The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D and multi-dimensions
LA,JS]Leonardo Abbrescia$^{* \dagger}$ and Jared Speck$^{** \dagger\dagger}$

$^{*}$Vanderbilt University, Nashville, TN, USA.

$^{**}$Vanderbilt University, Nashville, TN, USA.

$^\dagger$ LA gratefully acknowledges support from an NSF Postdoctoral Fellowship.

$^{\dagger\dagger}$JS gratefully acknowledges support from NSF grant # DMS-2054184 and NSF CAREER grant # DMS-1914537.

In this article, we provide notes that complement the lectures on
the relativistic Euler equations and shocks that were given by the second author at
the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime,
which was hosted by the Erwin Schrödinger International Institute for Mathematics and Physics
in Vienna in February, 2022.
We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and
providing a brief overview of local well-posedness in Sobolev spaces.
Then, using Riemann invariants, we provide
the first detailed construction of a localized subset of
the maximal globally hyperbolic developments
of an open set of initially smooth, shock-forming isentropic solutions in $1D$,
with a focus on describing the singular boundary and
the Cauchy horizon that emerges from the singularity.
Next, we provide an overview of the new second-order formulation of the $3D$ relativistic Euler equations
derived in <cit.>, its rich geometric and analytic structures,
their implications for the mathematical theory of shock waves,
and their connection to the setup we use in our $1D$ analysis of shocks.
We then highlight some key prior results on the study of shock formation and related problems.
Furthermore, we provide an overview of how the formulation of the flow derived in <cit.>
can be used to study shock formation in multiple spatial dimensions.
Finally, we discuss various open problems tied to shocks.

acoustical metric,
Cauchy horizon,
eikonal function,
genuinely nonlinear,
hyperbolic conservation law,
maximal globally hyperbolic development,
nonlinear geometric optics,
null form,
Riemann invariant,
singular boundary,
stable blowup,
wave breaking

Mathematics Subject Classification (2020) Primary: 35L67; Secondary: 35M11, 35L65, 35L81, 76L05, 76Y05

August 27, 2024

§ INTRODUCTION

In this article, we provide notes that complement the lectures on
the relativistic Euler equations and shocks that were given by the second author at
the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime,
which was hosted by the Erwin Schrödinger International Institute for Mathematics and Physics
in Vienna in February, 2022. Broadly speaking, our main goals are the following:

* To briefly introduce the equations.

* To provide an overview of the new formulation of the equations
derived in <cit.>.

* To describe the implications of the formulation from <cit.>
for the study of multi-dimensional shocks.

Although our main interest is solutions in $3D$ without symmetry assumptions,
we highlight Sect. <ref>, in which we rigorously study a family of initially smooth, simple isentropic shock-forming
plane-symmetric solutions and give a complete description of a portion of their  maximal (classical) globally hyperbolic developments
(MGHD for short), up to the boundary.
To the best of our knowledge,
this is the first article
on any quasilinear hyperbolic system in $1D$
that provides the detailed construction
of the MGHD within the vicinity of a shock singularity.
Sect. <ref> in particular
provides an introduction to some of the main ideas behind the study of shock formation
in a gentle, semi-explicit $1D$ setting, where energy estimates are not needed.
Roughly, the MGHD is the largest possible classical solution + globally hyperbolic
region that is launched by the initial data,
and when it exists and is unique
(for some solutions to some hyperbolic PDEs, it is not unique – see Sect. <ref>!),
it is the holy grail object, at least from the point of view of classical solution.

We stress outright that, although the relativistic Euler equations are posed on an ambient Lorentzian manifold $(\spacetimemanifold,\gfour)$, from the point of view of the causal structure of the fluid,
the correct notion of “globally hyperbolic”
is not with respect to the spacetime metric $\gfour$, but rather
with respect to the acoustical metric $\hfour$, introduced in
Def. <ref>. To avoid confusion, we will refer to this as $\hfour$-global hyperbolicity and the corresponding $\hfour$-MGHD.

Many of the techniques and geometric insights behind the modern approach to studying
shocks have roots in mathematical General Relativity,
notably the celebrated proof by Christodoulou–Klainerman
<cit.> of the nonlinear stability of Minkowski spacetime as a solution to Einstein's equations.
A key unifying theme between the global existence result <cit.> and the results on shocks that we discuss here is that
nonlinear geometric optics, implemented via an eikonal function $\eik$, plays a central role in the study of the flow.
We will discuss eikonal functions in detail
in Sects. <ref> and <ref>.

We highlight a key advantage of the formulation of the flow derived in <cit.>.

For general solutions (in $3D$, i.e., three spatial dimensions),
it allows one to implement a sharp version of nonlinear geometric optics.
Nonlinear geometric optics is crucial for the study of multi-dimensional shocks,
and it has other applications, such as low-regularity well-posedness
(see Sect. <ref>).

In line with points (1)–(3) above, this article is meant to give an introduction to the mathematical techniques and methods needed for the PDE analysis through the lens of nonlinear geometric optics. A comprehensive topical review of the rich history of shock singularities in the framework of hyperbolic conservation laws would encompass a work of strictly larger order of magnitude. We refer the reader to
cD2010, bressan2011nonlinear,gHsKjSwW2016 and the references therein for a good start.

§.§ Motivation, context, and the structure of the article

§.§.§ Mathematical and physical motivations

The relativistic Euler equations are an important matter model in mathematical General Relativity.
For example, relativistic fluids are often used in cosmology to model the average energy-matter
content of spacetime, and they in particular play a central role in the Standard Model of cosmology.

For a non-exhaustive yet comprehensive overview of these physical applications, we refer the reader to the texts
bO1983, rW1984,sW2008, and the references therein. There has also been exciting rigorous mathematical progress on the
global structure of solutions when the spacetime is expanding
iRjS2013,oliynyk2016future,fajman2021stabilizing,fajman2023stability,jS2012. We also highlight
<cit.> and <cit.>*Chapter 1 for a modern introduction to
the relativistic Euler equations and their connections to the laws of mechanics and thermodynamics.

Despite the above remarks,
the rigorous mathematical theory of multi-dimensional solutions is far from complete. Even if one considers the relativistic Euler equations
on a given spacetime background (i.e., without coupling to Einstein's equations),
there are many mathematically rich phenomena that have not been fully understood.
Chief among these is the fundamental issue,
going back to Riemann's foundational work <cit.>
on non-relativistic compressible fluids in $1D$,
that initially smooth solutions can develop shock singularities in finite time.
Shocks are singularities such that various fluid variables' gradients blow up in finite time
(in a precise, controlled fashion, as it turns out), though the fluid variables themselves (such as the velocity and density)
remain bounded. This phenomenon is also known as wave breaking.
The relatively mild nature of the singularity gives rise to the hope that one might
– at least for short times –
be able to uniquely continue the solution weakly
past the “first singularity,[In multi-dimensions, the correct notion of
“first singularity” is not a point in spacetime, but rather a co-dimension $2$ submanifold of points;
see Sect. <ref>. ]”
subject to suitable selection criteria, typically
in the form of Rankine–Hugoniot-type jump conditions across a shock hypersurface
(which is not known in advance) and an entropy-type condition.
This weak continuation problem is known as the shock development problem, and in full generality, it remains open, for both the $3D$
relativistic Euler equations and their non-relativistic analog, i.e., the $3D$ compressible Euler equations;
see Sect. <ref> for further discussion. The long-term goal is certainly the following:

Develop a rigorous global-in-time-and-space
theory of existence and uniqueness for open sets of multi-dimensional solutions
that are allowed to transition from classical to weak due to shock formation,
and describe the interactions of different
shock hypersurfaces, Cauchy horizons (see Sect. <ref>),
and other crucial qualitative features of the flow.

This goal is far out of reach as of present. If one could accomplish it even in a single perturbative regime,
that would represent truly remarkable progress.

While there have been many works on shock formation for the multi-dimensional non-relativistic compressible
Euler equations,
such as dCsM2014,jS2016b,jSgHjLwW2016,jLjS2018,jLjS2021,tBsSvV2022,tBsSvV2020,tBsSvV2019a,tBsI2022,lAjS2020,lAjS20XX,lAjS2022,
there have been relatively few works on multi-dimensional shock formation for the relativistic Euler equations.
The most notable work in the relativistic case is Christodoulou's breakthrough monograph <cit.>,
in which he proved shock formation and studied some aspects of the $\hfour$-MGHD for
open sets of irrotational and isentropic solutions
in $3D$ that are compactly supported perturbations of non-vacuum constant fluid states.
Unlike the non-relativistic case, there are currently no rigorous results that prove multi-dimensional shock formation
for the relativistic Euler equations in the presence of vorticity and entropy.
However, in Sect. <ref>, we provide a blueprint for how one can use the equations of
<cit.> and the analytic framework of dC2007,jS2016b,jSgHjLwW2016,jLjS2018,jLjS2021,lAjS2020,lAjS2022
to prove stable shock formation in the $3D$ relativistic case for open sets of solutions with vorticity and entropy.

The remainder of the article is organized as follows:

* In Sect. <ref>, we set up the study of the relativistic Euler equations and provide an overview
of basic ingredients that play a role in the proof of local well-posedness in Sobolev spaces.
In particular, we introduce the acoustical metric $\hfour$, which is the solution-dependent
Lorentzian metric that drives the propagation of sound waves;
the acoustical metric is fundamental for implementing nonlinear geometric optics and studying multi-dimensional shocks.
While we state the equations in the case of an arbitrary spacetime background $(\spacetimemanifold,\gfour)$,
the vast majority of the article concerns the case in which $\spacetimemanifold = \mathbb{R}^{1+3}$
and $\gfour = \mink$ is the Minkowski metric on $\spacetimemanifold$.
We again point out that nearly all notions of hyperbolicity will refer to the acoustical metric $\hfour$ and not $\gfour$ or $\mink$;
see also Sect. <ref> and Remark <ref>.

* In Sect. <ref>, to help readers gain intuition for the subsequent discussion,
we study the flow in one spatial dimension, which is equivalent to studying plane-symmetric solutions in $3D$.
In particular, in Theorem <ref>,
we provide a detailed analysis of the $\hfour$-MGHD for a large set of shock-forming relativistic Euler solutions in $1D$.
Some aspects of the theorem, notably the results concerning the formation of Cauchy horizons,
have not been proved in detail elsewhere in the literature. Although
Theorem <ref> specifically concerns the relativistic Euler equations,
the techniques could be applied to a wide variety of strictly hyperbolic genuinely nonlinear
PDEs in one spatial dimension.

* In Sect. <ref>, we provide an overview of
the new formulation of the $3D$ relativistic Euler equations derived in <cit.>.

* In Sect. <ref>, we describe some previous works on shock formation and related problems.

* In Sect. <ref>,
based in part on our prior experience jS2016b,jSgHjLwW2016,jLjS2018,jLjS2021,lAjS2020,lAjS2022
in studying shock formation in multi-dimensional non-relativistic compressible fluids,
we provide an overview of how the new formulation of the flow from <cit.>
can be used to study shock formation for open sets of solutions to the $3D$
relativistic Euler equations.
As of present, rigorous fluid shock formation
results in multi-dimensions with vorticity and entropy
have been proved only for the non-relativistic compressible Euler equations.
Nonetheless, from the perspective of shock formation, the relativistic Euler equations and
the non-relativistic compressible Euler equations enjoy many structural commonalities,
and we expect that the overview we provide in Sect. <ref> could be turned
(with substantial effort) into a complete proof.

* In Sect. <ref>, we describe various open problems.

§ STANDARD FORMULATIONS OF THE RELATIVISTIC EULER EQUATIONS AND LOCAL WELL-POSEDNESS

In this section, we introduce some standard first-order formulations of the relativistic Euler equations.
We also introduce the acoustical metric $\hfour$ and set up a corresponding
geometric version of the energy method that applies to a first-order formulation of the flow.
Finally, we state Prop. <ref>, which is a standard result on local well-posedness,
and briefly discuss how its proof is connected to the energy method.

§.§ The basic setup

We start by discussing standard formulations of the relativistic Euler equations on
an arbitrary four-dimensional Lorentzian manifold $(\spacetimemanifold,\gfour)$,
where $\gfour$ is the spacetime metric of signature $(-,+,+,+)$.
Our discussion in this section is motivated by Christodoulou's
presentation in <cit.>*Chapter 1.
We are somewhat terse in our presentation here, and thus we refer readers to <cit.>*Chapter 1 for additional details.

[Moving indices with $\gfour$ and $\mink$]

In this section, we will lower and raise indices with the spacetime metric $\gfour$
and its inverse $\gfour^{-1}$
respectively, i.e., $\fourvelocity_{\alpha} := \gfour_{\alpha \beta} \fourvelocity^{\beta}$
and $\upxi^{\alpha} := (\gfour^{-1})^{\alpha \beta} \upxi_{\beta}$. Similar comments apply to the bulk of the article,
once we fix $\spacetimemanifold = \R^{1+3}$ and $\gfour = \mink$ as the Minkowski metric on $\R^{1+3}$;
see Sect. <ref>.
This will become especially important later on, when we introduce the acoustical metric $\hfour$, i.e.,
we do not raise or lower indices with $\hfour$ or $\hfour^{-1}$.

§.§.§ The basic fluid variables and relations

We now introduce the basic fluid variables that will play a role in our study of the relativistic Euler equations:
$\uprho : \spacetimemanifold \rightarrow [0,\infty)$ denotes the fluid's (proper) energy density,
$p : \spacetimemanifold \rightarrow [0,\infty)$ denotes the fluid pressure,
$\Ent : \spacetimemanifold \rightarrow [0,\infty)$ denotes the entropy per particle (“entropy” for short from now on),
$n : \spacetimemanifold \rightarrow [0,\infty)$ denotes the proper number density,
$\Temp : \spacetimemanifold \rightarrow [0,\infty)$ denotes the temperature,

\begin{align} \label{E:ENTHDEF}
\Enth
& := \frac{\uprho + p}{n}
\end{align}

denotes the enthalpy per particle,
and $\fourvelocity^{\alpha}$ denotes the fluid's four-velocity,
which is a future-directed vectorfield on $\spacetimemanifold$ normalized by:

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

In the rest of the paper,

\begin{align} \label{E:FIXEDPOSITIVEENTHALPYCONSTANT}
\overline{\Enth} & > 0
\end{align}

denotes an arbitrary fixed, positive value of $\Enth$; we find it convenient to use $\Enth$ to normalize
various constructions.

We find it convenient to work with the natural log of the enthalpy.

We define
the (dimensionless) logarithmic enthalpy $\Lnenth$ as follows:

\begin{align} \label{E:LOGENTHALPY}
\Lnenth
&:= \ln \left(\Enth/\overline{\Enth} \right).
\end{align}

We define the quantity $\TempoverEnth$ as follows:

\begin{align} \label{E:TEMPDIVIDEDBYENTH}
\TempoverEnth
&:= \frac{\Temp}{\Enth}.
\end{align}

§.§.§ A first statement of the relativistic Euler equations

Relative to arbitrary coordinates,
the relativistic Euler equations on $(\spacetimemanifold,\gfour)$ can be expressed as
the following system of quasilinear hyperbolic conservation laws:

\begin{align} \label{E:INTRODIVOFREENERGYMOMENTUMTENSORIS0}
\nabla_{\kappa} \mathbf{T}^{\alpha \kappa}
& = 0,
& (\alpha & = 0,1,2,3),
\\
\nabla_{\kappa} (n \fourvelocity^{\kappa})
& = 0,
\label{E:INTROCONSERVATIONOFPARTICLENUMBER}
\end{align}

where $\nabla$ is the Levi-Civita connection of the spacetime metric $\gfour$ and:

\begin{align} \label{E:FLUIDENERGYMOMENTUMTENSOR}
\mathbf{T}^{\alpha \beta}
& := (\uprho + p) \fourvelocity^{\alpha} \fourvelocity^{\beta}
p (\gfour^{-1})^{\alpha \beta},
& (\alpha, \beta & = 0,1,2,3),
\end{align}

is the energy-momentum tensor of the fluid.
It turns out that under additional assumptions discussed in Sect. <ref>,
equation (<ref>) can be viewed as a constraint that is preserved by
the flow of the PDEs (<ref>); see
equation (<ref>) and the discussion surrounding it.
We also emphasize that later on, we will discuss several other formulations of the flow.

