp(-\LogDensity) \frac{p_{;\Ent;\Ent}}{\overline{\varrho}} \updelta_{ab}\GradEnt^a\GradEnt^b
\pm
F_{;\Ent} \mathfrak{L}_{(\Ent)},
\end{split}
\\
\mathfrak{L}_{(\LogDensity)}  \label{E:DENSITYLINEARORBETTER}
& \eqdef
\frac{5}{2} \exp(-\LogDensity) \frac{p_{;\Ent;\LogDensity}}{\overline{\varrho}} \GradEnt^a \partial_a \LogDensity
\exp(-\LogDensity) \frac{p_{;\Ent;\Ent}}{\overline{\varrho}} \updelta_{ab} \GradEnt^a \GradEnt^b,
\\
\mathfrak{L}_{(\Ent)}
& \eqdef
\Speed^2  \GradEnt^a \partial_a \LogDensity
\Speed \Speed_{;\LogDensity} \GradEnt^a \partial_a \LogDensity
\Speed \Speed_{;\Ent} \updelta_{ab} \GradEnt^a \GradEnt^b,
\label{E:ENTROPYLINEARORBETTER}
\\
\mathfrak{L}_{(\vortrenormalized)}^i
& \eqdef
\vortrenormalized^a \partial_a v^i
\exp(-2 \LogDensity) \Speed^{-2} \frac{p_{;\Ent}}{\overline{\varrho}} \upepsilon_{iab} (\Transport v^a) \GradEnt^b,
\label{E:SPECIFICVORTICITYLINEARORBETTER}
\\
\mathfrak{L}_{(\GradEnt)}^i
& \eqdef
\GradEnt^a \partial_a v^i
\upepsilon_{iab} \exp(\LogDensity) \vortrenormalized^a \GradEnt^b,
\label{E:ENTROPYGRADIENTLINEARORBETTER}
\\
\mathfrak{L}_{(\Flatdiv \vortrenormalized)}
& \eqdef - \vortrenormalized^a \partial_a \LogDensity,
\label{E:RENORMALIZEDVORTICITYDIVLINEARORBETTER}
\\
\begin{split} \label{E:RENORMALIZEDVORTICITYCURLLINEARORBETTER}
\mathfrak{L}_{(\VortVort)}^i
& \eqdef
2 \exp(-3 \LogDensity) \Speed^{-3} \Speed_{;\Ent} \frac{p_{;\Ent}}{\overline{\varrho}}
(\Transport v^i) \updelta_{ab} \GradEnt^a \GradEnt^b
\\
& \ \
2 \exp(-3 \LogDensity) \Speed^{-3} \Speed_{;\Ent} \frac{p_{;\Ent}}{\overline{\varrho}}
\updelta_{ab} \GradEnt^a (\Transport v^b) \GradEnt^i
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent;\Ent}}{\overline{\varrho}} \updelta_{ab} (\Transport v^a) \GradEnt^b \GradEnt^i
\\
& \ \
\exp(-3 \LogDensity) \Speed^{-2} \frac{p_{;\Ent;\Ent}}{\overline{\varrho}} (\Transport v^i) \updelta_{ab} \GradEnt^a \GradEnt^b.
\end{split}
\end{align}

Theorem <ref> was essentially proved as <cit.>*Theorem 1,
except that the wave equations (<ref>) for $\almostRiemann_{(\pm)}$ were not derived there.
In <cit.>*Theorem 5.1,
those wave equations were derived
as a straightforward consequence of <cit.>*Theorem 1.

§ THE ACOUSTIC GEOMETRY AND THE CONTROLLING QUANTITIES $\CONTROLVARS$ AND $\BADCONTROLVARS$

In this section, we construct the acoustic geometry,
reveal its basic properties,
and provide the evolution equations satisfied by various geometric tensors.
Our approach is based on the one pioneered by Christodoulou <cit.> in his study
of irrotational and isentropic solutions.
The fundamental object behind all the constructions is an acoustic eikonal function,
that is, a solution $u$ to the acoustic eikonal equation.
The eikonal function is fundamental for our approach because our proof shows that the fluid solution
remains rather smooth relative to the geometric coordinates $(t,u,x^2,x^3)$.
In particular, later on, we will use $u$ to construct suitable commutation and multiplier vectorfields
out of the geometric coordinates to control the solution and acoustic geometry up
to top-order.
This allows us, at least in some ways, to treat the problem of shock formation
as a long-time existence problem relative to the geometric coordinates.
We also introduce the solution variable arrays $\controlvars$ and $\badcontrolvars$,
which contain the wave variables and various components of the acoustic geometry.
We use these arrays throughout the paper to simplify the notation and to allow for convenient,
schematic expressions. We sometimes refer to
$\controlvars$ and $\badcontrolvars$ as the “controlling quantities” because all of the
quantities that we analyze can, in principle, be constructed out of them.

§.§ The eikonal function and inverse foliation density

In the following definition, we introduce the eikonal function $u$ and the inverse foliation density $\upmu$. The level sets of $u$ are the characteristic for the wave operator $\square_{\gfour}$,
while the inverse (i.e., reciprocal)
of the inverse foliation density measures the density of these characteristics.
In particular, the vanishing of $\upmu$ corresponds to the infinite density of the characteristics.
Our main results show that in the regime under study,
the vanishing of $\upmu$ coincides with the blowup of
$|\partial_1 \RRiemann|$.

The eikonal function $u$ is the solution of the following fully nonlinear hyperbolic initial value problem,
where $\gfour$ is the acoustical metric defined in (<ref>)
and the PDE is known as the acoustic eikonal equation:

\begin{align} \label{E:EIKONALEQUATION}
\begin{cases}
(\gfour^{-1})^{\alpha \beta} \partial_{\alpha} u \partial_{\beta} u = 0,
\\
\p_t u > 0,
\\
u|_{\Sigma_0} = - x^1.
\end{cases}
\end{align}

We define the inverse foliation density $\upmu$ by:

\begin{align} \label{E:MUDEF}
\upmu
\eqdef
\frac{1}{(\gfour^{-1})^{\alpha \beta} \partial_{\alpha} t \partial_{\beta} u} > 0.
\end{align}

§.§ Acoustical subsets of spacetime

We define the following “acoustical subsets” of spacetime:

\begin{align}
\Sigma_{t'}
& \eqdef
\{(t,x^1,x^2,x^3) \in \R\times\R\times\T^2 \ | \ t = t'\},
\label{E:SIGMAT}
\\
\nullhyparg{u'}
& \eqdef \{(t,x^1,x^2,x^3) \in \R\times\R\times\T^2 \ | \ u(t,x^1,x^2,x^3) = u'\},
\label{E:NULLHYPERSURFACES}
\\
\ell_{t',u'}
\eqdef
\Sigma_{t'} \cap \nullhyparg{u'}
\{(t,x^1,x^2,x^3) \in \R\times\R\times\T^2 \ | \ t= t',\ u(t,x^1,x^2,x^3) = u'\}.
\label{E:ELLTUSMOOTHTORI}
\end{align}