§.§.§ Equation of state and thermodynamic relations

The system (<ref>)
is not
closed because there are too many fluid variables and not enough equations.
The standard path to closing the system
begins with an equation of state, that is, an assumed
functional relationship of the form $p = p(\uprho,\Ent)$.
The basic hyperbolicity of the equations will require that
speed of sound $\speed$,
defined by:

\begin{align} \label{E:SPEEDOFSOUND}
\speed
& :=
\sqrt{\frac{\partial p}{\partial \uprho} \left|\right._{\Ent}},
\end{align}

should be real and non-negative,
where $\frac{\partial p}{\partial \uprho} \left|\right._{\Ent}$
denotes the partial derivative of $p$ (i.e., of the equation of state)
with respect to $\uprho$ at fixed $\Ent$.
The vanishing of $\speed$ causes a severe degeneracy in the system,
and we will therefore restrict our attention to
solution regimes in which:

\begin{align} \label{E:SPEEDOFSOUNDASSUMEDBOUNDS}
0 & < \speed \leq 1,
\end{align}

where we have normalized our setup so that the speed of light is unity
(i.e., the second inequality in (<ref>) implies that the speed
of sound is less than or equal to the speed of light).
We also restrict our attention to solutions such that:

\begin{align} \label{E:POSITIVITYOFSOMEFLUIDVARIABLES}
\uprho
& > 0,
\,
p > 0,
\,
n > 0,
\,
\Temp > 0,
\,
\Enth > 0.
\end{align}

The laws of thermodynamics demand that the fluid variables satisfy the following functional relations:

\begin{align} \label{E:THERMOLAWS}
\Enth
& =
\frac{\partial \uprho}{\partial n}\left|_{\Ent}\right.,
\Temp
& = \frac{1}{n} \frac{\partial \uprho}{\partial \Ent} \left|\right._n,
\mathrm{d} \Enth
& = \frac{\mathrm{d} p}{n} + \Temp \mathrm{d} \Ent,
\end{align}

$\frac{\partial}{\partial n}\left|_{\Ent}\right.$
denotes partial differentiation with respect to $n$ at fixed $\Ent$
$\frac{\partial}{\partial \Ent} \left|\right._n$
denotes partial differentiation with respect to $\Ent$ at fixed $n$.

§.§ Minkowski metric assumption and Minkowski-rectangular coordinates

In the rest of the paper, unless we explicitly state otherwise,
it should be understood that the spacetime manifold $\spacetimemanifold$ is equal to $\mathbb{R}^{1+3}$
and that $\gfour$ is equal to the Minkowski metric, which for clarity we denote by $\mink$.
We fix a standard global Minkowski-rectangular coordinate system $\lbrace x^{\alpha} \rbrace_{\alpha=0,1,2,3}$ on $\spacetimemanifold$,
relative to which $\mink = \mbox{\upshape diag}(-1,1,1,1)$.
We use the notation $\lbrace \partial_{\alpha} \rbrace_{\alpha=0,1,2,3}$ to denote the partial derivative vectorfields in this
coordinate system. We also use the alternate notation $t := x^0$ and $\partial_t := \partial_0$.

§.§.§ Index conventions

From now until the end of the paper, we use the following conventions for indices.

* (Lowercase Greek index conventions) Lowercase Greek spacetime indices
$\alpha$, $\beta$, etc. correspond to the Minkowski-rectangular coordinates
and vary over $0,1,2,3$. All lowercase Greek indices are lowered and raised with the Minkowski metric $\mink$
and its inverse $\mink^{-1}$.
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 Minkowski-rectangular coordinates.

* (Lowercase Latin index conventions)
Lowercase Latin spatial indices $a$, $b$, etc. correspond to the Minkowski-rectangular 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,
then $\left\lbrace \upxi_{b_1 \cdots b_n}^{a_1 \cdots a_m} \right\rbrace_{a_1,\cdots,a_m,b_1,\cdots,b_n = 1,2,3}$
denotes its components with respect to the Minkowski-rectangular spatial coordinates.

* (Uppercase Latin index conventions)
Uppercase Latin spatial indices $A,B$, etc. correspond to the spatial coordinates
$(x^2,x^3)$ and vary over $2,3$.
This will be important in Sect. <ref>, where
$(x^2,x^3)$ will correspond to perturbations away from plane-symmetry.

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

§.§ A well-posed first-order formulation

In this section, we provide a first-order formulation of the flow in
which the unknowns are taken to be
$\left(\Lnenth,\fourvelocity^0,\fourvelocity^1,\fourvelocity^2,\fourvelocity^3,\Ent \right)$.
Thanks to the assumptions state in Sect. <ref>,
the remaining fluid variables can be expressed[By (<ref>),
one can also express $\fourvelocity^0 = \sqrt{1 + \fourvelocity_a \fourvelocity^a}$. ]
as functions of the unknowns.

Throughout the paper, if $\genericvf$ is a vectorfield and $f$ is a scalar function,
then $\genericvf f := \genericvf^{\kappa} \partial_{\kappa}f$ denotes the derivative of $f$
in the direction of $\genericvf$.

With the help of (<ref>),
one can compute that for $C^1$ solutions,
is equivalent to the following first-order formulation of the flow:

\begin{align}
\fourvelocity^{\kappa} \partial_{\kappa} \Lnenth
\speed^2 \partial_{\kappa} \fourvelocity^{\kappa}
& = 0,
\label{E:ENTHALPYEVOLUTION}
\\
\fourvelocity^{\kappa} \partial_{\kappa} \fourvelocity^{\alpha}
\Pi^{\alpha \kappa} \partial_{\kappa} \Lnenth
\TempoverEnth
(\pmb{\mathfrak{m}}^{-1})^{\alpha \kappa} \partial_{\kappa} \Ent
& = 0,
(\alpha & = 0,1,2,3),
\label{E:VELOCITYEVOLUTIONWITHPROJECTION}
\\
\fourvelocity^{\kappa} \partial_{\kappa} \Ent
& = 0,
\label{E:ENTROPYEVOLUTION}
\\
\mink(\fourvelocity,\fourvelocity)
& = - 1,
\label{E:AGAINFLUIDFOURVELOCITYNORMALIZED}
\end{align}

\begin{align} \label{E:PROJECTTIONONTOORTHOGONALCOMPLEMENTOFFOURVELOCITY}
\Pi^{\alpha \beta}
:= (\mink^{-1})^{\alpha \beta}
\fourvelocity^{\alpha} \fourvelocity^{\beta},
(\alpha, \beta & = 0,1,2,3),
\end{align}

denotes projection onto the $\mink$-orthogonal complement of $\fourvelocity$.
In particular,

\begin{align} \label{E:PROJECTIONANNIHILATESFOURVELOCITY}
\Pi^{\alpha \beta} \fourvelocity_{\beta}
& = 0,
(\alpha & = 0,1,2,3).
\end{align}

Equations (<ref>)–(<ref>)
are a first-order quasilinear hyperbolic system and are locally well-posed in suitable Sobolev spaces;
see Prop. <ref>.
using (<ref>)
We also note by contracting (<ref>) against $\fourvelocity_{\beta}$ and
using (<ref>), we deduce – without using (<ref>) –
the following identity:

\begin{align}
\begin{split} \label{E:EVOUTIONOFMINKOWSKINORMOFFOURVELCOITY}
\fourvelocity^{\kappa} \partial_{\kappa}
\left\lbrace
\mink(\fourvelocity,\fourvelocity)
\right\rbrace
& = \fourvelocity^{\kappa} (\partial_{\kappa} \fourvelocity_{\lambda}) \fourvelocity^{\lambda}
\\
& =
\left\lbrace
\mink(\fourvelocity,\fourvelocity)
\right\rbrace
\fourvelocity^{\kappa} \partial_{\kappa} \Lnenth.
\end{split}
\end{align}

Using (<ref>), one can easily show via Grönwall's inequality
that if $\mink(\fourvelocity,\fourvelocity) + 1$ vanishes at $t=0$, then
$\mink(\fourvelocity,\fourvelocity) + 1$ vanishes on any slab $[0,T] \times \mathbb{R}^3$
of classical existence on which the solution is $C^1$.
Hence, (<ref>) can be viewed as a constraint on the initial data that
is preserved by the flow.

When posed on a general smooth spacetime background $(\spacetimemanifold,\gfour)$,
there are additional terms on
RHS (<ref>)–(<ref>)
of the schematic form $\bf{\Gamma} \cdot \smoothfunction(\solutionarray)$,
where $\bf{\Gamma}$ schematically denotes Christoffel symbols of
$\solutionarray$ denotes the array of fluid variable unknowns (see definition <ref>),
and $\smoothfunction$ schematically denotes a smooth function.
In the context of fluid shock formation, the dominant terms in the flow
are Riccati-type nonlinearities in the fluid obtained after differentiating the equations one time,
i.e., they are quadratic in the first derivatives of the fluid;
see Remark <ref>.
Note that upon differentiating the term
$\bf{\Gamma} \cdot \smoothfunction(\solutionarray)$,
one obtains a term that is linear in the derivatives of the fluid variables.
This means, in particular, that the for large-gradient solutions,
the differentiated terms $\partial_{\alpha} \left(\bf{\Gamma} \cdot \smoothfunction(\solutionarray) \right)$
will be small relative to the shock-driving Riccati-type fluid terms,
i.e., the “new terms” coming from the non-flat ambient geometry are relatively small.
This observation is a key reason behind our expectation that one can use the methods
described later in this article to prove shock formation
for the relativistic Euler equations on a general spacetime
at least for initial data with suitable large gradients.

§.§.§ Equations of variation

As a preliminary step in studying the well-posedness of
we will discuss the equations satisfied by the derivatives of solutions. This is important in the
sense that proofs of local well-posedness rely on differentiating the equations to obtain energy estimates
for the solution's higher derivatives.
Specifically, by differentiating the equations, it is straightforward to see that
given a solution $(\Lnenth,\fourvelocity^{\alpha},\Ent)$ to (<ref>)–(<ref>),
any derivative (of any order), which we denote by $(\dot{\Lnenth},\dot{\fourvelocity}^{\alpha},\dot{\Ent})$,
satisfies an inhomogeneous system, known as the equations of variation, of the following form:

\begin{align}
\fourvelocity^{\kappa} \partial_{\kappa} \dot{\Lnenth}
\speed^2 \partial_{\kappa} \dot{\fourvelocity}^{\kappa}
& = \dot{\mathfrak{F}},
\label{E:EOVENTHALPYEVOLUTION}
\\
\fourvelocity^{\kappa} \partial_{\kappa} \dot{\fourvelocity}^{\alpha}
\Pi^{\alpha \kappa} \partial_{\kappa} \dot{\Lnenth}
\TempoverEnth
(\pmb{\mathfrak{m}}^{-1})^{\alpha \kappa} \partial_{\kappa} \dot{\Ent}
& = \dot{\mathfrak{G}}^{\alpha},
(\alpha & = 0,1,2,3),
\label{E:EOVVELOCITYEVOLUTIONWITHPROJECTION}
\\
\fourvelocity^{\kappa} \partial_{\kappa} \dot{\Ent}
& = \dot{\mathfrak{H}},
\label{E:EOVENTROPYEVOLUTION}
\\
\mink(\fourvelocity,\dot{\fourvelocity})
& = \dot{\mathfrak{I}}.
\label{E:EOVFLUIDFOURVELOCITYNORMALIZED}
\end{align}

Although here we do not provide a detailed formula for the structure of
the inhomogeneous terms $\dot{\mathfrak{F}}$, $\cdots$, $\dot{\mathfrak{I}}$
on RHSs (<ref>)–(<ref>),
it is easy to see that they feature terms involving $\leq$ the number of derivatives represented by the
variations $(\dot{\Lnenth},\dot{\fourvelocity}^{\alpha},\dot{\Ent})$,
i.e., fewer derivatives than the principal terms on LHSs (<ref>)–(<ref>).

§.§.§ The acoustical metric

A basic question about the equations of variation is: what kinds of energy estimates are available for solutions?
It turns out that this question is connected to the intrinsic geometry of the flow via the acoustical metric,
which plays a fundamental role throughout the article.

We define:

\begin{align} \label{E:ACOUSTICALMETRIC}
\hfour_{\alpha \beta}
& : =
\normalizer
\left\lbrace
\speed^{-2} \mink_{\alpha \beta}
(\speed^{-2} - 1) \fourvelocity_{\alpha} \fourvelocity_{\beta}
\right
\rbrace,
\\
(\hfour^{-1})^{\alpha \beta}
& :
\normalizer^{-1}
\left\lbrace
\speed^2 (\mink^{-1})^{\alpha \beta}
(\speed^2 - 1)
\fourvelocity^{\alpha} \fourvelocity^{\beta}
\right\rbrace,
\label{E:INVERSEACOUSTICALMETRIC}
\\
\normalizer
& :=
\speed^2
(1 - \speed^2) (\fourvelocity^0)^2 \geq 1.
\label{E:ACOUSTICALMETRICNORMALIZER}
\end{align}

We refer to $\hfour$ as the acoustical metric.

With the help of (<ref>), one can compute that:

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

where $\updelta_{\alpha}^{\beta}$ is the Kronecker delta. That is,
(<ref>) shows that $\hfour^{-1}$ is indeed the inverse of $\hfour$.
Moreover, by construction, we have:

\begin{align} \label{E:GINVERSE00ISMINUSONE}
& = -1.
\end{align}

(<ref>) is a convenient normalization condition.
Other works (e.g. <cit.>) rely on a different normalization condition for the acoustical metric.
That represents a minor difference that typically has no important implications for the study of solutions.

Note that by (<ref>),
and (<ref>),
$\hfour(\genericvf,\genericvf) < 0 \implies \mink(\genericvf,\genericvf) < 0$,
i.e., the $\hfour$-null cones are inside the $\mink$-null cones.
Note also that by (<ref>),
and (<ref>),
we have:

\begin{align} \label{E:GFOURLENGTHOFFOURVELOCITY}
\hfour_{\kappa \lambda}
\fourvelocity^{\kappa}
\fourvelocity^{\lambda}
& = - \normalizer < 0.
\end{align}

In particular, $\fourvelocity$ is $\hfour$-timelike (and thus $\mink$-timelike, as one can explicitly see
via (<ref>)).

When studying the relativistic Euler equations in a general ambient spacetime $(\spacetimemanifold,\gfour)$
equipped with coordinates $(x^0,x^1,x^2,x^3)$,
where $t := x^0$ is a smooth time function
(see Sect. <ref>),
the acoustical metric is (modulo the comments of Remark <ref>):

\begin{align} \label{E:ACOUSTICALMETRICGENERALAMBINETSPACETIME}
\hfour_{\alpha \beta}
: =
\normalizer
\left\lbrace
\speed^{-2} \gfour_{\alpha \beta}
(\speed^{-2} - 1) \fourvelocity_{\alpha} \fourvelocity_{\beta}
\right
\rbrace,
\end{align}

\begin{align} \label{E:GENERALAMBIENTSPACETIMEACOUSTICALMETRICNORMALIZER}
\normalizer
& :=
\speed^2
(1 - \speed^2) (\fourvelocity^0)^2 > 0,
\end{align}

where the positivity on RHS (<ref>)
stems from (<ref>) and the assumption that $t$ is a time function for $\gfour$,
which in particular implies that relative to the coordinates $(x^0,x^1,x^2,x^3)$,
we have
(\gfour^{-1})^{\alpha \beta}
(\partial_{\alpha} t)
\partial_{\beta} t
< 0

§.§.§ The arrays $\solutionarray$ and $\variations$

For notational convenience, we define the solution array
$\solutionarray$ and the variation array $\variations$ as follows:

\begin{align}
\solutionarray
& := \left(\Lnenth,\fourvelocity^0,\fourvelocity^1,\fourvelocity^2,\fourvelocity^3,\Ent \right),
\label{E:SOLUTIONARRAY}
\\
\variations
& := \left(\dot{\Lnenth},\dot{\fourvelocity}^0,\dot{\fourvelocity}^1,\dot{\fourvelocity}^2,\dot{\fourvelocity}^3,\dot{\Ent} \right).
\label{E:VARIATIONARRAY}
\end{align}

§.§.§ Energy currents and energy identity in differential form

We now introduce energy currents, which form the basis for energy identities for the equations of variation.

Given a solution $\solutionarray$ to
variations $\variations = (\dot{\Lnenth},\dot{\fourvelocity}^{\alpha},\dot{\Ent})$
that satisfy (<ref>)–(<ref>),
and scalar functions $f_1$ and $f_2$,
we define the corresponding energy current vectorfield
as follows:

\begin{align}
\begin{split}
\label{E:ENERGYCURRENTVECTORFIELD}
\dot{\mathbf{J}}_{(\solutionarray,f_1,f_2)}^{\alpha}[\variations,\variations]
& :=
\fourvelocity^{\alpha} \dot{\Ent}^2
2 \speed^2 \TempoverEnth \dot{\fourvelocity}^{\alpha} \dot{\Ent}
2 \TempoverEnth \fourvelocity^{\alpha} \dot{\Ent} \dot{\Lnenth}
\fourvelocity^{\alpha} \dot{\Lnenth}^2
2 \speed^2 \dot{\fourvelocity}^{\alpha} \dot{\Lnenth}
\speed^2 \fourvelocity^{\alpha} \dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}
\\
& \ \
2 \speed^2 \fourvelocity^{\alpha} \fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} \dot{\Lnenth}
f_2 \fourvelocity^{\alpha} (\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa})^2.
\end{split}
\end{align}

Note that the components of $\dot{\mathbf{J}}$ can be viewed as quadratic forms in
$\variations$ with coefficients that depend on the solution $\solutionarray$.
Related energy currents were used by Christodoulou in <cit.>. A general framework for
constructing energy currents in the context of regularly hyperbolic PDEs was
developed by Christodoulou in <cit.>.

The following lemma provides a differential version of the energy identities
afforded by the energy current. The proof is a tedious but straightforward computation, omitted here
(c.f. <cit.>*Equation (1.41)).

Under the assumptions of Def. <ref>, with
denoting the energy current defined in (<ref>),
the following identity holds relative to the Minkowski-rectangular coordinates:

\begin{align}
\begin{split} \label{E:ENERGYCURRENTDIVERGENCEIDENTITY}
\partial_{\kappa} \dot{\mathbf{J}}_{(\solutionarray,f_1,f_2)}^{\kappa}[\variations,\variations]
& = 2 f_1 \dot{\Ent} \dot{\mathfrak{H}}
2 \TempoverEnth \dot{\Lnenth} \dot{\mathfrak{H}}
2 \TempoverEnth \dot{\Ent} \dot{\mathfrak{F}}
2 \dot{\Lnenth} \dot{\mathfrak{F}}
2 \speed^2 \dot{\fourvelocity}_{\kappa} \dot{\mathfrak{G}}^{\kappa}
2 \speed^2 \dot{\Lnenth} \fourvelocity_{\kappa} \dot{\mathfrak{G}}^{\kappa}
\\
& \ \
2 f_2 \TempoverEnth \fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} \dot{\mathfrak{H}}
2 f_2 \fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} \fourvelocity_{\lambda} \dot{\mathfrak{G}}^{\lambda}
\mathscr{Q}_{(\solutionarray,f_1,f_2,\bm{\partial} \solutionarray,\bm{\partial} f_1, \bm{\partial} f_2)}[\variations,\variations],
\end{split}
\end{align}

$\mathscr{Q}_{(\solutionarray,f_1,f_2,\bm{\partial} \solutionarray,\bm{\partial} f_1, \bm{\partial} f_2)}[\variations,\variations]$
is a quadratic form in $\variations$ with coefficients that depend on
$\solutionarray$, $f_1$, $f_2$, and their first-order partial derivatives;
it is straightforward to compute $\mathscr{Q}$
in detail, though we do not do so here.

By integrating (<ref>) over a spacetime region and applying the divergence theorem,
one obtains an energy identity, where the energies are defined along the boundary of the region. A fundamental question is:
when are the energies obtained in this fashion positive definite[In general, energies are not useful if they are indefinite.
The answer to this question, provided by Lemma <ref>,
is tied to the acoustic geometry of the system, captured by the acoustical metric.
In short, the energies are positive definite whenever the boundary of the region
has $\hfour$-timelike normals, i.e., whenever the boundary is spacelike with respect to $\hfour$.

Before proving the lemma, we first introduce a subset of state-space in which the
relativistic Euler equations are hyperbolic.

We define the regime of hyperbolicity $\mathcal{H}$ to be the following subset of solution variable space:

\begin{align} \label{E:REGIMEOFHYPERBOLICITY}
\mathcal{H}
& := \left\lbrace
\left(\Lnenth,\fourvelocity^0,\fourvelocity^1,\fourvelocity^2,\fourvelocity^3,\Ent \right) \in \mathbb{R}^6
\ | \ 0 < \speed(\Lnenth,\Ent) \leq 1
\right\rbrace.
\end{align}

Let $\upxi$ be a past-directed $\hfour$-timelike one-form,
i.e., assume that $(\hfour^{-1})^{\kappa \lambda} \upxi_{\kappa} \upxi_{\lambda} < 0$ and $\upxi_0 := \upxi(\partial_t) > 0$.
Let $\mathfrak{K}$ be a compact subset of the set $\mathcal{H}$ defined in (<ref>),
and assume that the solution array $\solutionarray$ defined in (<ref>) satisfies
$\solutionarray \in \mathfrak{K}$.
Then if $f_1$ and $f_2$ are sufficiently large (in a manner that depends on $\upxi$ and $\mathfrak{K}$),
there exists a $C = C(\upxi,\mathfrak{K},f_1,f_2) > 0$ such that
the quadratic form $\upxi_{\alpha} \dot{\mathbf{J}}_{(\solutionarray,f_1,f_2)}^{\alpha}[\variations,\variations]$
is positive definite in the following sense:

\begin{align} \label{E:POSITIVEDEFINITENESSOFENERGYCURRENT}
\upxi_{\kappa} \dot{\mathbf{J}}_{(\solutionarray,f_1,f_2)}^{\kappa}[\variations,\variations]
& \geq
C |\variations|^2.
\end{align}

Although Lemmas <ref>–<ref> might seem at first glance like unmotivated technical results, they sit within the general framework of multiplier methods for hyperbolic equations. To give a rough idea of the multiplier method, consider a Schwartz-class solution $\phi$ to the linear wave equation
$-\p_t^2\phi + \Delta_x \phi = 0$ on Minkowski spacetime $(\R^{1+3},\mink)$.
We define the energy current vectorfield $\mathbf{J}$ to be the vectorfield with the following components relative
the standard Minkowski-rectangular coordinates:

\begin{align}
\mathbf{J}^0
& := \frac{1}{2} (\partial_t \phi)^2 + \frac{1}{2} |\nabla_x \phi|^2,
\label{E:WAVEENERGYCURRENT0COMPONENT}
\\
\mathbf{J}^i
& := - (\p_i \phi) \p_t \phi,
&& (i=1,2,3).
\label{E:WAVEENERGYCURRENTSPACECOMPONENTS}
\end{align}

In (<ref>), $\nabla_x \phi := (\p_1 \phi,\phi_2,\phi_3)$ denotes the spatial gradient of $\phi$
and $| \cdot |$ is the standard Euclidean norm on $\R^3$.
Then straightforward calculations yield that $\partial_{\alpha} \mathbf{J}^\alpha = 0$ and that $(\p_t)_\alpha \mathbf{J}^\alpha = \tfrac{1}{2} (\p_t \phi)^2 + \tfrac{1}{2} |\nabla_x \phi|^2$. These two identities are analogs of (<ref>) and
(<ref>) respectively.
The upshot is that by integrating $\partial_{\alpha} \mathbf{J}^\alpha = 0$
over $[0,t] \times \mathbb{R}^3$, one obtains the celebrated conservation of energy formula for solutions
to the linear wave equation:

\begin{align} \label{E:CONSERVATIONOFENERGYINLINEARWAVEEQ}
\frac{1}{2}
\int_{\{t\} \times \R^3}
\left\{(\p_t \phi)^2 + |\nabla_x \phi|^2
\right\}\mathrm{d} x
& =
\frac{1}{2}
\int_{\{0\} \times \R^3}  \left\{(\p_t\phi)^2 + |\nabla_x \phi|^2 \right\} \mathrm{d} x.
\end{align}

The approach is called the “multiplier method” because it is a geometric version of the more standard proof of
in which one multiplies the wave equation by $\p_t \phi$ and integrates by parts over $[0,t] \times \mathbb{R}^3$.

Throughout, $C =C(\upxi,\mathfrak{K},f_1,f_2)$
denotes a positive constant (often smaller than $1$) that can vary from line to line.

We start by defining the scalar function $\upxi_{\parallel}$ by:

\begin{align} \label{E:XICONTRACTIWITHFOURVELOCITY}
\upxi^{\parallel}
& := \upxi_{\kappa} \fourvelocity^{\kappa}.
\end{align}

Then, using (<ref>), we decompose:

\begin{align} \label{E:DEOCMPOSEXIINTOPARALLELANDPERPENDICULAR}
\upxi_{\alpha}
& =
\upxi^{\parallel} \fourvelocity_{\alpha}
\upxi_{\alpha}^{\perp},
\end{align}

where the one-form $\upxi_{\alpha}^{\perp}$ satisfies
$\upxi_{\kappa}^{\perp} \fourvelocity^{\kappa} = 0$.
In particular, $\upxi_{\alpha}^{\perp}$ is $\mink$-orthogonal to the $\mink$-timelike vectorfield
$\fourvelocity^{\alpha}$ and thus $\upxi_{\alpha}^{\perp}$ is $\mink$-spacelike whenever it is non-zero,
i.e., $(\mink)^{-1} \upxi_{\kappa}^{\perp} \upxi_{\lambda}^{\perp} \geq 0$.
Since $(\hfour^{-1})^{\kappa \lambda} \upxi_{\kappa} \upxi_{\lambda} < 0$ and
$\hfour_{\kappa \lambda} \fourvelocity^{\kappa} \fourvelocity^{\lambda} < 0$
(see (<ref>)),
and since $\upxi_{\alpha}$ is past-directed while $\fourvelocity^{\alpha}$ is future-directed,
it follows that there is a $C$ with $0 < C < 1$ such that:

\begin{align} \label{E:BOUNDFORXIPARALLEL}
& \leq
\upxi^{\parallel}
\leq \frac{1}{C}.
\end{align}

We can similarly decompose $\dot{\fourvelocity}^{\alpha}$ into a variation that is parallel to
$\fourvelocity^{\alpha}$ and one that vanishes upon contraction with $\fourvelocity_{\alpha}$.
We claim that prove (<ref>),
it suffices to show that when $f_1 = f_2 = 0$, $\dot{\Ent} = 0$,
and $\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} = 0$,
we have:

\begin{align} \label{E:REDUCEDPOSITIVEDEFINITENESSSTATEMENT}
\upxi_{\kappa} \dot{\mathbf{J}}^{\kappa}[\variations,\variations]
& \geq C (\dot{\Lnenth}^2 + \dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}).
\end{align}

The estimate (<ref>) would yield control over $\dot{\Lnenth}^2$ and variations such that
$\dot{\Ent} = \fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} = 0$
(note that $\dot{\fourvelocity}^{\alpha}$ is $\mink$-spacelike for such variations and thus the
term $\dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}$ on RHS (<ref>)
is positive-definite in such variations).
to handle general variations,
since the first and last
terms on RHS (<ref>)
yield the two coercive (in view of (<ref>)) terms
$f_1 \upxi^{\parallel} \dot{\Ent}^2$ and $f_2 \upxi^{\parallel} (\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa})^2$,
we can choose $f_1$ and $f_2$ to be sufficiently large
(for example, we can choose them to be large, positive constants)
to obtain control over
$\dot{\Ent}^2$ and $(\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa})^2$, and large enough so that these
two coercive terms
and the coercivity (<ref>)
can collectively be used to absorb (via Young's inequality)
all the remaining terms on RHS (<ref>).
In total, this would yield (<ref>).

Hence, to complete the proof of (<ref>), we can
assume that $f_1 = f_2 = 0$, $\dot{\Ent} = 0$,
and $\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} = 0$.
This implies that $\dot{\fourvelocity}^{\alpha}$ is $\mink$-spacelike.
We use (<ref>),
to compute that:

\begin{align} \label{E:REDUCEDPOSITIVEDEFINITENESSEQUALITY}
\upxi_{\kappa} \dot{\mathbf{J}}^{\kappa}[\variations,\variations]
& =
\upxi^{\parallel} \dot{\Lnenth}^2
2 \speed^2 \upxi_{\kappa}^{\perp} \dot{\fourvelocity}^{\kappa} \dot{\Lnenth}
\speed^2 \upxi^{\parallel} \dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}.
\end{align}

Since $(\hfour^{-1})^{\kappa \lambda} \upxi_{\kappa} \upxi_{\lambda} < - C < 0$,
$C < \normalizer < \frac{1}{C}$,
and $\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} = 0$,
we compute using
and (<ref>)
that there is a $C' > 0$ such that:

\begin{align} \label{E:KEYBOUNDFOR}
\normalizer
(\hfour^{-1})^{\kappa \lambda} \upxi_{\kappa} \upxi_{\lambda}
& =
\speed^2
(\mink^{-1})^{\kappa \lambda}
\upxi_{\kappa}^{\perp}
\upxi_{\lambda}^{\perp}
< - C'.
\end{align}

We are denoting the positive constant on RHS (<ref>) by “$C'$” (rather than “$C$”) for clarity.
Since $\upxi_{\alpha}^{\perp}$ is $\mink$-spacelike, we have
(\mink^{-1})^{\kappa \lambda}
\upxi_{\kappa}^{\perp}
\upxi_{\lambda}^{\perp} \geq 0$
and thus $C' < (\upxi^{\parallel})^2$.
Since $\upxi_{\alpha}^{\perp}$ and $\dot{\fourvelocity}^{\alpha}$ are
for any $\upsigma > 0$,
we can use
the Cauchy–Schwarz inequality with respect to $\mink$,
and Young's inequality
to bound the cross term on RHS (<ref>) as follows:

\begin{align}
\begin{split} \label{E:KEYBOUNDFORCROSSTERM}
|2 \speed^2 \upxi_{\kappa}^{\perp} \dot{\fourvelocity}^{\kappa} \dot{\Lnenth}|
& \leq
\frac{1}{\upsigma}
\speed^4
(\mink^{-1})^{\alpha \beta}
\upxi_{\alpha}^{\perp}
\upxi_{\beta}^{\perp}
\dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}
\upsigma
\dot{\Lnenth}^2
\\
\leq
\frac{1}{\upsigma}
\speed^2
\left\lbrace
\right\rbrace
\dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}
\upsigma
\dot{\Lnenth}^2
\\
\leq
\frac{1}{\upsigma}
\speed^2
\left\lbrace
C' C^2
\right\rbrace
\dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}
\upsigma
\dot{\Lnenth}^2,
\end{split}
\end{align}

where the last $C$ on RHS (<ref>) is from (<ref>).
We now set $\upsigma := \upxi^{\parallel}
\left\lbrace
C' C^2
\right\rbrace^{1/2}$. It follows that:

\begin{align} \label{E:FINALKEYBOUNDFORCROSSTERM}
|2 \speed^2 \upxi_{\kappa}^{\perp} \dot{\fourvelocity}^{\kappa} \dot{\Lnenth}|
& \leq
\speed^2
\upxi^{\parallel}
\left\lbrace
C' C^2
\right\rbrace^{1/2}
\dot{\fourvelocity}_{\kappa} \dot{\fourvelocity}^{\kappa}
\upxi^{\parallel}
\left\lbrace
C' C^2
\right\rbrace^{1/2}
\dot{\Lnenth}^2.
\end{align}

since $1
C' C^2 < 1$,
we deduce from (<ref>)
that the cross term on RHS (<ref>) can be absorbed
into the two positive definite terms with room to spare.
We have therefore proved (<ref>)
assuming $f_1 = f_2 = 0$, $\dot{\Ent} = 0$,
$\fourvelocity_{\kappa} \dot{\fourvelocity}^{\kappa} = 0$,
which completes the proof of the lemma.