Given real numbers $u_1 \leq u_2$ and $t_1 \leq t_2$, we define the following “truncated” subsets of spacetime:

\begin{align}
\Sigma_{t'}^{[u_1,u_2]}
& \eqdef \Sigma_{t'} \cap \{(t,x^1,x^2,x^3) \in \R\times\R\times\T^2 \ | \ u_1 \leq u(t,x^1,x^2,x^3) \leq u_2 \},
\label{E:TRUNCATEDSIGMAT}
\\
\nullhyparg{u'}^{[t_1,t_2]}
& \eqdef \nullhyparg{u'} \cap \{(t,x^1,x^2,x^3) \in \R\times\R\times\T^2 \ | \ t_1 \leq t \leq t_2 \}.
\label{E:TRUNCATEDNULLHYPERSURFACES}
\end{align}

We refer to the $\Sigma_t$ as “constant Cartesian-time hypersurfaces,”
the $\nullhyparg{u}$ as “null hypersurfaces,”
“acoustic characteristics,”
or “characteristics,” and the $\ell_{t,u}$ as “smooth tori.”
We emphasize that with the exception of the appendices,
in this paper,
we will not derive estimates on $\Sigma_t$
or the $\ell_{t',u'}$. Instead, we will control the solution
on the rough hypersurfaces
and the rough tori of Def. <ref>.

§.§ Projection tensorfields and related differential operators

* We define the type $\binom{1}{1}$
$\Sigma_t$-projection tensorfield $\Sigmatproject$ and the type $\binom{1}{1}$ $\ell_{t,u}$-projection tensorfield
$\smoothtorusproject$ as follows, where $\updelta_{\beta}^{\alpha}$ denotes the Kronecker delta:

\begin{align}
\Sigmatproject_{\beta}^{\ \alpha}
& \eqdef
\updelta_{\beta}^{\alpha}
\Transport^{\alpha}
\Transport_\beta,
\label{E:CARTESIANSIGMAPROJECT}
\\
\smoothtorusproject^{\ \alpha}_\beta
\eqdef
\updelta_{\beta}^{\alpha}
\Transport^{\alpha} \Transport_\beta
X^{\alpha} X_\beta
\updelta_{\beta}^{\alpha}
\Lunit^{\alpha} \updelta^0_\beta
X^{\alpha} \Lunit_{\beta}.
\label{E:SMOOTHTORUSPROJECT}
\end{align}

* Given any type $\binom{m}{n}$ spacetime tensorfield $\upxi$,
we respectively define its $\gfour$-orthogonal projection onto $\Sigma_t$, denoted by $\Sigmatproject \upxi$,
and its $\gfour$-orthogonal projection onto $\ell_{t,u}$, denoted by $\smoothtorusproject \upxi$,
as follows:

\begin{align}
(\Sigmatproject \upxi)_{\beta_1 \cdots \beta_n}^{\alpha_1 \cdots \alpha_m}
\eqdef
\Sigmatproject_{\widetilde{\alpha}_1}^{\ \alpha_1}
\cdots
\Sigmatproject_{\widetilde{\alpha}_m}^{\ \alpha_m}
\Sigmatproject_{\beta_1}^{\ \widetilde{\beta}_1}
\cdots
\Sigmatproject_{ \beta_n}^{\ \widetilde{\beta}_n}
\upxi_{\widetilde{\beta}_1 \cdots \widetilde{\beta}_n}^{\widetilde{\alpha}_1 \cdots \widetilde{\alpha}_m},
\label{E:PROJECTIONOFTENSORONTOCARTESIANSIGMAT}
\\
(\smoothtorusproject \upxi)_{\beta_1\cdots\beta_n}^{\alpha_1\cdots\alpha_m}
& \eqdef
\smoothtorusproject_{\widetilde{\alpha}_1}^{\ \alpha_1}
\cdots
\smoothtorusproject_{\widetilde{\alpha}_m}^{\ \alpha_m}
\smoothtorusproject_{ \beta_1}^{\ \widetilde{\beta}_1}
\cdots
\smoothtorusproject_{ \beta_n}^{\ \widetilde{\beta}_n}
\upxi_{\widetilde{\beta}_1 \cdots \widetilde{\beta}_n}^{\widetilde{\alpha}_1 \cdots \widetilde{\alpha}_m}.
\label{E:PROJECTIONOFTENSORONTOFLATTORUS}
\end{align}

* We say that a spacetime tensorfield $\upxi$ is $\Sigma_t$-tangent if $\Sigmatproject \upxi = \upxi$.
We say that a spacetime tensorfield $\upxi$ is $\ell_{t,u}$-tangent if $\smoothtorusproject \upxi = \upxi$.

* If $\upxi$ is a symmetric type $\binom{0}{2}$-spacetime tensor and $V$ is a vectorfield,
then we define $\slashed{\upxi}_V \eqdef \smoothtorusproject(\upxi \cdot V)$,
where $\upxi \cdot V$ is the one-form with components
$(\upxi \cdot V)_{\alpha} \eqdef \upxi_{\alpha \beta} V^{\beta}$.

* If $\upxi$ is a spacetime tensor, then we define $\slashed{\upxi} = \smoothtorusproject \upxi$.
From 3 above, it follows that $\xi$ is $\ell_{t,u}$-tangent if and only if $\slashed{\upxi} = \xi$.

It is straightforward to check that
$\Sigmatproject \Transport = 0$, while if $\SigmatTan$ is $\Sigma_t$-tangent, then
$\Sigmatproject \SigmatTan = \SigmatTan$, i.e, in view of the properties of $\Transport$ from Lemma <ref>,
we see that $\Sigmatproject$ is the $\gfour$-orthogonal projection onto $\Sigma_t$.
Similarly, $\smoothtorusproject \Lunit = \smoothtorusproject X = \smoothtorusproject \Transport = 0$,
while if $Y$ is $\ell_{t,u}$-tangent, then $\smoothtorusproject Y = Y$.

§.§ First fundamental forms

* We define $g$, the first fundamental form of $\Sigma_t$ relative to $\gfour$,
to be the symmetric type $\binom{0}{2}$ tensorfield $\Sigmatproject \gfour$. Note that
$g(Y,Z) = \gfour(Y,Z)$ for
all pairs $(Y,Z)$ of $\Sigma_t$-tangent vectorfields.
We define the corresponding inverse first fundamental form $g^{-1}$
to be the symmetric type $\binom{2}{0}$ tensorfield that is $\gfour$-dual to $g$
$(g^{-1})^{\alpha \beta}
\eqdef
(\gfour^{-1})^{\alpha \widetilde{\alpha}}
(\gfour^{-1})^{\beta \widetilde{\beta}}
g_{\widetilde{\alpha} \widetilde{\beta}}$.
Note that the restriction of $g$ to $\Sigma_t$-tangent tensorfields is the Riemannian[It is Riemannian
is because the $\gfour$-normal to $\Sigma_t$ is the $\gfour$-timelike vectorfield $\Transport$.]
metric on $\Sigma_t$ induced by $\gfour$. In particular,
relative to the Cartesian spatial coordinates, we have:

\begin{align} \label{E:CARTESIANCOMPONENTSOFFIRSTFUNDAMENTLAFORMOFSIGMAT}
& = \gfour_{ij}
= \Speed^{-2} \updelta_{ij},
\end{align}

where $\updelta_{ij}$ denotes the Kronecker delta,
and to obtain the last equality in (<ref>),
we have used (<ref>).

* We define $\gtorus$, the first fundamental form of $\ell_{t,u}$ relative to $\gfour$,
to be the symmetric type $\binom{0}{2}$ tensorfield $\smoothtorusproject \gfour$. Note that
$\gtorus(Y,Z) = \gfour(Y,Z)$ for
all pairs $(Y,Z)$ of $\ell_{t,u}$-tangent vectorfields.
Note that the restriction of $\gtorus$ to $\ell_{t,u}$-tangent tensorfields is the Riemannian[It is Riemannian
is because $\ell_{t,u}$ is a submanifold of the spacelike hypersurface $\Sigma_t$.]
metric on $\ell_{t,u}$ induced by $\gfour$.
We define the corresponding inverse first fundamental form $\gtorus^{-1}$
to be the symmetric type $\binom{2}{0}$ tensorfield that is $\gfour$-dual to $\gtorus$
$(\gtorus^{-1})^{\alpha \beta}
\eqdef
(\gfour^{-1})^{\alpha \widetilde{\alpha}}
(\gfour^{-1})^{\beta \widetilde{\beta}}
\gtorus_{\widetilde{\alpha} \widetilde{\beta}}$.

§.§ Geometric coordinates and related vectorfields

§.§.§ Geometric coordinates

We define the geometric coordinate system to be $(t,u,x^2,x^3)$. We define
$\left\lbrace\geop{t},\geop{u},\geop{x^2},\geop{x^3} \right\rbrace$ to be the coordinate
partial vectorfields in the geometric coordinate system.

Note that $(x^2,x^3)$ form a coordinate system on the smooth tori $\ell_{t,u}$
and that $\left\lbrace \geop{x^2},\geop{x^3} \right\rbrace$ span the tangent space of $\ell_{t,u}$.
Similarly, $(t,x^2,x^3)$ form a coordinate system on the null hypersurfaces $\nullhyparg{u}$
and $\left\lbrace \geop{t}, \geop{x^2},\geop{x^3} \right\rbrace$ span the tangent space of
$\nullhyparg{u}$. We will silently use these basic facts throughout the rest of the article.

[Conventions used with $(x^2,x^3)$ and $\lbrace \geop{x^2}, \geop{x^3} \rbrace$]

* If $V$ is a vectorfield, then for $A = 2,3$, $V^A \eqdef V x^A = V^{\alpha} \partial_{\alpha} x^A$.
In particular, if $V$ is $\ell_{t,u}$-tangent, then $V = V^A \geop{x^A}$, and
$V^A$ are the components of $V$ with respect to geometric coordinates $(x^2,x^3)$ on $\ell_{t,u}$.

* If $\upxi$ is a one-form, then we denote its contraction with $\geop{x^A}$ by using the abbreviated notation
$\upxi_A \eqdef \upxi\left( \geop{x^A} \right) = \upxi_{\alpha} (\geop{x^A})^{\alpha}$ for $A = 2,3$.

* We adopt a similar convention for contractions involving higher order tensorfields,
i.e., $\gtorus_{AB} = \gtorus(\geop{x^A},\geop{x^B})$ for $A,B = 2,3$.

* We sum repeated uppercase Latin indices over $A = 2,3$, e.g.,
$\upxi_{AA} \eqdef \upxi_{22} + \upxi_{33}$.

§.§.§ The important acoustic vectorfields

The vectorfields in the next definition are fundamental for the rest of the paper.
We will use them to control the solution up to top-order.

* We define the geodesic null vectorfield by:

\begin{align} \label{E:LGEO}
\Lgeo^{\alpha}
& \eqdef - (\gfour^{-1})^{\alpha \beta}\p_\beta u,
\end{align}

and the rescaled null vectorfield
as follows, where $\upmu$ is the inverse foliation density defined in (<ref>):

\begin{align} \label{E:LUNIT}
\Lunit
& \eqdef \upmu \Lgeo.
\end{align}

For $i = 1,2,3$, we define the scalar functions $\Lsmall^i$ as follows,
where throughout the paper, $\Lunit^i$ denotes the Cartesian component $\Lunit x^i$:

\begin{align} \label{E:LSMALLDEF}
\Lsmall^1
& \eqdef L^1 - 1, \qquad \Lsmall^2 \eqdef L^2,\qquad \Lsmall^3 \eqdef L^3.
\end{align}

* We define $X$ to be the unique vectorfield that is $\Sigma_t$-tangent and $\gfour$-orthogonal to the
smooth tori $\ell_{t,u}$, and normalized by:

\begin{align} \label{E:X}
\gfour(\Lunit,X)
& = -1,
\end{align}

and we define the rescaled vectorfield $\muX$ by:

\begin{align} \label{E:MUX}
\muX
& \eqdef \upmu X.
\end{align}

For $i = 1,2,3$, we define the scalar functions $\Xsmall^i$ as follows,
where throughout the paper, $X^i$ denotes the Cartesian component $X x^i$:[For the solutions covered by our main results, the functions $\Lsmall^i$ and $\Xsmall^i$ will be $\ll 1$.]

\begin{align} \label{E:XSMALL}
\Xsmall^1
& \eqdef X^1 + 1, \qquad \Xsmall^2 \eqdef X^2, \qquad \Xsmall^3 \eqdef X^3.
\end{align}

* We define $\Yvf{2}$, $\Yvf{3}$, to be the following $\ell_{t,u}$-tangent vectorfields:

\begin{align} \label{E:YCOMMUTATOR}
\Yvf{2} & \eqdef \p_2 - \gfour(\p_2,X) X,  & \Yvf{3} & \eqdef \p_3 - \gfour(\p_3,X) X.
\end{align}

We also define $\Yvfsmall{2}$, $\Yvfsmall{3}$, to be the following vectorfields
(which are not generally $\ell_{t,u}$-tangent):

\begin{align} \label{E:YSMALL}
\Yvfsmall{2} & \eqdef \Yvf{2} - \partial_2, & \Yvfsmall{3} & \eqdef \Yvf{3} - \partial_2.
\end{align}

We similarly define the Cartesian component functions $\Yvf{2}^i, \Yvfsmall{3}^i$,
analogously to (<ref>).

* We define the commutation vectorfields $\Fullset$, the
$\nullhyparg{u}$-tangential subset $\Tanset$, and the $\ell_{t,u}$-tangential subset
$\Angularset$ as follows:

\begin{align} \label{E:COMMUTATIONVECTORFIELDS}
\Fullset
& \eqdef \lbrace \Lunit, \muX, \Yvf{2}, \Yvf{3} \rbrace,
\Tanset
& \eqdef \lbrace \Lunit, \Yvf{2}, \Yvf{3} \rbrace,
\Angularset
\eqdef \lbrace \Yvf{2}, \Yvf{3} \rbrace.
\end{align}

Lemma <ref> shows that
$\Fullset$ spans the tangent spaces of spacetime equipped with
the differential structure corresponding to the geometric coordinates $(t,u,x^2,x^3)$.
We sometimes refer to $\Fullset$ as the rescaled frame
because the vectorfield $\muX = \upmu X$ degenerates with respect to the Cartesian differential
structure as $\upmu \downarrow 0$, i.e., $\muX^i = \upmu X^i$ tends to $0$.
Similarly, the lemma shows
that $\Tanset$ spans the tangent spaces of the characteristics $\nullhyparg{u}$
and that $\Angularset$ spans the tangent spaces of the $\ell_{t,u}$.
To derive $L^{\infty}$ and Hölder estimates, we commute various PDEs with
elements of $\Fullset$. To derive energy estimates,
we will commute various PDEs with the elements of $\Tanset$.
For a handful of key estimates, we will refer to the set $\Angularset$.