§.§.§ Local well-posedness and continuation principle in Sobolev spaces

In Prop. <ref>, we discuss local well-posedness for the relativistic Euler equations in Sobolev spaces.

Recall the notation $\solutionarray$ introduced in (<ref>).
\mathring{\fourvelocity}^3,\mathring{\Ent})
be initial data for the relativistic Euler equations (<ref>)–(<ref>)
(in particular, assume that $\mathring{\fourvelocity}$ satisfies (<ref>)).
Assume that there is an integer $N \geq 3$ such that
\mathring{\Lnenth},\mathring{\fourvelocity}^1,\mathring{\fourvelocity}^2,\mathring{\fourvelocity}^3, \mathring{\Ent}
\in H^N(\mathbb{R}^3)
(and thus (<ref>) and the standard Sobolev–Moser calculus imply that
$\mathring{\fourvelocity}^0 - 1 \in H^N(\mathbb{R}^3)$)
and that $\mathring{\solutionarray}(\mathbb{R}^3)$ is contained in a compact subset
of the interior of set $\mathcal{H}$ defined in (<ref>).
Then there is a maximal $T_{\textnormal{Lifespan}} > 0$ such that a unique solution to
exists and satisfies
$\Lnenth, \fourvelocity^0 - 1, \fourvelocity^1, \fourvelocity^2, \fourvelocity^3, \Ent
\in
C^0\left([0,T_{\textnormal{Lifespan}}),H^N(\mathbb{R}^3) \right)
\cap
C^1\left([0,T_{\textnormal{Lifespan}}), H^{N-1}(\mathbb{R}^3) \right)
Moreover, either $T_{\textnormal{Lifespan}} = \infty$, or one of the following two breakdown scenarios occurs:

* (Blowup). The solution's $C^1$ norm blows up:

\begin{align} \label{E:C1BLOWUPSCENARIO}
\lim_{t \uparrow T_{\textnormal{Lifespan}}}
\| \solutionarray \|_{C^1([0,t] \times \mathbb{R}^3)}
= \infty.
\end{align}

* (Exiting the regime of hyperbolicity).
There exists a sequence of points
$\lbrace (t_m,x_m) \rbrace_{m \in \mathbb{N}} \subset [0,T_{\textnormal{Lifespan}} ) \times \mathbb{R}^3$
such that for every compact subset $\mathfrak{K}$ of the interior of $\mathcal{H}$,
there exists an $m_{\mathfrak{K}} \in \mathbb{N}$ such that
\notin \mathfrak{K}
whenever $m \geq m_{\mathfrak{K}}$.

We make the following remarks.

* In Sects. <ref> and <ref>,  we will study the formation
of shock singularities, in which the solution remains in the interior of the regime of hyperbolicity and the breakdown is precisely
due to the scenario (<ref>).

* The techniques of <cit.>, which rely only on energy estimates, Sobolev embedding, and the fractional
Sobolev–Moser calculus, can be used to save half a derivative compared to Prop. <ref>, that is, to
prove local well-posedness for the $3D$ relativistic Euler equations whenever
\mathring{\Lnenth},\mathring{\fourvelocity}^1,\mathring{\fourvelocity}^2,\mathring{\fourvelocity}^3, \mathring{\Ent}
\in H^{{5/2}^+}(\mathbb{R}^3)

* In <cit.>, a low-regularity
local well-posedness result for the $3D$ relativistic Euler equations was proved. The low-regularity assumption
was on the “wave-part” of the initial data, which was assumed to belong to $H^{2^+}(\mathbb{R}^3)$,
while the “transport-part” (including the vorticity and entropy) was assumed to satisfy some
additional smoothness assumptions, which were shown to be propagated by the flow;
the terminology “wave-part” and “transport”
is tied to the formulation of the flow provided by Theorem <ref>
and is explained in <cit.>.
Lindblad's work <cit.> implies that the assumptions on the wave-part
of the initial data are optimal in the scale of Sobolev spaces due to the
phenomenon of instantaneous shock formation for initial data in $H^2(\mathbb{R}^3)$.
The analysis in <cit.>
relies on the new formulation of the flow presented in Theorem <ref>,
which allows one to employ analytical techniques based on nonlinear geometric optics and Strichartz estimates.
The techniques used in <cit.> have roots in the works
mDcLgMjS2022,qW2022 (see also zH2020,zHlA2021,zHlA2022)
on low-regularity well-posedness for the non-relativistic compressible Euler equations,
which in turn have roots in the works
on low-regularity well-posedness for quasilinear wave equations.

This is a standard result that can be proved by linearizing the equations,
deriving a priori estimates for solutions to the linearized equations, and
then invoking a standard iteration scheme.
We refer to jS2008a,jS2008b,jS2008c for the main ideas, where an analog of Prop. <ref>
was proved for the coupling of the relativistic Euler equations to
Nordström's theory of gravity. The main idea behind
the a priori estimates (in $H^N$) is to construct energy currents for the linearized systems
such that analogs of (<ref>)
and (<ref>) hold,
and to control RHS (<ref>)
in $L^2$ using the Sobolev–Moser calculus (which is standard in $3D$ when $N \geq 3$).

§ A STUDY OF THE MAXIMAL DEVELOPMENT IN SIMPLE ISENTROPIC PLANE-SYMMETRY

Our goal in this section is to prove Theorem <ref>, which yields
a complete description of a localized portion of the boundary of the maximal (classical) $\hfour$-globally hyperbolic development ($\hfour$-MGHD for short)
for open sets of shock-forming solutions to the relativistic Euler equations (<ref>)–(<ref>)
in simple isentropic plane-symmetry; see Fig. <ref>.
Roughly, the $\hfour$-MGHD is the largest possible classical solution + $\hfour$-globally hyperbolic region that is launched by the initial data.
By a $\hfour$-globally hyperbolic region $\mathfrak{R}$ of classical existence launched by the initial data, we mean that every inextendible
$\hfour$-causal curve in $\mathfrak{R}$ intersects $\Sigma_0 := \lbrace t = 0 \rbrace$,
where $\hfour$ is the acoustical metric from Def. <ref>.
In particular, the $\hfour$-MGHD cannot contain any points lying in the $\hfour$-causal future of any singularity,
where the $\hfour$-causal future of a point is effectively determined by integral curves of
the two transversal null vectorfields $\Lunit$ and $\uLunit$ from
Def. <ref>.

The portion of the boundary that we study includes a singular boundary, where the fluid's gradient blows up in a shock singularity,
and a Cauchy horizon, along which the solution remains smooth.
For the solutions under study, the past boundary of the Cauchy horizon intersects the past boundary of the singular boundary
in a distinguished point that we refer to as “the crease;” see Fig. <ref>.
The crease plays the role of the true initial singularity in
a shock-forming relativistic Euler solution.

Sect. <ref> is intended to provide a gentle introduction to the main ideas behind the study of the $\hfour$-MGHD
in a simplified setting in which the geometry is easy to study and one can avoid the burden of energy estimates.
It will also help set the stage for Sect. <ref>, in which we
outline some of the main ideas needed to study shock formation in $3D$.

§.§ A description of the symmetry class

As we mentioned, we will study simple isentropic plane-symmetric solutions, which we now describe.
First, by “isentropic solutions,” we mean that $\Ent \equiv 0$;
for any constant $\Ent_0$, we could treat the case $\Ent \equiv \Ent_0$ using the same arguments.
From (<ref>), we infer that for $C^1$ isentropic solutions,
the condition $\Ent = \Ent_0$
is preserved by the flow if it is satisfied at $t = 0$. By “plane-symmetry,” we mean that
$\fourvelocity^2 = \fourvelocity^3 \equiv 0$ and that $\Lnenth = \Lnenth(t,x^1)$, $\fourvelocity^1 = \fourvelocity^1(t,x^1)$, and $\fourvelocity^0 = \fourvelocity^0(t,x^1)$.
Such solutions can be viewed as solutions in $1D$ in which $\Ent \equiv 0$, and throughout
the rest of the paper,
we make that identification. By “simple,” we mean that only one Riemann invariant
(see Sect. <ref>) is non-vanishing.

§.§ The basic setup and conventions in plane-symmetry

In Sect. <ref>, since we are studying isentropic plane-symmetric solutions,
we will slightly abuse notation and use $\hfour$ to denote the restriction of
the acoustical metric (<ref>)–(<ref>),
to the $(t,x^1)$-plane. That is, in Sect. <ref>, relative to the $(t,x^1)$-coordinates,
we have:

\begin{align} \label{E:1DACOUSTICALMETRIC}
\hfour_{\alpha \beta}
& : =
\normalizer
\left\lbrace
\speed^{-2} \mink_{\alpha \beta}
(\speed^{-2} - 1) \fourvelocity_{\alpha} \fourvelocity_{\beta}
\right
\rbrace,
(\alpha, \beta
& = 0,1),
\\
(\hfour^{-1})^{\alpha \beta}
& :
\normalizer^{-1}
\left\lbrace
\speed^2 (\mink^{-1})^{\alpha \beta}
(\speed^2 - 1)
\fourvelocity^{\alpha} \fourvelocity^{\beta}
\right\rbrace,
(\alpha, \beta
& = 0,1),
\label{E:1DINVERSEACOUSTICALMETRIC}
\end{align}

where $\normalizer$ is defined by (<ref>),
and we recall that (<ref>) holds.

For future use, we note that for $C^1$  isentropic plane-symmetric solutions,
the relativistic Euler equations (<ref>)–(<ref>)
in Minkowski spacetime are equivalent to:

\begin{align}
\fourvelocity^{\kappa} \partial_{\kappa} \Lnenth
\speed^2 \partial_{\kappa} \fourvelocity^{\kappa}
& = 0,
\label{E:1DENTHALPYEVOLUTION}
\\
\fourvelocity^{\kappa} \partial_{\kappa} \fourvelocity^1
\Pi^{1 \kappa} \partial_{\kappa} \Lnenth
& = 0,
\label{E:1DVELOCITYEVOLUTIONWITHPROJECTION}
\\
\fourvelocity^0
& = \sqrt{1 + (\fourvelocity^1)^2}.
\label{E:1DFLUIDFOURVELOCITYNORMALIZED}
\end{align}

For future use, we also note the following algebraic identity, which holds in plane-symmetry due to

\begin{align} \label{E:NORMAIZERFACTORSIN1D}
\normalizer
& = (\fourvelocity^0 + \fourvelocity^1 \speed)(\fourvelocity^0 - \fourvelocity^1 \speed).
\end{align}

To ensure ensure that shocks can form in solutions near constant fluid states with constant enthalpy $\overline{\Enth} > 0$,
we assume the following non-degeneracy assumption:

\begin{align} \label{E:NONDEGENARCYASSUMPTION}
1 - (\overline{\speed})^2 + \frac{\overline{\speed}'}{\overline{\speed} \overline{\Enth}}
& \neq 0,
\end{align}

where in (<ref>), we are viewing $\speed = \speed(\Enth)$,
$\speed' := \frac{\mathrm{d}}{\mathrm{d} \Enth} \speed$,
and LHS (<ref>) is defined to be the constant obtained by evaluating
$1 - \speed^2 + \frac{\speed'}{\Enth \speed}$ at $\Enth = \overline{\Enth}$.
The condition (<ref>) implies in particular that
the isentropic plane-symmetric relativistic Euler equations are genuinely nonlinear near the
constant-state solution $(\Enth,\fourvelocity^1) \equiv (\overline{\Enth},0)$;
see also Sect. <ref>.
Christodoulou showed <cit.>*Chapter 1 that irrotational relativistic fluids
(of which isentropic plane-symmetric fluids are a special case)
admit a Lagrangian formulation for a potential function,
and the only equation of state leading to
$1 - \speed^2 + \frac{\speed'}{\Enth \speed} \equiv 0$
is such that the potential function solves the
timelike minimal surface equation.[For the minimal surface equation,
$\speed \equiv 1$ corresponds to the trivial embedding $\R^{1+d} \hookrightarrow \R^{1 + (d+1)}$ with respect to the Minkowski metric.]
For that equation, which satisfies a nonlinear version of the null condition that is stronger than the
classic null condition identified by Klainerman <cit.>,
the first author proved that perturbations of plane-symmetric data
(which are similar to the data we study here in Sect. <ref>)
lead to global, nonlinearly stable $C^2$ solutions in both $\R^{1+1}$ and $\R^{1+3}$

§.§ Null frame

In $1D$, it is possible to explicitly write down a pair of transversal null vectorfields that dictate the geometry of sound waves.
As we explain in Sect. <ref>, in more than one spatial dimension, it is not possible to explicitly write down
null vectorfields in terms of the fluid variables. Rather, in that context, the null vectorfields depend on the gradient
of an eikonal function $\eik$.

Relative to the Minkowski-rectangular coordinates $(t,x^1)$,
we define the vectorfields $\Lunit$ and $\uLunit$ as follows:

\begin{align}
\Lunit
& :=  \partial_t
\frac{\frac{\fourvelocity^1}{\fourvelocity^0} + \speed}{1 + \frac{\fourvelocity^1}{\fourvelocity^0} \speed} \partial_1,
\label{E:1DLUNIT} \\
\uLunit
& :=  \partial_t
\frac{\frac{\fourvelocity^1}{\fourvelocity^0} - \speed}{1 - \frac{\fourvelocity^1}{\fourvelocity^0} \speed} \partial_1.
\label{E:1DULUNIT}
\end{align}

Straightforward calculations
based on (<ref>)
and (<ref>)–(<ref>)
yield the following identities:

\begin{align} \label{E:GFOURINNERPRODUCTOFNULLVECTORFIELDSIN1D}
\hfour(\Lunit,\Lunit)
\hfour(\uLunit,\uLunit)
& = 0,
\hfour(\Lunit,\uLunit)
& = -2.
\end{align}

In particular, (<ref>) shows that $\Lunit$ and $\uLunit$ are $\hfour$-null.
The pair $\lbrace \Lunit, \uLunit \rbrace$ is known as a $\hfour$-null frame.