Throughout the paper, we will often silently use the identities
featured in the following lemma.

The follow results hold.

* The vectorfield $\Lgeo$ is a geodesic and $\gfour$-null, i.e.,

\begin{align} \label{E:LGEOISGEODESICANDNULL}
\gfour(\Lgeo,\Lgeo) & = 0,
\Dfour_{\Lgeo}\Lgeo & = 0.
\end{align}

The rescaled vectorfield $\Lunit$ is also $\gfour$-null:

\begin{align} \label{E:LUNITISNULL}
\gfour(\Lunit,\Lunit)
& = 0,
\end{align}

and it satisfies the following identity,
where $\upmu$ is the inverse foliation density defined in (<ref>):

\begin{align} \label{E:COVARIANTLUNITDERIVATIVEOFLUNIT}
\Dfour_{\Lunit} \Lunit
& = \frac{(\Lunit \upmu)}{\upmu} \Lunit.
\end{align}

* $\Lunit$ is $\gfour$-orthogonal to the characteristics $\nullhyparg{u}$,
that is, for any vectorfield $\Singletan$ tangent to $\nullhyparg{u}$,
we have:

\begin{align} \label{E:LUNITISNORMALTOCHARACTERISTICS}
\gfour(\Lunit,\Singletan)
& = 0.
\end{align}

* The following identities hold:

\begin{align}
\Lunit u & = 0, & \Lunit t = \Lunit^0 & = 1, & \muX u & = 1, & \muX t = \muX^0 &= 0, \label{E:LULTMUXUMUXT} \\
\gfour(X,X) & = 1, & \gfour(\muX,\muX) & = \upmu^2, & \gfour(\Lunit,X) & = -1, & \gfour(\Lunit,\muX) &= -\upmu. \label{E:GOFLANDX}
\end{align}

* The material vectorfield $\Transport$ is future directed, $\gfour$-orthogonal to $\Sigma_t$ (and hence also to $\ell_{t,u}$),
and it is of $\gfour$-unit size:

\begin{align} \label{E:TRANSPORTISUNITLENGTH}
\gfour(\Transport,\Transport)
& = -1.
\end{align}

Moreover, we have:

\begin{align} \label{E:BISLPLUSX}
\Transport & = \Lunit + X,
\end{align}

and relative to the Cartesian coordinates, we have:

\begin{align}
\Transport_{\alpha}
& =
\updelta_\alpha^0, \label{E:MATERIALDERIVATIVELOWEREDCARTESIANCOORDINATES}
\end{align}

where $\updelta_{\alpha}^{\beta}$ is the Kronecker delta.

* Finally, the following identities hold for $i=1,2,3$ and $A=2,3$:

\begin{align}
\Xsmall^i  & = - \Lsmall^i + v^i,
\label{E:XSMALLINTERMSOFLSMALLANDVELOCITY}
\\
\Yvfsmall{A}^i
& =
-\Speed^{-2} \Xsmall^A
(- \Lsmall^A + v^A)
(-\Lunit^i + v^i).
\label{E:YSMALLINTERMSOFLSMALLANDVELOCITY}
\end{align}

All aspects of the lemma except for
(<ref>) follow from
minor modifications of the proofs of <cit.>.
The identity (<ref>) follows from
definition (<ref>),
and the Leibniz rule for the connection $\Dfour$.
The identity (<ref>) follows from definitions
(<ref>)–(<ref>), the form (<ref>) of $\gfour_{\alpha \beta}$,
and (<ref>).

§.§ Controlling quantities $\controlvars$ and $\badcontrolvars$

In the next definition, we introduce the solution variable arrays
$\controlvars$ and $\badcontrolvars$. These arrays allow us to provide
simple, schematic formulas in contexts where the precise details are not important
for the PDE analysis.

We define $\controlvars$ and $\badcontrolvars$ to be the following arrays of scalar functions:

\begin{align}
\controlvars
\eqdef (\wavearray, \Lsmall^1, \Lsmall^2, \Lsmall^3),
\label{E:CONTROLVARS}
\\
\badcontrolvars
& \eqdef
(\wavearray, \upmu - 1, \Lsmall^1, \Lsmall^2, \Lsmall^3).
\label{E:BADCONTROLVARS}
\end{align}

For the solutions that we study in our main results, along the data hypersurface
$\Sigma_0$, $\controlvars$ and $\badcontrolvars$ are small in $L^{\infty}$.

§.§ Identities for the $\ell_{t,u}$-projection tensorfield and the first fundamental forms

The following lemma provides useful identities for
the $\ell_{t,u}$-projection tensorfield $\smoothtorusproject$,
the first fundamental form $\gtorus$ of $\ell_{t,u}$,
and the first fundamental form $g$ of $\Sigma_t$.

Recall that $\smoothtorusproject$ is the $\ell_{t,u}$-projection tensorfield from Def. <ref>,
that $\gtorus$ is the first fundamental form of $\ell_{t,u}$ from Def.<ref>,
and that $g$ is the first fundamental form of $\Sigma_t$ from Def.<ref>.
Let $X$ be the vectorfield defined in Def. <ref>.
Then the following identities hold relative to the geometric coordinates ($A,B = 2,3$):

\begin{align}
\gtorus_{AB} & = \Speed^{-2} \updelta_{AB} +\Speed^{-2} \frac{X^AX^B}{(X^1)^2},
\label{E:SMOOTHTORIGABEXPRESSION}
\\
= \Speed^2 \updelta^{AB} - X^A X^B,
\label{E:SMOOTHGINVERSEABEXPRESSION}
\\
\det \gtorus & = \frac{1}{\Speed^2 (X^1)^2}.
\label{E:DETERMSMOOTHGTORUSRELTOGEOMETRICCOORDS}
\end{align}

Moreover, the following identities hold relative to arbitrary coordinates:

\begin{align} \label{E:FIRSTFUNDOFSIGMATINTERMSOFACOUSTICALMETRICANDTRANSPORT}
g_{\alpha \beta}
& =
\gfour_{\alpha \beta}
\Transport_{\alpha} \Transport_{\beta},
\\
(g^{-1})^{\alpha \beta}
(\gfour^{-1})^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}.
\label{E:INVERSEFIRSTFUNDOFSIGMATINTERMSOFACOUSTICALMETRICANDTRANSPORT}
\end{align}

Furthermore, the following identities hold relative to the Cartesian coordinates:

\begin{align}
& = \Speed^{-2} \sum_{a=1}^3 (\rmD x^a - v^a \rmD t)\otimes  (\rmD x^a - v^a \rmD t),
\label{E:FIRSTFUNDOFSIGMATIDENTITY}
\\
& = \Speed^2 \sum_{a=1}^3 \p_a \otimes \p_a.
\label{E:INVERSEFIRSTFUNDOFSIGMATIDENTITY}
\end{align}

In addition, the following identities hold relative to arbitrary coordinates:

\begin{align}
\gtorus_{\alpha \beta}
& =
g_{\alpha \beta}
X_{\alpha} X_{\beta}
\gfour_{\alpha \beta}
\Transport_{\alpha} \Transport_{\beta}
X_{\alpha} X_{\beta}
= \gfour_{\alpha \beta}
\Lunit_{\alpha} \Lunit_{\beta}
\Lunit_{\alpha} X_{\beta}
X_{\alpha} \Lunit_{\beta},
\label{E:SMOOTHTORUSMETRICINTERMSOFSIGMATMETRICANDX}
\\
(\gtorus^{-1})^{\alpha \beta}
& =
(g^{-1})^{\alpha \beta}
X^{\alpha} X^{\beta}
(\gfour^{-1})^{\alpha \beta}
\Transport^{\alpha} \Transport^{\beta}
X^{\alpha} X^{\beta}
(\gfour^{-1})^{\alpha \beta}
\Lunit^{\alpha} \Lunit^{\beta}
\Lunit^{\alpha} X^{\beta}
X^{\alpha} \Lunit^{\beta},
\label{E:SMOOTHTORUSINVERSEMETRICINTERMSOFINVERSESIGMATMETRICANDX}
\\
\smoothtorusproject^{\ \alpha}_\beta
& =
\gfour_{\beta \gamma}
(\gtorus^{-1})^{\alpha \gamma}
\gtorus_{\beta \gamma}
(\gtorus^{-1})^{\alpha \gamma}.
\label{E:LTUSPROJECTIONISONEINDEXLOWERINGOFGTORUSINVERSE}
\end{align}

Finally, relative to the Cartesian coordinates, the following identities hold for $\alpha, \beta = 0,1,2,3$:

\begin{align} \label{E:GTORUSINVERSE0COMPONENTSVANISH}
(\gtorus^{-1})^{0 \alpha}
(\gtorus^{-1})^{\alpha 0}
= 0,
\\
(g^{-1})^{0 \alpha}
(g^{-1})^{\alpha 0}
= 0,
\label{E:FIRSTFUNDSIGMAT0COMPONENTSVANISH}
\\
\smoothtorusproject^{\ 0}_{\beta}
& = 0.
\label{E:SMOOTHTORUSUPPER0COMPONENTSVANISH}
\end{align}

The identities (<ref>)–(<ref>) were proved in <cit.>.
The identity (<ref>) follows from (<ref>) and the
following identity:

\begin{align} \label{E:CARTESIANCOMPONENTSOFXSUMTOCSQUARED}
\Speed^2
& = \sum_{i=1}^3 (X^i)^2,
\end{align}

which follows from (<ref>) and $\gfour(X,X) = 1$ (see (<ref>)).

follows from definition (<ref>)
and the fact that $g = \Sigmatproject \gfour$.
follows from raising the indices in (<ref>)
with $\gfour^{-1}$.

Since $\ell_{t,u} \subset \Sigma_t$, the first equality in (<ref>)
follows from the fact that $X$ is $\Sigma_t$-tangent, $\gfour$-orthogonal to $\ell_{t,u}$, and normalized by
$\gfour(X,X) = g(X,X) = 1$. The second equality in (<ref>)
follows from (<ref>).
The last equality in (<ref>)
follows from (<ref>).
(<ref>) follows from raising
the indices in (<ref>) with $\gfour^{-1}$.

The first equality in (<ref>)
follows from definition (<ref>) and the second equality in
The second equality in (<ref>)
follows from the second equality in (<ref>)
and the fact that $\gtorus^{-1}$ vanishes when contracted against $\Transport$ or $X$.

follows from (<ref>)
and the last equality in (<ref>).
(<ref>) follows from (<ref>)
and (<ref>), which in particular implies that in Cartesian coordinates,
the $\gfour$-dual of $\partial_a$
is $dx^a - v^a dt$.

(<ref>) follows
from (<ref>),
the first equality in (<ref>),
and the fact that $X^0 = 0$,
i.e., $X$ is $\Sigma_t$-tangent.
(<ref>) follows from (<ref>).
(<ref>) follows from (<ref>)
and (<ref>).

If $Y = Y^B \geop{x^B}$ is an $\ell_{t,u}$-tangent vectorfield,
then $Y_{\flat}$ denotes the $\ell_{t,u}$-tangent one-form that is $\gtorus$-dual to $Y$,
i.e., for $A=2,3$, $(Y_{\flat})_A = \gtorusdoublearg{A}{B} Y^B$.
Similarly, if $\upxi$ is an $\ell_{t,u}$-tangent one-form,
then $\upxi^{\sharp}$ denotes the $\ell_{t,u}$-tangent vectorfield that is $\gtorus$-dual to $\upxi$,
i.e., for $A=2,3$, $(\upxi^{\#})^A = (\gtorus^{-1})^{AB} \upxi_B$, where
$\upxi_B = \upxi \cdot \geop{x^B}$.
Similarly, if $\upxi$ is a symmetric type $\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfield,
then $\upxi^{\sharp}$ denotes the type $\binom{1}{1}$ $\ell_{t,u}$-tangent tensorfield
obtained by raising one index of $\upxi$ with $\gtorus^{-1}$,
$\upxi^{\sharp \sharp}$ denotes the type $\binom{2}{0}$ $\ell_{t,u}$-tangent tensorfield
obtained by raising both indices of $\upxi$ with $\gtorus^{-1}$.

If $\varphi$ is a scalar function, then we define $\angrmD \varphi$
to be the following $\ell_{t,u}$-tangent one-form:

\begin{align} \label{E:ANGULARDIFFERENTIAL}
\angrmD \varphi
& \eqdef \smoothtorusproject \rmD \varphi.
\end{align}

Note that $\argangrmD{A} \varphi = \angrmD \varphi \cdot \geop{x^A}  = \geop{x^A} \varphi$ for $A = 2,3$
and that $\argangrmD{\alpha} \varphi
\angrmD \varphi \cdot \partial_{\alpha}
\smoothtorusproject^{\ \beta}_{\alpha} \partial_{\beta} \varphi$
for $\alpha = 0,1,2,3$.

* We denote the Levi-Civita connection of $\gfour$ by $\Dfour$.

* We denote the Levi-Civita connection of $\gtorus$ by $\newangD$. In particular,
for $\ell_{t,u}$-tangent tensorfields $\upxi$, we have $\newangD \upxi = \smoothtorusproject \Dfour \upxi$.

* If $\upxi$ is an $\ell_{t,u}$-tangent one-form, then we define its $\ell_{t,u}$-divergence
to be the scalar function
$\angdiv \upxi \eqdef \gtorus^{-1}\cdot \newangD \upxi$. Similarly, if $Y$ is an $\ell_{t,u}$-tangent vectorfield,
then we define its $\ell_{t,u}$-divergence to be the scalar function $\angdiv Y \eqdef \gtorus^{-1} \cdot \newangD Y_{\flat}$.
where $Y_{\flat}$ is the $\ell_{t,u}$-tangent one-form that is $\gfour$-dual to $Y$.