From (<ref>) and straightforward calculations,
we deduce the following lemma.

In $1 + 1$ spacetime dimensions,
the inverse acoustical metric $\hfour^{-1}$ from (<ref>) satisfies:

\begin{align} \label{E:1DACOUSTICALMETRICINVERSEINTERMSOFNULLVECTORFIELDS}
(\hfour^{-1})^{\alpha \beta}
& = -
\frac{1}{2} \Lunit^{\alpha} \uLunit^{\beta}
\frac{1}{2} \uLunit^{\alpha} \Lunit^{\beta},
(\alpha, \beta
& = 0,1),
\end{align}

i.e., $\hfour^{-1} = - \frac{1}{2} \Lunit \otimes \uLunit
\frac{1}{2} \uLunit \otimes \Lunit

§.§ Riemann invariants

In this section, we introduce the Riemann invariants, which are a pair of fluid variables, one of which is
constant along the integral curves of $\Lunit$ and the other along the integral curves of $\uLunit$. For
isentropic plane-symmetric solutions to the relativistic Euler equations,
the Riemann invariants can be used as state-space variables for the fluid, i.e., the other fluid variables are determined by them.
Riemann invariants were discovered by Riemann in his proof of shock formation for the non-relativistic
compressible Euler equations in $1D$.
In <cit.>, Taub discovered Riemann invariants for the isentropic relativistic Euler equations in $1D$.
Studying the flow with respect to the Riemann invariants allows one to study the $\hfour$-MGHD using
techniques based on transport equations and ODE theory. In particular, energy estimates are not needed in $1D$.
Although Riemann invariants are not available in multi-dimensions, approximate Riemann invariants are available, and they
are convenient for studying perturbations of isentropic plane-symmetric solutions; see Sect. <ref>.

To proceed, we let $\RIfunction = \RIfunction(\Enth)$ be the solution to
the initial value problem:

\begin{align} \label{E:ODEFORRIEMANNINVARIANT}
\frac{\mathrm{d}}{\mathrm{d} \Enth}
\RIfunction(\Enth)
& = \frac{1}{\Enth \speed},
\RIfunction(\overline{\Enth})
= 0,
\end{align}

where $\overline{\Enth} > 0$ is the constant
and in (<ref>),
we are viewing $\speed$ as a function of $\Enth$.

Note that since $\speed(\overline{\Enth}) \neq 0$ by assumption,
it follows from (<ref>) that the function $\RIfunction$
has a local inverse $\RIfunction^{-1}$ defined near the origin such that $\RIfunction^{-1}(0) = \overline{\Enth}$.

We define the Riemann invariants $\RRiemannPS$ and $\LRiemannPS$ as follows:

\begin{align} \label{E:RRIEMANNPS}
\RRiemannPS
& :=
\RIfunction(\Enth)
\frac{1}{2} \ln \left( \frac{1 + \frac{\fourvelocity^1}{\fourvelocity^0}}{1 - \frac{\fourvelocity^1}{\fourvelocity^0}} \right),
\\
\LRiemannPS
& :=
\RIfunction(\Enth)
\frac{1}{2} \ln \left( \frac{1 + \frac{\fourvelocity^1}{\fourvelocity^0}}{1 - \frac{\fourvelocity^1}{\fourvelocity^0}} \right).
\label{E:LRIEMANNPS}
\end{align}

In the next proposition, for isentropic plane-symmetric solutions, we express the relativistic Euler equations
in terms of $(\RRiemannPS,\LRiemannPS)$, and we provide explicit formulae
for various quantities in terms of $(\RRiemannPS,\LRiemannPS)$.

Let $(\Lnenth,\fourvelocity^0,\fourvelocity^1)$ be a $C^1$ solution to (<ref>)–(<ref>),
and let $\Lunit$ and $\uLunit$ be the vectorfields from Def. <ref>.
Then the Riemann invariants of Def. <ref> satisfy the
following quasilinear transport system:

\begin{align} \label{E:RRIEMANNPSTRANSPORT}
\Lunit \RRiemannPS
& = 0,
\\
\uLunit \LRiemannPS
& = 0.
\label{E:LRIEMANNPSTRANSPORT}
\end{align}

Moreover, the components of the fluid velocity
in the Minkowski-rectangular coordinate system $(t,x^1)$
can be expressed in terms of the Riemann invariants as follows:

\begin{align}
\fourvelocity^0  & =
\cosh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right), \qquad \fourvelocity^1  = \sinh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right).
\label{E:1DFLUIDVELOCITYINTERMSOFRIEMANNINVARIANTS}
\end{align}

In addition, with $\RIfunction^{-1}$ denoting the inverse function of the map $\RIfunction$ from Def. <ref>,
we have:

\begin{align} \label{E:1DENTHALPYINTERMSOFRIEMANNINVARIANTS}
\Enth
\RIfunction^{-1}\left( \frac{1}{2}\left(\RRiemannPS + \LRiemannPS\right)\right).
\end{align}

Finally, the following identities hold:

\begin{align}
\Lunit & = \p_t + \frac{\sinh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right) + \speed \cosh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right)}{\cosh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right) + \speed \sinh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right)} \p_1,
\label{E:1DLUNITINTERMSOFRIEMANNINVARIANTS}
\\
\uLunit
\p_t
\frac{\sinh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right)
\speed \cosh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right)}{
\cosh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right) - \speed \sinh\left(\frac{\RRiemannPS - \LRiemannPS}{2}\right)} \p_1.
\label{E:1DULUNITINTERMSOFRIEMANNINVARIANTS}
\end{align}

Equations (<ref>)–(<ref>) follow from straightforward but tedious calculations involving
Def. <ref>, equations (<ref>)–(<ref>),
the identity $\fourvelocity^0 \p_{\alpha} \fourvelocity^0 = \fourvelocity^1 \p_{\alpha} \fourvelocity^1$
(which follows from differentiating (<ref>) with $\p_{\alpha}$),
and the identity
$\p_{\alpha} \RIfunction(\Enth) = \frac{1}{c} \p_{\alpha} \Lnenth$
(which follows from (<ref>) and (<ref>)).

To prove the identities in (<ref>),
we first subtract
and use
(<ref>) to compute that
\left(
\RRiemannPS - \LRiemannPS
\right)
\ln(\fourvelocity^0 + \fourvelocity^1)$.
Rewriting (<ref>) as $(\fourvelocity^0 + \fourvelocity^1)(\fourvelocity^0-\fourvelocity^1) = 1$
and carrying out straightforward calculations based on the definitions of $\cosh$ and $\sinh$,
we conclude (<ref>).
(<ref>) follows from adding (<ref>) to (<ref>),
dividing by $2$,
and then applying $\RIfunction^{-1}$ to each side of the resulting identity.

Finally, (<ref>)–(<ref>)
follow from definitions (<ref>)–(<ref>)
and the already proven identity (<ref>).

[The fluid variables as functions of the Riemann invariants]
In view of (<ref>)–(<ref>),
throughout Sect. <ref>,
we often view all the fluid variables as smooth functions of the Riemann invariants
(or of only $\RRiemannPS$ when we are studying simple isentropic plane-symmetric solutions),
e.g., $\fourvelocity^0,\fourvelocity^1,\Lnenth,\Enth = \smoothfunction(\RRiemannPS,\LRiemannPS)$,
where throughout the paper, $\smoothfunction$ schematically denotes a smooth function that can vary from
line to line.
For example, we use
the inverse function theorem,
and the chain rule
to compute the following identity, which holds for isentropic plane-symmetric solutions:

\begin{align} \label{E:1DARBITRARYDERIVATIVEOFSPEEDOFSOUNDINTERMSOFRIEMANNINVARIANTS}
\p_{\alpha} \speed
& =
\frac{\speed \Enth}{2}
\speed'
\left\lbrace
\p_{\alpha} \RRiemannPS - \p_{\alpha} \LRiemannPS
\right\rbrace,
\end{align}

where on RHS (<ref>),
$\speed' := \frac{\mathrm{d} \speed}{\mathrm{d} \Enth}$.

§.§ Brief remarks on Riccati-type blowup and genuine nonlinearity

Consider a simple isentropic plane-symmetric solution $\RRiemannPS$ to (<ref>) (with $\LRiemannPS = 0$).
By differentiating the equation with respect to $\partial_1$, one obtains a Riccati-type equation
for $\partial_1 \RRiemannPS$ of the form
$\Lunit \partial_1 \RRiemannPS = - \frac{\mathrm{d} }{\mathrm{d} \RRiemannPS} \Lunit^1 \cdot (\partial_1 \RRiemannPS)^2$,
where here we are viewing $\Lunit^1$ as a function of $\RRiemannPS$, which is possible thanks to
Prop. <ref>
and Convention <ref>.
Hence, as long as the genuine nonlinearity condition
$\frac{\mathrm{d} }{\mathrm{d} \RRiemannPS} \Lunit^1 \neq 0$ holds,
then for solutions $\RRiemannPS$ launched by smooth compactly supported initial data,
one can easily show that $\partial_1 \RRiemannPS$ typically blows up along the integral curves of $\Lunit$,
much like in the case of Burgers' equation $\partial_t \Psi + \Psi \partial_1 \Psi = 0$;
we refer to <cit.>*Section 7.5 for a detailed discussion of the notion of genuine nonlinearity
in the context of $1D$ hyperbolic conservation laws.
One can check that the non-degeneracy assumption (<ref>) indeed implies that
\frac{\mathrm{d} }{\mathrm{d} \RRiemannPS} \Lunit^1|_{\RRiemannPS = 0}
\neq 0
i.e., that equation (<ref>) is genuinely nonlinear
near $(\RRiemannPS,\LRiemannPS) = (0,0)$.

Our study of the blowup in Theorem <ref> does not rely on differentiating
the equations with $\partial_1$ to obtain a Riccati-type PDE.
Instead, it relies on a much sharper approach tied to nonlinear geometric optics, which we set up in
Sect. <ref>.
In $3D$, such a sharpened approach seems essential, as
there is no known simple approach for studying gradient-blowup based only on
differentiating the equations with respect to a standard, solution-independent partial derivative operator.
In particular, in our discussion of shock formation in $3D$ in Sect. <ref>,
nonlinear geometric optics will play a fundamental role.

§.§ Simple isentropic plane symmetry and nonlinear geometric optics

Our study of the $\hfour$-MGHD in Theorem <ref>
intimately relies on nonlinear geometric optics, implemented via an acoustic eikonal function $\eik$,
and a careful study of the corresponding acoustic geometry. The main idea of the proof of
Theorem <ref>
is to show that the solution remains smooth relative to the “geometric coordinates” $(t,\eik)$ all the way up to the shock
and to recover the formation of the shock as a degeneracy between the geometric coordinates
and the Minkowski-rectangular coordinates $(t,x^1)$.

In this section, we construct the eikonal function and corresponding acoustic geometry for
the isentropic relativistic Euler equations in $1D$,
i.e., to the following initial value problem:

\begin{align} \label{E:1DCAUCHYIVPFORRIEMANNINVARIANTS}
& \begin{cases}
\Lunit \RRiemannPS = 0,
\\
\RRiemannPS|_{t=0} =  \dataRRiemannPS,
\end{cases} & &
\begin{cases}
\uLunit \LRiemannPS = 0,
\\
\LRiemannPS|_{t=0} =  \LRiemannPSdata.
\end{cases}
\end{align}

More precisely, we restrict our attention to simple isentropic plane-symmetric solutions,
defined here to be those solutions such that $\LRiemannPS \equiv 0$.
From (<ref>) and (<ref>),
it follows that such solutions arise from initial data with $\LRiemannPSdata \equiv 0$. In this case,
(<ref>) yields the identity
$\fourvelocity^0 = \cosh(\frac{1}{2} \RRiemannPS)$.
As we alluded to in Convention <ref>,
also yield similar formulas
for $\fourvelocity^1,\, \Enth$ as functions of $\RRiemannPS$.

We start by defining the acoustic eikonal function, which plays a central role in all of the forthcoming analysis.

Let $\hfour^{-1}$ be the inverse acoustical metric defined in (<ref>). Then we define the acoustic eikonal function (eikonal function for short) to be the solution $\eik$ to the following fully nonlinear hyperbolic PDE with Cauchy data:

\begin{align}
(\hfour^{-1})^{\alpha\beta} \partial_{\alpha} \eik \p_\beta \eik & = 0, \label{E:1DEIKONALNULLGRADIENTVANISHES} \\
\p_t \eik & > 0, \label{E:MINKPARTIALTOF1DEIKISPOSITIVE} \\
\eik|_{t=0} & = - x^1. \label{E:1DEIKONALDATAFORFULLYNONLINEARIVP}
\end{align}

We refer to the level sets of $\eik$ as “the characteristics.”

Although understanding the fully nonlinear structure of (<ref>) is necessary in multiple space dimensions, in $1D$,  (<ref>) is equivalent to the linear (in $\eik$) transport equation of $\eik$ derived in the following lemma. This makes the analysis of the acoustic geometry in this section considerably simpler than in multi-$D$. It is also unique to $1D$ that the $\hfour$-null vectorfield $\Lunit$ can be explicitly expressed in terms of the fluid via the formula (<ref>). The correct analog of (<ref>) in higher dimensions is precisely $\Lunit = \frac{1}{\Lgeo^0} \Lgeo$, where $\Lgeo^{\alpha} = - (\hfour^{-1})^{\alpha \kappa} \p_{\kappa} \eik$.

Let $\Lunit$ be the vectorfield defined in (<ref>).
For simple isentropic plane-symmetric solutions,
if $\RRiemannPS$ is sufficiently small,[The solutions from Theorem <ref>
satisfy the needed smallness. The key point is that if $\RRiemannPS$ is large, then
$\Lunit$ and $\uLunit$ could both be right-pointing or both be left-pointing, in which case the initial
condition (<ref>) would not be sufficient for identifying the “correct root of the eikonal equation,”
namely the “root” $\Lunit \eik = 0$.
the scalar function $\eik$ solves (<ref>)–(<ref>)
if and only if
it solves the following transport equation initial value problem:

\begin{align}
\Lunit \eik & = 0, \label{E:1DEIKONALTRANSPORTEQUATION} \\
\eik|_{t=0} & = -x^1. \label{E:1DEIKONALDATA}
\end{align}

Suppose $\eik$ solves (<ref>)–(<ref>).
Then by (<ref>),
we find that $0 = (\Lunit \eik) \cdot \uLunit \eik$.
If $\Lunit \eik = 0$, there is nothing to prove. Hence, we assume that $\Lunit \eik \neq 0$,
and we will show that this leads to a contradiction.
Then by (<ref>), we have $\uLunit \eik = 0$.
and our smallness assumption on $\RRiemannPS$,
we have:

\begin{align}\label{E:CONTRADICTIONSTEPIFULUNITEIKONALVANISHES}
\p_t \eik\big|_{t=0}
& = - \frac{\sinh(\tfrac{1}{2}\RRiemannPS)
\speed \cosh(\tfrac{1}{2} \RRiemannPS)}{\cosh(\tfrac{1}{2}\RRiemannPS)
\speed \sinh(\tfrac{1}{2}\RRiemannPS)} \p_1 \eik \big|_{t=0}
\frac{\sinh(\tfrac{1}{2}\RRiemannPS)
\speed \cosh(\tfrac{1}{2} \RRiemannPS)}{\cosh(\tfrac{1}{2}\RRiemannPS)
\speed \sinh(\tfrac{1}{2}\RRiemannPS)}
< 0.
\end{align}

(<ref>) contradicts (<ref>).