* If $\upxi$ is a symmetric type $\binom{0}{2}$ $\ell_{t,u}$-tangent tensorfield,
then we define its $\ell_{t,u}$-divergence
to be the $\ell_{t,u}$-tangent one-form with the following $\ell_{t,u}$ components for $A=2,3$:
$(\angdiv \upxi)_A \eqdef (\gtorus^{-1})^{BC} \cdot \newangDarg{B} \upxi_{CA}$.

* We denote the covariant wave operator of $\gfour$ by $\Box_{\gfour} \eqdef \gfour^{-1}\cdot\Dfour^2
= (\gfour^{-1})^{\alpha \beta} \Dfour_{\alpha} \Dfour_{\beta}

* We denote the $\ell_{t,u}$-Laplacian associated to $\gtorus$ by
$\angLap \eqdef \gtorus^{-1} \cdot \newangD^2
(\gtorus^{-1})^{AB} \angDarg{A} \angDarg{B}

Given a spacetime tensorfield $\upxi$
and a vectorfield $Z$,
we define $\SigmatLie_Z \upxi$ and $\angLie_Z \upxi$ to respectively be the following $\Sigma_t$-tangent and $\ell_{t,u}$-tangent tensorfields:

\begin{align} \label{E:PROJECTEDLIEDERIVATIVES}
\SigmatLie_Z \upxi
&	\eqdef \Sigmatproject \Lie_Z \upxi,
\angLie_Z \upxi
& \eqdef \smoothtorusproject \Lie_Z \upxi.
\end{align}

We will use the following simple commutation lemma when deriving various equations.

Let $f$ be a scalar function and let
$Z \in \Fullset$ (see definition (<ref>)).
Then the following identity holds:

\begin{align} \label{E:ANGULARDIFFERENTIALCOMMUTESWITHANGLIE}
\angLie_Z \angrmD f
& = \angrmD Z f.
\end{align}

The same proof of <cit.> holds.

§.§ Traces of tensorfields

In our analysis, we will encounter various traces of tensorfields.

* If $\upxi$ is a type $\binom{0}{2}$ spacetime tensorfield,
then we define its $\gfour$-trace as follows:

\begin{align} \label{E:SPACETIMETRACE}
\mytr_{\gfour}\upxi
& \eqdef (\gfour^{-1})^{\alpha \beta} \upxi_{\alpha \beta}.
\end{align}

* If $\upxi$ is a type $\binom{0}{2}$ spacetime tensorfield, then we define its $\gtorus$-trace as follows:

\begin{align} \label{E:SMOOTHTORUSTRACE}
\mytr_{\gtorus} \upxi
& \eqdef
(\gtorus^{-1})^{\alpha \beta} \upxi_{\alpha \beta}.
\end{align}

§.§ Pointwise norms and semi-norms of tensorfields

In the next definition, we define various pointwise norms and semi-norms
that we will use to measure the size of tensorfields.

* If $\upxi$ is a type $\binom{m}{n}$ spacetime tensorfield
such that $\gfour_{\alpha_1 \widetilde{\alpha}_1} \cdots \gfour_{\alpha_m \widetilde{\alpha}_m} (\gfour^{-1})^{\beta_1 \widetilde{\beta}_1} \cdots (\gfour^{-1})^{\beta_n \widetilde{\beta}_n} \upxi_{\beta_1\cdots\beta_n}^{\alpha_1\cdots\alpha_n} \upxi_{\widetilde{\beta}_1 \cdots \widetilde{\beta}_n}^{\widetilde{\alpha}_1\cdots\widetilde{\alpha}_m} \geq 0$,
then we define $|\upxi|_{\gfour}\geq 0$ by:

\begin{align} \label{E:SQUAREPOINTWISELORENTZIANNORMWITHRESPECTTOACOUSTICALMETRIC}
& \eqdef \gfour_{\alpha_1 \widetilde{\alpha}_1} \cdots \gfour_{\alpha_m \widetilde{\alpha}_m} (\gfour^{-1})^{\beta_1 \widetilde{\beta}_1} \cdots (\gfour^{-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}

* If $\upxi$ is a type $\binom{m}{n}$ tensorfield, then we define $|\upxi|_{\gtorus} \geq 0$ by:

\begin{align} \label{E:SQUAREPOINTWISESEMINORMWITHRESPECTTOFIRSTFUNDOFSMOOTHTORI}
& \eqdef \gtorus_{\alpha_1 \widetilde{\alpha}_1} \cdots \gtorus_{\alpha_m \widetilde{\alpha}_m} (\gtorus^{-1})^{\beta_1 \widetilde{\beta}_1} \cdots (\gtorus^{-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}

* If $\upxi$ is a type $\binom{m}{n}$ tensorfield, then we define $|\upxi|_{g} \geq 0$ by:

\begin{align} \label{E:SQUAREPOINTWISESEMINORMWITHRESPECTTOFIRSTFUNDOFSIGMAT}
|\upxi|_{g}^2 \eqdef g_{\alpha_1 \widetilde{\alpha}_1} \cdots g_{\alpha_m \widetilde{\alpha}_m} (g^{-1})^{\beta_1 \widetilde{\beta}_1} \cdots (g^{-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}

$|\cdot|_{\gfour}$ is a pointwise norm on the space of $\gfour$-spacelike tensorfields.
$|\cdot|_g$ is a pointwise norm on the space of $\Sigma_t$-tangent tensorfields and a pointwise seminorm on the space of all
$|\cdot|_{\gtorus}$ is a pointwise norm on the space of $\ell_{t,u}$-tangent tensorfields and a pointwise seminorm on the space of all

Similarly, the function $|\cdot|_{\gtorusroughfirstfund}$ from Def. <ref> below
is a pointwise norm on the space of $\twoargroughtori{\timefunction,u}{\muxmulevelsetvalue}$-tangent tensorfields
a pointwise semi-norm on the space of all tensorfields.

In view of
Def. <ref>,
we see that if $\upxi$ is a type $\binom{m}{n}$ $\Sigma_t$-tangent tensorfield,
then relative to the Cartesian coordinates, we have:

\begin{align} \label{E:NO0COMPONENTSQUAREPOINTWISESEMINORMWITHRESPECTTOFIRSTFUNDOFSMOOTHTORI}
\gtorus_{a_1 \widetilde{a}_1}
\cdots
\gtorus_{a_m \widetilde{a}_m}
(\gtorus^{-1})^{b_1 \widetilde{b}_1}
\cdots
(\gtorus^{-1})^{b_n \widetilde{b}_n}
\upxi_{b_1 \cdots b_n}^{a_1 \cdots a_n}
\upxi_{\widetilde{b}_1 \cdots \widetilde{b}_n}^{\widetilde{a}_1 \cdots \widetilde{a}_m},
\\
& =
g_{a_1 \widetilde{a}_1}
\cdots
g_{a_m \widetilde{a}_m}
(g^{-1})^{b_1 \widetilde{b}_1}
\cdots
(g^{-1})^{b_n \widetilde{b}_n}
\upxi_{b_1 \cdots b_n}^{a_1 \cdots a_n}
\upxi_{\widetilde{b}_1 \cdots \widetilde{b}_n}^{\widetilde{a}_1 \cdots\widetilde{a}_m},
\label{E:NO0COMPONENTSQUAREPOINTWISESEMINORMWITHRESPECTTOFIRSTFUNDOFSIGMAT}
\end{align}

i.e., we can omit all “$0$” components on
RHSs (<ref>)–(<ref>).

Similarly, taking into account definition (<ref>) and (<ref>),
we see that if $\SigmatTan$ is a $\Sigma_t$-tangent vectorfield,
then relative to the Cartesian coordinates, we have
$\smoothtorusproject_{\beta}^{\ \alpha} \partial_{\alpha} \SigmatTan^{\beta}
\smoothtorusproject_b^{\ a} \partial_a \SigmatTan^b

In the rest of the paper, we will use these basic facts without always explicitly mentioning them.

§.§ Second fundamental forms and the torsion

In this section, we provide the standard definitions
of the second fundamental form $k$ of $\Sigma_t$,
the null second fundamental form $\upchi$ of $\ell_{t,u}$,
and the one-form $\upzeta$.
These quantities will appear in various PDEs throughout the article.
It is well-known that there are many technical difficulties that have to be overcome to obtain
top-order energy estimates for
$\mytr_{\gtorus}\upchi$ and $\upchi$.
To achieve control, we will use the modified quantities defined in Sect. <ref>
and elliptic estimates on the rough tori, which we derive in Sect. <ref>.

* We define the second fundamental form $k$ of $\Sigma_t$ as follows:

\begin{align} \label{E:SIGMATSECONDFUND}
\eqdef \frac{1}{2} \SigmatLie_{\Transport} g.
\end{align}

* We define the null second fundamental form of $\ell_{t,u}$ as follows:

\begin{align} \label{E:NULLSECONDFUND}
\upchi
\eqdef \frac{1}{2} \angLie_\Lunit \gtorus.
\end{align}

* We define $\upzeta$ to be the $\ell_{t,u}$-tangent one-form with the following components:

\begin{align} \label{E:TORISONTENSORFIELD}
\upzeta_A
\eqdef
\gfour(\Dfour_A \Lunit, X).
\end{align}

§.§ Transport equations for the eikonal function quantities

To control the eikonal function quantities $\upmu$ and $\Lunit^i$, we will use the following transport equations.

$\upmu$ and $\Lunit^i$ satisfy the following transport equations:

\begin{align} \label{E:MUTRANSPORT}
\Lunit \upmu
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray
\frac{1}{2} \upmu \vec{G}_{\Lunit \Lunit} \diamond \Lunit \wavearray
\upmu \vec{G}_{\Lunit X} \diamond \Lunit \wavearray,
\\
\Lunit \Lsmall^i
= \frac{1}{2} (\vec{G}_{\Lunit \Lunit}\diamond \Lunit \wavearray) X^i
(\angG_{\Lunit}^{\#} \diamond \Lunit \wavearray)\cdot \angrmD x^i
\frac{1}{2} (\vec{G}_{\Lunit \Lunit} \diamond \angrmD^{\#}\wavearray)\cdot \angrmD x^i.
\label{E:LUNITITRANSPORT}
\end{align}

The same proof of <cit.> holds.

§.§ The factor driving the shock formation and formulas involving $G_{\Lunit \Lunit}$

In the following lemma, we compute
an expression for the product $\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray$
on the RHS of the evolution equation (<ref>) for $\upmu$.
For every smooth equation of state besides that of a Chaplygin gas,
there exist open sets of background densities $\overline{\varrho} > 0$ such that
the non-degeneracy condition (<ref>) holds.
The identity (<ref>) then shows
that for solutions that are close to the trivial solution $\wavearray \equiv 0$,
the expansion of $\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray$
features a non-zero term proportional to $\muX \RRiemann$;
the presence of this term is crucial for our main results, as it drives the formation of the shock,
i.e., it drives $\upmu$ to $0$.
In contrast, for the equation of state $p = C_0 - C_1 \exp(- \LogDensity)$ of a Chaplygin gas,
one can compute that $\Speed^{-1} \Speed_{;\LogDensity} + 1 \equiv 0$, and the non-degeneracy condition
(<ref>) is therefore impossible.
In this case, equation (<ref>) shows that
the product $\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray$
does not depend on the solution's $\muX$ derivative, and hence our main results do not apply.
We note that one can show that for irrotational and isentropic solutions,
the equation $\Speed^{-1} \Speed_{;\LogDensity} + 1 = 0$
is equivalent to the statement that the quasilinear wave equation for a potential function
satisfies Klainerman's null condition <cit.>.

For solutions to the compressible Euler equations
the following identity holds,
where $\almostRiemannfunction(\LogDensity,\Ent)$
is the scalar function from (<ref>):

\begin{align} \label{E:IDENTITYFORMAINTERMDRIVINGTHESHOCK}
\begin{split}
\frac{1}{2} \vec{G}_{\Lunit \Lunit} \diamond \muX \wavearray
& = - \frac{1}{2} \Speed^{-1}(\Speed^{-1} \Speed_{;\LogDensity} + 1)
\left\lbrace
\muX \RRiemann - \muX \LRiemann
\right\rbrace
\\
& - \frac{1}{2} \upmu \Speed^{-1} X^1
\left\lbrace \Lunit \RRiemann + \Lunit \LRiemann\} - \upmu \Speed^{-2} \lbrace X^2 \Lunit v^2 + X^3 \Lunit v^3
\right\rbrace
\\
& - \upmu \Speed^{-1} \Speed_{;\Ent} X^a \GradEnt^a + \upmu \Speed^{-1} (\Speed^{-1} c_{;\LogDensity} + 1) F_{;\Ent} X^a \GradEnt^a.
\end{split}
\end{align}

The same proof of Lemma <cit.> holds.

In the next lemma, we derive expressions for $G_{\Lunit \Lunit}^2$ and $G_{\Lunit \Lunit}^3$.
When deriving estimates, we will use the expressions to track smallness.

The following identities hold for $A= 2,3$:

\begin{align}
G_{\Lunit \Lunit}^A
& = 2 \Speed^{-2}(v^A - \Lunit^A) = 2\Speed^{-2} X^A. \label{E:GLLAEXPRESSION}
\end{align}

The identities follow from
the expression (<ref>) for the Cartesian component $\gfour_{\alpha \beta}$
viewed as a function of $(\RRiemann,\LRiemann,v^2,v^3,\Ent)$,
the identity $G_{\Lunit \Lunit}^A = (\frac{\p}{\p v^A} \gfour_{\alpha \beta}) \Lunit^{\alpha} \Lunit^{\beta}$,
and the identities $\Lunit^0 = 1$ and $\Lunit^A + X^A = v^A$,
which follow from Lemma <ref> and (<ref>).

§.§ Useful geometric decompositions

In this section, we provide some geometric decompositions that we will use throughout the article.

We start with the following alternate expressions for $\upchi$ and $\angk$, which are useful for computations.