Conversely, if (<ref>)–(<ref>) hold,
then since (<ref>) implies that
$(\hfour^{-1})^{\alpha \beta} \partial_{\alpha} \eik \p_\beta \eik = - (\Lunit \eik) \cdot \uLunit \eik$,
it immediately follows that (<ref>) and (<ref>) hold.
To prove (<ref>),
we use
and the smallness of $\RRiemannPS$,
arguing as in the proof of (<ref>)
to deduce that $\partial_t \eik > 0$, as is desired.

Some remarks are in order.

* The minus sign in (<ref>) is just a convention imposed for consistency with related results that have
appeared in the literature, such as jSgHjLwW2016,jLjS2018,jLjS2021,lAjS2022.

* The precise initial condition in (<ref>) is not important; small perturbations of this initial condition
could also be used to study the $\hfour$-MGHD.

* We have adapted our eikonal function to $\Lunit$ because in
Theorem <ref>, we will study initial data that lead to shock-forming solutions
such that the singularity is tied to the infinite density of the $\Lunit$-characteristics and the blowup of
$\partial_1 \RRiemannPS$. There is nothing special about $\Lunit$ compared to $\uLunit$, and it is only for convenience that we have
adapted our constructions to it. That is,
using the methods of this section, one could also construct an eikonal function $\underline{\eik}$ adapted to $\uLunit$
and study different initial data that lead to a singularity
tied to the infinite density of the $\uLunit$-characteristics and the blowup of
$\partial_1 \LRiemannPS$. One could also try to study richer problems in which singularities occur
along both families of characteristics.

We now define $\upmu$, the inverse foliation density of the characteristics.
The name is justified by the fact that $\frac{1}{\upmu}$ is a measure of the density of the stacking of the
characteristics relative to the level-sets $\Sigma_{t'} := \lbrace t = t' \rbrace$.
Roughly speaking, $\upmu$ is initially positive, while shock formation corresponds to $\upmu \downarrow 0$,
which signifies the infinite density of the integral curves of $\Lunit$
(or equivalently, the level sets of $\eik$ in $(t,x^1)$-space);
we refer to Fig. <ref>, a context in which
$\upmu$ vanishes along the curve portion denoted by “$\Upsilon(\singularcurve_{[-\interestingu,0]})$”
(we will explain these issues in much more detail later on).

We define the inverse foliation density of the characteristics, denoted by $\upmu$, as follows:

\begin{align} \label{E:INVERSEFOLIATIONDENSITYINPLANESYMMETRY}
\upmu
\frac{1}{\frac{\speed}{\normalizer}
\partial_{1} \eik}.
\end{align}

In Sect. <ref>, which concerns solutions in $3D$,
we will define $\upmu$ to be $- \frac{1}{(\hfour^{-1})^{\kappa \lambda} \p_{\kappa} t \p_{\lambda} \eik}$;
see (<ref>).
Under the initial condition (<ref>) that we will choose for $\eik$,
that definition of $\upmu$ can be shown to agree with the one for plane-symmetric solutions
stated in (<ref>).

We also note the following identities, which follow from (<ref>)

\begin{align} \label{E:DATAIDENTITIESFORMU}
\upmu|_{t=0}
& = \frac{\normalizer}{\speed}\big|_{t=0}, \qquad \Longrightarrow \qquad \frac{\speed}{\normalizer}\upmu \big|_{t=0}  = 1.
\end{align}

We now define a collection of vectorfields associated to the eikonal function.

Let $\Lunit$ and $\uLunit$ be the vectorfields from Def. <ref>
(they are $\hfour$-null by (<ref>)).
We define:

\begin{align} \label{E:SEVERALACOUSTICVECTORFIELDS}
X &:= \frac{1}{2} (\uLunit - \Lunit), & \muX & := \upmu X, & \newuL &:= \upmu \uLunit.
\end{align}

and (<ref>),
it is straightforward to see that $X$ is $\Sigma_t$-tangent (i.e., $X t = 0$) and satisfies
$\hfour(X,X) = 1$, that $\hfour(\muX,\muX) = \upmu^2$, and that $\hfour(\newuL,\newuL) = 0$.
We also note that in plane-symmetry,
using Def. <ref>,
and Def. <ref>,
one can compute the following identity:

\begin{align} \label{E:EXPLICITFORMOFXIN1D}
& = -\frac{\speed}{\normalizer} \p_1.
\end{align}

Our main results concern solutions in which the factor $\frac{\speed}{\normalizer}$
on RHS (<ref>) satisfies $\frac{\speed}{\normalizer} \approx 1$.

We define $(t,\eik)$ to be the geometric coordinates,
and we denote the corresponding geometric coordinate partial derivative vectorfields
by $\left\lbrace \geop{t},\geop{\eik} \right\rbrace$
(which are not to be confused with the partial derivatives $\lbrace \partial_t, \partial_1 \rbrace$
in the $(t,x^1)$-coordinate system).

Using (<ref>), (<ref>), (<ref>),
and (<ref>), it is straightforward to compute that:

\begin{align} \label{E:1DGEOMETRICVECTORFIELDSAPPLIEDTOGEOMETRICCOORDIANTEFUNCTIONS}
\Lunit t
& = \muX  \eik = 1,
\\
\Lunit \eik
& = \muX t = 0.
\label{E:SECOND1DGEOMETRICVECTORFIELDSAPPLIEDTOGEOMETRICCOORDIANTEFUNCTIONS}
\end{align}

it follows that in geometric coordinates, the following identities hold:

\begin{align} \label{E:PLANESYMMETRYCOMMUTATORVECTORFIELDSAREGEOCOORDINATEVECTORFIELDS}
\Lunit
& = \geop{t},
\muX & = \geop{\eik},
\\
[\Lunit, \muX] &
= 0. \label{E:LANDMUXCOMMUTEINPLANESYMMETRY}
\end{align}

From (<ref>),
and the identities in (<ref>),
we deduce the following expression for $\newuL$ relative to the geometric coordinates:

\begin{align} \label{E:NEWULINGEOMETRICCOORDINATES}
\newuL
& = \upmu \geop{t} + 2 \geop{\eik}.
\end{align}

We will often silently use
in the rest of Sect. <ref>.

The following sets play a fundamental role in our analysis of solutions.

Given intervals $I,J \subset \mathbb{R}$,
relative to the geometric coordinates, we define:

\begin{align}
\nullhyparg{\eik'}^{I}
& := \big\{ (t,\eik) \in \R^2 \ | \ t \in I, \, \eik = \eik' \big\}, \label{E:CHARACTERISTICSURFACES1D} \\
\Sigma_{t'}^{J} & := \big\{ (t,\eik) \in \R^2 \ | \ t = t', \, \eik \in J \big\}. \label{E:TRUNCATEDSIGMAT}
\end{align}

We often write $\nullhyparg{\eik}$ instead of $\nullhyparg{\eik}^{[0,\infty)}$ and
$\Sigma_t$ instead of $\Sigma_t^{(-\infty,\infty)}$.
We often refer to the $\nullhyparg{\eik}$ as the characteristics.

The following lemma shows that the geometric coordinates degenerate with respect to the Minkowski-rectangular coordinates precisely when
$\upmu = 0$.
However, Theorem <ref> shows that for appropriate initial data,
$\Upsilon$ will still be a homeomorphism on the subset of the boundary of the $\hfour$-MGHD where $\upmu$ vanishes.

\begin{align} \label{E:1DCHOVGEOTORECT}
\Upsilon(t,\eik)
& := (t,x^1)
\end{align}

denote the change of variables map from geometric coordinates to Minkowski-rectangular coordinates.
Then the following identity holds:

\begin{align} \label{E:JACOBIANDETERMINANTOFCHANGEOFVARIABLESMAP1D}
\det \mathrm{d} \Upsilon
& = - \frac{\speed}{\normalizer} \upmu.
\end{align}

(<ref>) follows from straightforward calculations based
on the identity $\frac{\p}{\p \eik} x^1 = -\frac{\speed}{\normalizer} \upmu$,
which is implied by (<ref>)–(<ref>) and

§.§.§ Explicit solution formula in geometric coordinates

From (<ref>), we see
that in simple isentropic plane-symmetry,
the transport equation (<ref>)
takes the following form in geometric coordinates:

\begin{align} \label{E:RRIEMANNTRIVIALTRANSPORTEQUATIONINGEOMETRICCOORDINATES}
\geop{t} \RRiemannPS(t,\eik)
& = 0.
\end{align}

From (<ref>) and (<ref>),
we see that in geometric coordinates, the solution to (<ref>) is:

\begin{align} \label{E:SOLUIONRRIEMANNTRIVIALTRANSPORTEQUATIONINGEOMETRICCOORDINATES}
\RRiemannPS(t,\eik)
& = \dataRRiemannPS(\eik).
\end{align}

§.§.§ The evolution equation for $\upmu$, the non-degeneracy condition, and explicit expressions for various solution variables

In this section, we derive various identities involving $\upmu$. In particular,
we will later use the identities to show that suitable assumptions on the initial data of $\RRiemannPS$
lead to the vanishing of $\upmu$ in finite time, i.e., the formation of a shock.

We begin with the following lemma, which provides the evolution equation for $\upmu$.

For simple isentropic plane-symmetric solutions,
the inverse foliation density, defined in (<ref>),
satisfies the following transport equation,
\speed'(\Enth) := \frac{\mathrm{d} }{\mathrm{d} \Enth} \speed(\Enth)

\begin{align} \label{E:LMUPS}
\Lunit \left( \frac{\speed}{\normalizer} \upmu\right)
& = \PSLmusourcetermfunction,
\\
\PSLmusourcetermfunction
& :=
\frac{1 - \speed^2 + \frac{\speed'}{\speed \Enth}}{2(\fourvelocity^0 + \speed \fourvelocity^1)^2} \muX \RRiemannPS.
\label{E:PSLMUSOURCETERMFUNCTION}
\end{align}

\begin{align}\label{E:LLUPS}
\Lunit \Lunit \upmu
& =
\Lunit \Lunit \left( \frac{\speed}{\normalizer} \upmu\right)
= 0.
\end{align}

First, using (<ref>),
the quotient rule,
the identity $\cosh^2(z) - \sinh^2(z) = 1$,
and straightforward calculations, we deduce the following commutator identity, valid for simple isentropic plane-symmetric solutions:

\[ [\Lunit,\p_1] = \frac{\normalizer}{\speed} \left\lbrace \frac{1 - \speed^2 + \frac{\speed'}{\speed \Enth}}{2(\fourvelocity^0 + \speed \fourvelocity^1)^2} \right\rbrace X \RRiemannPS \p_1.\]

Equation (<ref>) then follows from multiplying (<ref>) by $\speed/\normalizer$, differentiating both sides with respect to $\Lunit$, and using the following commutator identity,
which is a consequence the eikonal equation $\Lunit \eik = 0$:

\[ \Lunit \left( \frac{\speed}{\normalizer} \upmu\right) = \frac{1}{(\p_1 \eik)^2} \Lunit \p_1 \eik = \left( \frac{\speed}{\normalizer} \upmu \right)^2 \Lunit \p_1 \eik =  \left( \frac{\speed}{\normalizer} \upmu \right)^2  [\Lunit,\p_1 ] \eik.\]

Equation (<ref>) then follows from differentiating (<ref>) with respect to $\Lunit$
and using (<ref>),
and the assumption of simple isentropic plane-symmetry.

We find it convenient to work with an anti-derivative of the nonlinearity $\PSLmusourcetermfunction$
defined in (<ref>).
To this end,
viewing the quantity
$- \frac{1 - \speed^2 + \frac{\speed'}{\speed \Enth}}{2(\fourvelocity^0 +  \speed \fourvelocity^1)^2}$ as a function of $\RRiemannPS$
(see Convention <ref>),
we define the function $\RRiemannPS \rightarrow \antiderivativePSLmusourcetermfunction[\RRiemannPS]$ as follows:

\begin{align} \label{E:ANTIDERIVATIVESOURCETERM}
\antiderivativePSLmusourcetermfunction[\RRiemannPS]
& :=
\int_0^{\RRiemannPS}
\left\lbrace
-\frac{1 - \speed^2 + \frac{\speed'}{\speed \Enth}}{2(\fourvelocity^0 +  \speed \fourvelocity^1)^2}
\right\rbrace(z)
\, \mathrm{d} z.
\end{align}

From (<ref>),
the fundamental theorem of calculus,
and the chain rule, it follows that:

\begin{align}
\geop{u} \antiderivativePSLmusourcetermfunction[\RRiemannPS]
\muX \antiderivativePSLmusourcetermfunction[\RRiemannPS]
\PSLmusourcetermfunction, \label{E:ANTIDERIVATIVEIDENTITYFORSOURCETERM}
\\
\antiderivativePSLmusourcetermfunction[0] & = 0. \label{E:NORMALIZATIONOFANTIDERIVATIVEFORLMUSOURCETERM}
\end{align}

We highlight the important fact that in simple isentropic plane-symmetry,
the non-degeneracy assumption (<ref>),
the initial condition of (<ref>),
and (<ref>)
collectively imply that:

\begin{align} \label{E:1DNONDEGENERACYONEOSCONSEQUENCE}
\frac{\mathrm{d} }{\mathrm{d} \RRiemannPS}
\antiderivativePSLmusourcetermfunction[\RRiemannPS]|_{\RRiemannPS = 0}
- \frac{1}{2}
\left\lbrace
\frac{\overline{\speed}'}{\overline{\speed} \overline{\Enth}}
\right\rbrace
\neq 0.
\end{align}

it follows that the function
$\RRiemannPS \mapsto \antiderivativePSLmusourcetermfunction [\RRiemannPS]$ is
a diffeomorphism from a neighborhood of the origin
(i.e., near $\RRiemannPS = 0$) onto a neighborhood of the origin;
we will silently use this basic fact in the rest of Sect. <ref>.
As we mentioned in Sect. <ref>,
(<ref>) is fundamental for ensuring the
formation of shocks, for it implies that near $(\RRiemannPS,\LRiemannPS) = (0,0)$,
equation (<ref>) is genuinely nonlinear.

Next, we note that in simple isentropic plane-symmetry,
in view of (<ref>),
the quantities $\PSLmusourcetermfunction$, $\speed$, and $\normalizer$ can be viewed as functions of $\eik$ alone.
We sometimes emphasize this fact by using notation such as $\PSLmusourcetermfunction(\eik)$, $\speed(\eik)$, etc.
The following corollary is a straightforward application of this fact and the above discussion;
we omit the straightforward proof.