The following identities hold:

\begin{align} \label{E:SECONDFUNDSALTERNATE}
\upchi_{AB}
\gfour\left(\Dfour_A \Lunit, \geop{x^B}\right),
\angk_{AB}
& = \gfour\left(\Dfour_A \Transport, \geop{x^B}\right).
\end{align}

The same proof of <cit.> holds in this setting.

We will use the following identities and decompositions when deriving estimates for
$\upchi$, $\angk$, and $\upzeta$.

The following[Here, $\otimesarray$ is defined by
$\angG_{\Lunit} \otimesarray \angrmD \wavearray \eqdef \sum_{\iota = 0}^4
\angG_{\Lunit}^\iota \otimes \angrmD \Psi_\iota$,
and similarly for $\angrmD \Psi \otimesarray\angG_{\Lunit}$, $\angG_X\otimesarray \angrmD \wavearray$, etc.] identities hold:

\begin{align}
\upchi & = \gfour_{ab} \angrmD \Lunit^a \otimes \angrmD x^b + \frac{1}{2} \angG \diamond \Lunit \wavearray + \frac{1}{2} \angrmD \wavearray \otimesarray \angG_{\Lunit} - \frac{1}{2} \angG_{\Lunit} \otimesarray \angrmD \wavearray,
\label{E:CHIEXPRESSSIONINTERMSOFDERIVATIVESOFLUNITI}
\\
\mytr_{\gtorus}\upchi & = \gfour_{ab} \gtorus^{-1} \cdot \left\lbrace \angrmD \Lunit^a \otimes \angrmD x^b\right\rbrace + \frac{1}{2} \gtorus^{-1} \cdot \angG \diamond \Lunit \wavearray. \label{E:TRCHIEXPRESSSIONINTERMSOFDERIVATIVESOFLUNITI}
\end{align}

Moreover, we can decompose $\angk$ and $\upzeta$ into $\upmu^{-1}$-singular and $\upmu^{-1}$-regular pieces as follows:

\begin{align} \label{E:CONNECTIONCOEFFICIENTDECOMPOSITIONS}
\upzeta
& = \zetatan + \upmu^{-1} \zetatrans, & \angk & = \angktan + \upmu^{-1}\angktrans,
\end{align}

\begin{align}
\angktan & \eqdef
\frac{1}{2} \angG \diamond \Lunit \wavearray
\frac{1}{2} \angG_{\Lunit} \otimesarray \angrmD \wavearray
\frac{1}{2} \angrmD \wavearray \otimesarray \angG_{\Lunit}
\frac{1}{2} \angG_X \otimesarray \angrmD \wavearray
\frac{1}{2} \angrmD \wavearray \otimesarray \angG_X, \label{E:ANGKTAN}
\\
\angktrans & \eqdef
\frac{1}{2} \upmu^{-1} \angG\diamond \muX \wavearray,
\label{E:ANGKTRANS}
\\
\zetatan
& \eqdef
\frac{1}{2} \angG_X \diamond \Lunit \wavearray
\frac{1}{2} \vec{G}_{\Lunit X} \diamond \angrmD \wavearray
\frac{1}{2} \vec{G}_{XX} \diamond \angrmD \wavearray,
\label{E:ZETATAN}
\\
\zetatrans
\eqdef
- \frac{1}{2} \upmu^{-1} \angG_{\Lunit} \diamond \muX \wavearray.
\label{E:ZETATRANS}
\end{align}

The same proofs of <cit.> holds with minor modifications accounting for the third spatial dimension.

In the next lemma, we decompose the $\upmu$-weighted covariant
wave operator relative to the rescaled frame $\{\Lunit,\muX, \Yvf{2}, \Yvf{3}\}$
and the second fundamental forms.

Let $f$ be a scalar function. Then relative to the
rescaled frame $\{\Lunit,\muX, \Yvf{2}, \Yvf{3}\}$,
the following identities hold:

\begin{align}
\begin{split}
\upmu \square_{\gfour(\wavearray)} f
& = - \Lunit (\upmu \Lunit f + 2 \muX f)
\upmu \angLap f
(\mytr_{\gtorus} \upchi) \muX f
\label{E:BOXDECOMPLOUTSIDE}
\\
& -
\upmu \mytr_{\gtorus} \angk \Lunit f
2 \upmu \upzeta^{\#} \cdot \angrmD f,
\end{split}
\\
\begin{split}
\upmu \square_{\gfour(\wavearray)} f
& =
- (\upmu \Lunit + 2 \muX)(\Lunit f)
\upmu \angLap f
(\mytr_{\gtorus} \upchi)\muX f
(\Lunit \upmu) \Lunit f
\label{E:BOXDECOMPMULBAROUTSIDE}
\\
& - \upmu \mytr_{\gtorus} \angk \Lunit f +
2 \upmu \upzeta^{\#} \cdot \angrmD f
2 (\angrmD^{\#} \upmu) \cdot \angrmD f.
\end{split}
\end{align}

The same proof of <cit.> holds with $\muX$
in the role of the vectorfield denoted by “$\breve{R}$” there.

§ ROUGH TIME FUNCTIONS, ROUGH ADAPTED COORDINATES, AND ROUGH SUBSETS

The geometric coordinates $(t,u,x^2,x^3)$ from Sect. <ref>
are fundamental for our construction of commutation and multiplier vectorfields.
However, these coordinates, in particular the Cartesian time function $t$,
are not adapted to the shape of the singular boundary.
For this reason,
in this section, we construct a one-parameter family of rough time functions
$\lbrace \timefunctionarg{\muxmulevelsetvalue} \rbrace_{\muxmulevelsetvalue \in [0,\muxmulevelsetvalue_0]}$
that are adapted to structure of the singular boundary,
where $\muxmulevelsetvalue_0 > 0$ is a constant depending on the initial data on $\Sigma_0$.

We refer to $(\timefunctionarg{\muxmulevelsetvalue},u,x^2,x^3)$ as rough adapted coordinates.

In our forthcoming PDE analysis, we will derive estimates on
the level sets of the $\timefunctionarg{\muxmulevelsetvalue}$,
which we will prove are $\gfour$-spacelike.
We construct $\timefunctionarg{\muxmulevelsetvalue}$ by
solving a well-chosen transport equation (see Def. <ref>) with
data equal to $- \upmu$ on the  “initial hypersurface"
$\lbrace \muX \upmu = -\muxmulevelsetvalue \rbrace$.
Standard well-posedness and Cauchy stability results
(see Appendices <ref> and <ref>)
imply that for perturbations of simple isentropic plane-symmetric solutions,
our construction is well-defined
on short “rough time” intervals of the form $[\timefunction_0,\timefunctionboot)$,
where $\timefunction_0 < 0$ is a data-dependent constant (independent of $\muxmulevelsetvalue \in \muxmulevelsetvalue_0$)
and $\timefunction_0 < \timefunctionboot < 0$ is a “bootstrap parameter.”
Our main results will show that each $\timefunctionarg{\muxmulevelsetvalue}$ exists
in a neighborhood of the singular boundary and has range $[\timefunction_0,0]$,

Finally, we remark that there are several subtleties tied to the analysis and regularity of