Let $\PSLmusourcetermfunction$ be the source term from (<ref>), viewed as a function of $\eik$.
Let $\antiderivativePSLmusourcetermfunction = \antiderivativePSLmusourcetermfunction[\RRiemannPS]$
be the function of $\RRiemannPS$ defined by
Then for simple isentropic plane-symmetric solutions, the following identities hold
relative to the geometric coordinates,
\antiderivativePSLmusourcetermfunction'[z]
\frac{\mathrm{d} }{\mathrm{d} z}
\antiderivativePSLmusourcetermfunction[z]

\begin{align}
\Lunit
\left\lbrace
\frac{\speed}{\normalizer} \upmu(t,\eik)
\right\rbrace
& = \PSLmusourcetermfunction(\eik)
\frac{ \mathrm{d} }{ \mathrm{d}  \eik}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
= \antiderivativePSLmusourcetermfunction'[\dataRRiemannPS(\eik)]
\frac{\mathrm{d}}{\mathrm{d} \eik} \dataRRiemannPS(\eik),
\label{E:LSPEEDTIMESMUPLANESYMMETRYWITHEXPLICITSOURCE}
\\
\begin{split} \label{E:SPEEDTIMESMUPLANESYMMETRYWITHEXPLICITSOURCE}
\frac{\speed}{\normalizer} \upmu(t,\eik)
& =
t \PSLmusourcetermfunction(\eik)
= 1
\frac{ \mathrm{d} }{ \mathrm{d}  \eik}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
\\
= 1
\antiderivativePSLmusourcetermfunction'[\dataRRiemannPS(\eik)]
\frac{\mathrm{d}}{\mathrm{d} \eik}
\dataRRiemannPS(\eik),
\end{split}
\\
\begin{split} \label{E:MUXMUPLANESYMMETRYWITHEXPLICITSOURCE}
\muX \upmu(t,\eik)
& =
\left(
\frac{\normalizer}{\speed}
\PSLmusourcetermfunction'
\right)(\eik)
\left\lbrace
\frac{ \mathrm{d}}{ \mathrm{d}  \eik}
\left( \frac{\normalizer}{\speed}\right)(\eik)
\right\rbrace
\left\lbrace
t \PSLmusourcetermfunction(\eik)
\right\rbrace
\\
& =
\frac{\normalizer}{\speed} t
\frac{ \mathrm{d}^2 }{ \mathrm{d}  \eik^2}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
\left\lbrace
\frac{ \mathrm{d}}{ \mathrm{d}  \eik}
\left( \frac{\normalizer}{\speed}\right)(\eik)
\right\rbrace
\left\lbrace
\frac{\mathrm{d} }{\mathrm{d} \eik}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
\right\rbrace,
\end{split}
\\
\begin{split}  \label{E:CLOSEDPARTIAL1DERIVATIVEOFRPLUSPS}
[\partial_1 \RRiemannPS](t,\eik)
& = - \frac{1}{\frac{\speed}{\normalizer} \upmu (t,\eik)} \geop{\eik} \RRiemannPS(t,\eik)
- \frac{\frac{ \mathrm{d} }{ \mathrm{d}  \eik} \dataRRiemannPS(\eik)}{1
t \PSLmusourcetermfunction(\eik)}
\\
- \frac{\frac{ \mathrm{d} }{ \mathrm{d}  \eik} \dataRRiemannPS(\eik)}{
\antiderivativePSLmusourcetermfunction'[\dataRRiemannPS(\eik)]
\frac{ \mathrm{d} }{ \mathrm{d}  \eik}
\dataRRiemannPS(\eik)}.
\end{split}
\end{align}

§.§ Initial data that lead to admissible shock-forming solutions

In this section, we construct examples of initial data that fall under the scope of
Theorem <ref>,
i.e., initial data that launch shock-forming solutions
such that we can derive the structure of a portion of the maximal classical development that includes a neighborhood of its boundary.
Our construction could be substantially generalized to exhibit a much larger class of initial data for which the results of
Theorem <ref> hold; here we have striven for simplicity.

§.§.§ Assumptions on the “seed” profile

To start, we fix any scalar “seed profile” $\seed$ with the following properties
(it is straightforward to show that such functions exist):

* $\seed = \seed(\eik)$ is a[Our proof of Theorem <ref>
does not actually require $\seed$ to be $C^{\infty}$; $C^4$ would suffice. ]
$C^{\infty}$ function that is
compactly supported in
an interval $[-\rightu,\leftu]$ of $\eik$-values, where $\rightu, \leftu > 1$.

* $\frac{\mathrm{d} }{\mathrm{d} \eik} \seed(\eik)$
has a unique, non-degenerate minimum at $\eik=0$ (our choice $\eik = 0$ is only for convenience).
In particular, $\frac{\mathrm{d}^3 }{\mathrm{d} \eik^3} \seed(0) > 0$.

* Modifying $\seed$ by multiplying it by a constant
and composing it with a linear map of the form $\eik \rightarrow z \eik$
for some constant $z \in \mathbb{R}$ if necessary, we assume that
(for the modified $\seed$), there is a constant $\blowupdelta$ satisfying:

\begin{align} \label{E:1DBLOWUPDELTAISPOSITIVE}
\blowupdelta
& > 0
\end{align}

such that:

\begin{align} \label{AE:CONVENIENTRESCALINGOFSEED}
\frac{\mathrm{d} }{\mathrm{d} \eik} \seed(\eik)\bigg|_{\eik=0}
& = - \blowupdelta,
& &
\frac{\mathrm{d}^2 }{\mathrm{d} \eik^2} \seed(\eik)\bigg|_{\eik=0} = 0;
\end{align}

that there is a constant $\PSdatamuHessianTaylorcoefficient$ satisfying:

\begin{align}  \label{AE:PSKEYMUHESSIANTAYLORCOEFFICIENT}
\PSdatamuHessianTaylorcoefficient
& > 0
\end{align}

such that:

\begin{align} \label{E:1DROLEOFPSKEYMUHESSIANTAYLORCOEFFICIENT}
\frac{\mathrm{d}^3 }{\mathrm{d} \eik^3} \seed(0)
& = \PSdatamuHessianTaylorcoefficient,
\end{align}

and such that:

\begin{align} \label{E:1SEEDESTIMATETIEDTOPSKEYMUHESSIANTAYLORCOEFFICIENT}
\frac{1}{2} \PSdatamuHessianTaylorcoefficient
& \leq \frac{d^3}{\mathrm{d} \eik^3} \seed(\eik)
\leq
2 \PSdatamuHessianTaylorcoefficient,
\mbox{if }
|\eik| \leq 1;
\end{align}

and that there is a constant $\PSLmunottoonegativeparameter$ satisfying:

\begin{align} \label{AE:PSLOWERBOUNDONNULLDERIVATIVEOFMUINBORINGREGION}
\leq
\PSLmunottoonegativeparameter
< 1
\end{align}

such that:

\begin{align} \label{AE:PSMUNOTTOONEGATIVEINBORINGREGION}
\frac{\mathrm{d} }{\mathrm{d} \eik} \seed(\eik)
& > - \PSLmunottoonegativeparameter \blowupdelta,
& & \mbox{if } |\eik| \geq 1.
\end{align}

See Fig. <ref> for the graph
of a representative seed profile.

[scale=.6, grid = false, tics=5, trim=-.5cm -1cm -1cm -.5cm, clip]graph_of_seed.pdf
(9,15) $\eik=\leftu$
(26,15) $\eik=1$
(38,15) $\eik=0$
(51,15) $\eik=-1$
(82,15) $\eik = - \rightu$
(36,7) $\eik$
(65,32) $\seed(\eik)$

The graph of a representative “seed profile,” with $\eik$ increasing from right to left.

§.§.§ The initial data for $\RRiemannPS$ and amplitude smallness assumption

We now construct the initial data for $\RRiemannPS$.
Let $\seed$ be as in Sect. <ref>.
We define the initial data function $\dataRRiemannPS := \RRiemannPS|_{t=0}$ to be the following function of

\begin{align}
\dataRRiemannPS(\eik)
& := \antiderivativePSLmusourcetermfunction^{-1}[\seed(\eik)],
\label{AE:RIEMANNINVARIANTRESCALEDSEEDDATA}
\end{align}

where $\antiderivativePSLmusourcetermfunction^{-1}$ is the inverse function of the map
$\RRiemannPS \rightarrow \antiderivativePSLmusourcetermfunction[\RRiemannPS]$
defined by (<ref>).

Amplitude smallness assumption on the initial data

By (<ref>) and continuity,
we can further modify $\seed$ by multiplying it by a constant
to shrink the amplitude of $\dataRRiemannPS$ and ensure that
the following two conditions hold:

* The isentropic plane-symmetric initial data array
$\mathring{\solutionarray} := (\Lnenth,\fourvelocity^0,\fourvelocity^1,0,0,0)|_{t=0}$
determined by $\dataRRiemannPS$ via
(with $\LRiemannPS = 0$ in those formulas)
satisfies $\mathring{\solutionarray} \in \mbox{\upshape interior}(\mathcal{H})$,
where $\mathcal{H}$ is the regime of hyperbolicity defined in (<ref>).

* The following
non-degeneracy condition holds:

\begin{align} \label{E:ANOTHER1DNONDEGENERACYONEOSCONSEQUENCE}
\frac{\mathrm{d} }{\mathrm{d} \RRiemannPS}
\antiderivativePSLmusourcetermfunction[\RRiemannPS]\bigg|_{\RRiemannPS = \dataRRiemannPS(\eik)}
\neq 0,
& & \eik \in [-\rightu,\leftu].
\end{align}

In the rest of Sect. <ref>, we will use the above two assumptions without always explicitly mentioning them.

Taking into account (<ref>) and (<ref>),
we deduce from Taylor expansions and standard calculus
that the following conclusions
hold, where $C > 0$ depends on $\seed$,
and for functions $f = f(\eik)$ on $\Sigma_0 := \lbrace t = 0 \rbrace$,
$\| f \|_{C^0(\Sigma_0)} := \max_{\eik \in \mathbb{R}} |f(\eik)|$:

* $\dataRRiemannPS$ is compactly supported in $[-\rightu,\leftu]$
and satisfies the following bounds:

\begin{align} \label{AE:PSRESCALEDDATALINFINITYBOUND}
\left\| \frac{\mathrm{d}^M}{\mathrm{d} \eik^M} \dataRRiemannPS \right\|_{C^0(\Sigma_0)}
& \leq C,
M = 0,1,2,3,4.
\end{align}

* $\frac{\mathrm{d}}{\mathrm{d} \eik} \antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]$
has a unique, negative, non-degenerate minimum at $\eik=0$.
In particular,
$\frac{\mathrm{d}^3}{\mathrm{d} \eik^3} \antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)] |_{\eik=0} > 0$.

* There is a differentiable function $\PSthirdorderTaylorremaindercoefficientfunction: [-\rightu,\leftu] \rightarrow \mathbb{R}$
such that
\left\|
\PSthirdorderTaylorremaindercoefficientfunction
\right\|_{C^1(\Sigma_0^{[-\rightu,\leftu]})}
\leq C
and such that:

\begin{align} \label{E:TAYLOREXPANSIONOFKEYMUTERMABOUTCREASE}
\frac{\mathrm{d}}{\mathrm{d} \eik} \antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
& = -
\blowupdelta
\frac{1}{2}  \PSdatamuHessianTaylorcoefficient \eik^2
\PSthirdorderTaylorremaindercoefficientfunction(\eik) \eik^3.
\end{align}

* For $j=1,2,3$, there exists a continuous function
$\PSinitialkeymufunctioncoefficients_j : [-\rightu,\leftu] \rightarrow (0,\infty)$
such that:

\begin{align} \label{E:INITIALMUFUNCTIONSKEYEXPANSION}
\PSinitialkeymufunctioncoefficients_j(\eik)
& \approx 1,
&& \eik \in [-\rightu,\leftu],
\end{align}

and such that for $M=0,1,2$, we have:

\begin{align}
\frac{\mathrm{d}^M}{\mathrm{d} \eik^M}
\left(
\frac{\mathrm{d}}{\mathrm{d} \eik}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
\blowupdelta
\right)
& = \PSinitialkeymufunctioncoefficients_{M+1}(\eik)  \eik^{2-M},
\label{E:RESCALEDMUFUNCTIONANDDERIVATIVESEXPANSIONSNEAR0}
& \mbox{if } |\eik | \leq 1,
\\
\frac{\mathrm{d}}{\mathrm{d} \eik} \antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
& \geq - \PSLmunottoonegativeparameter \blowupdelta,
& \mbox{if } |\eik | \geq 1,
\label{E:RESCALEDFIRSTDERIVATIVEOFMUFUNCTIONUNIFORMLYPOSITIVEAWAYFROMINTERESTINGREGION}
\end{align}

where $\PSLmunottoonegativeparameter \in [0,1)$ is the non-negative constant in (<ref>),
and in (<ref>) and throughout,
$A \approx B$ means that there is a $C \geq 1$ such that $\frac{1}{C} B \leq A \leq C B$.

* The constant $\blowupdelta$ appearing in (<ref>) satisfies:

\begin{align}  \label{E:1DUSEFULCHARACTERIZATIONOFBLOWUPDELTA}
\blowupdelta
\max_{\eik \in [- \rightu,\leftu]}
\max_{\eik \in [- \rightu,\leftu]}
\left[\frac{\mathrm{d}}{\mathrm{d} \eik} \antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]\right]_-,
\end{align}

where $[z]_- := \max\lbrace -z, 0 \rbrace$ is the negative part of $z$.

§.§.§ The time of first blowup

Given any initial data function $\dataRRiemannPS(\eik)$ constructed above,
with $\blowupdelta > 0$ as in (<ref>),
we define:

\begin{align} \label{E:BLOWUPTIME}
\blowuptime
& := \frac{1}{\blowupdelta}.
\end{align}

Note that
(<ref>) imply that $\blowuptime$ is the Minkowskian time
at which $\upmu$ first vanishes, which by (<ref>) is
the positive time of first blowup for $|\partial_1 \RRiemannPS|$.

§.§.§ Sharp estimates for the inverse foliation density

In Lemma <ref>,
we provide various estimates for $\upmu$ and its derivatives
in a neighborhood of $[0,\blowuptime] \times \{0\} \subset \mathbb{R}_t \times \mathbb{R}_{\eik}$.
These estimates will be useful in the description of the boundary of the maximal development.

Let $\dataRRiemannPS = \dataRRiemannPS(\eik)$
be compactly supported initial data constructed in
and satisfying the assumptions stated in
Sect. <ref>.
Recall that for the corresponding simple isentropic plane-symmetric solution,
relative to the geometric coordinates $(t,\eik)$,
$\RRiemannPS$, $\speed$, and $\normalizer$ are functions of $\eik$.
With this in mind, we define:

\begin{align} \label{E:UPPERANDLOWERBOUNDSONSPEEDOVERNORMALIZER}
\minimumspeed
& := \min_{\eik \in \mathbb{R}} \frac{\speed(\eik)}{\normalizer(\eik)},
& \maximumspeed & := \max_{\eik \in \mathbb{R}} \frac{\speed(\eik)}{\normalizer(\eik)}.
\end{align}

Note that by (<ref>) and (<ref>), we have:

\begin{align} \label{E:EASYSPEEDOVERNORMALIZERBOUNDS}
0 & < \minimumspeed \leq \maximumspeed < 1.
\end{align}

Then there exists[With minor additional effort, one could explicitly compute such a constant $\interestingu$ in terms of the constants associated to $\seed$ such as $\blowupdelta, \, \PSdatamuHessianTaylorcoefficient$, etc.] a
constant $\interestingu$ depending only on the seed profile $\seed$
and satisfying $0 < \interestingu  < 1$
such that the following hold.

Behavior of $\upmu$ on $[-\interestingu,\interestingu]$.
Let $\blowuptime > 0$ be as defined in (<ref>).
Then the following estimates hold for $t \in [0,\blowuptime]$:

\begin{align} \label{AE:BOUNDSONLMUINTERESTINGREGION}
\minimumspeed^{-1}
\blowupdelta
\leq
\min_{\Sigma_t^{[-\interestingu,\interestingu]}} \Lunit \upmu
\leq
\max_{\Sigma_t^{[-\interestingu,\interestingu]}} \Lunit \upmu
\leq
- \maximumspeed^{-1} \blowupdelta.
\end{align}

Moreover, for
\in
\times [-\interestingu,\interestingu]
the following estimates hold, where here and throughout,
$A = \positivebigO(B)$ means that there exist constants $C_1, C_2$
(depending on $\seed$)[For example, the implicit constants present on RHS (<ref>) can be taken to be
$C_1 = \maximumspeed^{-1}$ and $C_2 = \minimumspeed^{-1}$; see (<ref>).] such that
$0 < C_1 \leq C_2$ and $C_1 B \leq A \leq C_2 B$,
$A = \mathcal{O}(B)$ means that there exists a $C > 0$ (depending on $\seed$)
such that $|A| \leq C |B|$:

\begin{align}
\upmu(t,\eik)
\positivebigO(1)
\frac{\PSdatamuHessianTaylorcoefficient}{2} \eik^2
\positivebigO(1) \blowupdelta
(\blowuptime - t),
\label{AE:PSMUTAYLOREXPANSIONININTERESTINGREGION}
\\
\Lunit \upmu(t,\eik)
= -
\positivebigO(1)
\blowupdelta,
\label{AE:PSLUNTMUTAYLOREXPANSIONININTERESTINGREGION}
\\
\muX \upmu(t,\eik)
\positivebigO(1)
\PSdatamuHessianTaylorcoefficient \eik + \mathcal{O}(1) (\blowuptime - t),
\label{AE:PSMUFIRSTDERIVATIVETAYLOREXPANSIONININTERESTINGREGION}
\\
\Lunit \muX \upmu(t,\eik)
\mathcal{O}(1),
\label{AE:PSLUNITMUXMUTAYLOREXPANSIONININTERESTINGREGION}
\\
\muX \muX \upmu(t,\eik)
& = \positivebigO(1)
\PSdatamuHessianTaylorcoefficient
\mathcal{O}(1)
(\blowuptime - t).
\label{AE:PSMUSECONDDERIVATIVETAYLOREXPANSIONININTERESTINGREGION}
\end{align}

In addition, we have:

\begin{align} \label{AE:PSLOCATIONOFXMUEQUALSMINUSKAPPA}
\lbrace \muX \upmu = 0 \rbrace
\cap
\Sigma_{\blowuptime}^{[-\interestingu,\interestingu]}
& = \lbrace (\blowuptime,0) \rbrace
\subset
\Sigma_{\blowuptime}^{[-\frac{1}{4}\interestingu,\frac{1}{4}\interestingu]},
\\
\min_{\Sigma_{\blowuptime}^{[-\interestingu,\interestingu]}
\backslash
\Sigma_{\blowuptime}^{[-\frac{1}{2} \interestingu,\frac{1}{2} \interestingu]}}
|\muX \upmu |
& \geq \frac{\PSdatamuHessianTaylorcoefficient \interestingu}{8},
\label{AE:PSREGIONWHEREMUXMUKAPPALEVELSETISNOTLOCATED}
\end{align}

\begin{align}  \label{AE:PSMUTRANSVERSALCONVEXITY}
\begin{split}
\frac{1}{2} \maximumspeed^{-1} \blowuptime \PSdatamuHessianTaylorcoefficient
& \leq
\min_{\Sigma_{\blowuptime}^{[-\interestingu,\interestingu]}}
\muX \muX \upmu,
\leq
\max_{\Sigma_{\blowuptime}^{[-\interestingu,\interestingu]}}
\muX \muX \upmu,
\leq
2 \minimumspeed^{-1} \blowuptime \PSdatamuHessianTaylorcoefficient.
\end{split}
\end{align}

$\muPS$ is uniformly positive away from the interesting region.
With $\PSLmunottoonegativeparameter \in [0,1)$ denoting the non-negative constant on
RHS (<ref>),
we have the following estimate:

\begin{align} \label{AE:PSMUISLARGEAWAYFROMINTERESTINGREGION}
\min_{\lbrace (t,\eik) \ | \ t \in [0,\blowuptime], \,  |u| \geq \interestingu \rbrace} \upmu(t,\eik)
\geq \positivebigO(1) (1 - \PSLmunottoonegativeparameter).
\end{align}

Properties of $\PSLmusourcetermfunction$ in the interesting region.
Let $\PSLmusourcetermfunction$ be the source term from (<ref>),
viewed as a function of $\eik$, and let $\PSLmusourcetermfunction' := \frac{\mathrm{d}}{\mathrm{d} \eik} \PSLmusourcetermfunction$.
Then for $\eik \in [-\interestingu,\interestingu]$, the following estimates hold:

\begin{align}  \label{E:ALTERNATERESCALEDMUFUNCTIONEXPANSIONSNEAR0}
\PSLmusourcetermfunction(\eik)
& =
\blowupdelta
\positivebigO(1) \eik^2
-\positivebigO(1) \blowupdelta,
\\
\frac{1}
& =
\frac{1}{\blowupdelta}
\positivebigO(1) \eik^2,
\label{E:ALTERNATERERECIPROCALSCALEDMUFUNCTIONEXPANSIONSNEAR0}
\\
\PSLmusourcetermfunction'(\eik)
& =
\positivebigO(1) \eik.
\label{E:ALTERNATERESCALEDMUFUNCTIONANDDERIVATIVESEXPANSIONSNEAR0}
\end{align}

In particular:

* $\PSLmusourcetermfunction(\eik) < 0$ for $\eik \in [-\interestingu,\interestingu]$

* $\PSLmusourcetermfunction'(0) = 0$

* $\PSLmusourcetermfunction'(\eik) > 0$ for $\eik \in [0,\interestingu]$

* $\PSLmusourcetermfunction'(\eik) < 0$ for $\eik \in [-\interestingu,0]$.

All of the results except for those concerning $\PSLmusourcetermfunction$
are straightforward consequences of Cor. <ref>,
the properties of the initial data $\dataRRiemannPS(\eik)$ from Sect. <ref>,
and Taylor expansions; we omit the details.

To prove the results for $\PSLmusourcetermfunction$,
we first use (<ref>)
and (<ref>)
to deduce that
\frac{\mathrm{d}}{\mathrm{d} \eik}
\antiderivativePSLmusourcetermfunction[\dataRRiemannPS(\eik)]
Using this identity and
(<ref>), we conclude
provided $\interestingu > 0$ is taken to be sufficiently small.

§.§ The crease, singular curve, and Cauchy horizon in $1D$ simple isentropic plane-symmetry

In this section,
we provide a description of a localized portion of the boundary of the $\hfour$-MGHD.
We emphasize the following key point, which follows from
and (<ref>):

For the initial data from Sect. <ref>,
the corresponding fluid solution $\RRiemannPS$ and the inverse foliation density
$\upmu$ are explicit smooth functions on all[In multi-dimensions, the best known estimates allow for the possibility of high order energy-blowup; see Sect. <ref>.
However, even in multi-dimensions, the solution remains smooth at the low-to-mid order derivative levels.]
of $\R^2$ in the $(t,\eik)$ differential structure, even though
the Minkowskian partial derivative $\partial_1 \RRiemannPS$ blows up in finite time.

Therefore, when we speak of the $\hfour$-MGHD, we are referring to the standard differential structure corresponding to the $(t,x^1)$ coordinates.
Nonetheless, we prefer to carry out most our analysis of the $\hfour$-MGHD in $(t,\eik)$ coordinates,
thanks to the explicit structure of the equations and solutions in geometric coordinates revealed in
Cor. <ref>. Hence, to reach conclusions about the behavior of solution in the $(t,x^1)$-differential structure, we have to control the change of variables map $\Upsilon(t,\eik) := (t,x^1)$; indeed,
in Theorem <ref>, we reveal various diffeomorphism and homeomorphism properties of $\Upsilon$.

For the solutions and localized region under study,
when described in geometric coordinates,
the boundary of the $\hfour$-MGHD consists of two pieces, depicted in Fig. <ref>:

* A singular curve portion
$\singularcurve_{[-\interestingu,0]}$, emanating from the first shock point – called the crease and denoted by $\crease$ –
such that $\upmu$ vanishes on $\singularcurve_{[-\interestingu,0]}$.

* A Cauchy horizon portion, denoted by $\cauchyhor_{[0,\interestingu]}$, which is a portion of
the integral curve of $\uLunit$ that
that emanates from $\crease$, and along which $\upmu$ is positive,
except at $\crease$.

In particular, the image $\Upsilon(\singularcurve_{[-\interestingu,0]})$,
depicted in Fig. <ref>,
is a boundary of the $\hfour$-MGHD in the differential structure of the Minkowski-rectangular coordinates $(t,x^1)$ in the sense that $|\partial_1 \RRiemannPS|$ blows up when $\upmu$ vanishes.
In contrast, even though $\RRiemannPS$ remains smooth all the way up to
$\Upsilon(\cauchyhor_{[0,\interestingu]} \backslash \crease)$,
$\Upsilon(\cauchyhor_{[0,\interestingu]})$ is a boundary of the $\hfour$-MGHD
because the crease $\Upsilon(\crease)$ – a point at which $|\partial_1 \RRiemannPS|$ blows up –
lies in the $\hfour$-causal past of any point in $\Upsilon(\cauchyhor_{[0,\interestingu]})$.
We highlight again that the singular behavior is only visible relative to the differential structure of Minkowski space in $(t,x^1)$ coordinates. The discrepancy between the behavior of the solution in the two coordinate systems
is tied to the breakdown in the diffeomorphism property of $\Upsilon$
when $\upmu$ vanishes; see (<ref>).

§.§.§ Singular curve, crease, Cauchy horizon, and developments

We now define some subsets of geometric coordinate space $\mathbb{R}_t \times \mathbb{R}_{\eik}$ that,
in Theorem <ref>,
will play an important role in describing the shape of the $\hfour$-MGHD.
Many of our definitions are localized to the subset $\lbrace |\eik| \leq \interestingu \rbrace$,
where $\interestingu > 0$ is the constant from Lemma <ref>.
This is mainly for convenience, since Lemma <ref> provides sharp
control over the solution when $|\eik| \leq \interestingu$.

For the solution launched by the compactly supported initial data $\dataRRiemannPS= \dataRRiemannPS(\eik)$
constructed in Sect. <ref>,
let $\PSLmusourcetermfunction$ be the corresponding source term from (<ref>),
viewed as a function of $\eik$.
We define:

\begin{align} \label{E:TISAFUNCTIONALONGSINGULARCURVE}
\timeisafunctionofeikonalinsingularcurve(\eik)
& : =
- \frac{1}{\PSLmusourcetermfunction(\eik)}.
\end{align}

Let $I \subset [-\interestingu,\interestingu]$ be a subset of $\eik$-values,
where $\interestingu > 0$ is the constant from Lemma <ref>.
We define the truncated singular curve to be the
following subset of geometric coordinate space
$\mathbb{R}_t \times \mathbb{R}_{\eik}$:

\begin{align} \label{E:TRUNCATEDPORTIONOFSINGULARCURVE}
\begin{split}
\singularcurve_I
:= \big\{ (t,\eik) \in [0,\infty) \times I \ | \ \upmu(t,\eik) = 0 \big\}
\cap
\big\{ (t,\eik) \in [0,\infty) \times I \ | \ \muX \upmu(t,\eik) \leq 0 \big\}
\\
& = \big\{
\left( \timeisafunctionofeikonalinsingularcurve(\eik),\eik \right)
\ | \
\eik \in I
\big\}
\cap
[0,\infty) \times \mathbb{R}
\cap
\big\{ (t,\eik) \in [0,\infty) \times I \ | \ \muX \upmu(t,\eik) \leq 0 \big\},
\end{split}
\end{align}

where $\muX$ is the vectorfield defined in (<ref>),
and the second identity in (<ref>) follows from

We define the crease, which we denote by $\crease$,
to be the following subset of geometric coordinate space $\mathbb{R}_t \times \mathbb{R}_{\eik}$
(Theorem <ref> implies that $\crease$ is a single point):

\begin{align} \label{E:1DCREASE}
\crease
:= \big\{ (t,\eik) \in [0,\infty) \times [-\interestingu,\interestingu] \ | \ \upmu(t,\eik) = 0 \big\}
\cap
\big\{ (t,\eik) \in [0,\infty) \times [-\interestingu,\interestingu] \ | \ \muX \upmu(t,\eik) = 0 \big\}.
\end{align}

and (<ref>),
it follows that if
$(t',\eik') \in \big\{ (t,\eik) \in  [0,\infty) \times [-\interestingu,\interestingu] \ | \ \upmu(t,\eik) = 0 \big\}$,
then $\muX \upmu(t',\eik') = \positivebigO(1) \eik'$ and thus $\muX \upmu(t',\eik') \leq 0 \iff \eik' \in [-\interestingu,0]$.
We therefore have the following alternate characterization of $\singularcurve_I$:

\begin{align} \label{E:ALTERNATETRUNCATEDPORTIONOFSINGULARCURVE}
\begin{split}
\singularcurve_I
& = \big\{
\left( \timeisafunctionofeikonalinsingularcurve(\eik),\eik \right)
\ | \
\eik \in I \cap [-\interestingu,0]
\big\}
\cap
[0,\infty) \times \mathbb{R}.
\end{split}
\end{align}

We make the following remarks.

* One might wonder why in the singular curve definition (<ref>),
there is a restriction to regions in which $\muX \upmu \leq 0$.
This is because the portion of $\lbrace \upmu = 0 \rbrace$ on which we formally have $\muX \upmu > 0$
never actually dynamically develops in the $\hfour$-MGHD;
that “fictitious portion” is cut off by the Cauchy horizon before it has a chance to dynamically form; see also
Remark <ref>.

* Our terminology “crease” is motivated by picture of the $\hfour$-MGHD in multi-dimensions, where the
analog of $\crease$ is a co-dimension $2$ hypersurface given by the intersection of the singular
boundary and the Cauchy horizon; see Fig. <ref>.

* From (<ref>),
and (<ref>),
we deduce the following key result: in regions where we can justify the use of geometric coordinates
for reaching conclusions about the behavior of the solution in $(t,x^1)$ space,
$\singularcurve_I$ is the subset of points $(t,\eik) \in [0,\infty) \times \left(I \cap [-\interestingu,0] \right)$
such that $|\p_1 \RRiemannPS(t,\eik)|$ blows up at $\Upsilon(t,\eik)$.
In Theorem <ref>, we will indeed justify the use of the geometric coordinates
when $I = [-\interestingu,0]$.

* (<ref>),
imply that $\blowuptime$ is the smallest value of $t$ along $\singularcurve_{[-\interestingu,0]}$.

* To further flesh out the previous point, we note that for the solutions
we study in Theorem <ref>,
the smallest value of $t$ on $\singularcurve_{[-\interestingu,0]}$ will occur precisely at $\eik = 0$, and
we will have $\timeisafunctionofeikonalinsingularcurve(0) = \blowuptime$.

Assume that $\crease = (\blowuptime,0)$, where $\blowuptime$ is defined by (<ref>)
(Theorem <ref> shows that this assumption is satisfied for the solutions under study here).
Let $\newuL$ be the vectorfield defined in (<ref>).
Let $\intcurvenewuL$ be the integral curve of $\frac{1}{2} \newuL$ in geometric coordinate space $\mathbb{R}_t \times \mathbb{R}_{\eik}$
parametrized by $\eik$ (note that $\frac{1}{2} \newuL \eik = 1$ by (<ref>)),
with the initial condition $\intcurvenewuL(0) = \crease$.
That is, $\intcurvenewuL$ solves the following initial value problem:

\begin{align}
\frac{\mathrm{d}}{\mathrm{d} \eik} \intcurvenewuL(\eik)
\frac{1}{2}
\newuL \circ \intcurvenewuL(\eik),
\label{E:INTEGRALCURVEOFNEWULODE}
\\
\intcurvenewuL(0) & = \left(\blowuptime,0 \right). \label{E:INTEGRALCURVEOFNEWULINITIALCONDITION}
\end{align}

We use the notation $\timeisafunctionofeikonalcauchyhorizon$ to denote the $t$-component of the curve $\intcurvenewuL$.
That is, by (<ref>), we have:

\begin{align}  \label{E:1DUPARAMETERIZEDINTEGRALCURVEOFMULUNIT}
\intcurvenewuL(\eik)
& = \left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right),
\end{align}

where $\timeisafunctionofeikonalcauchyhorizon$ solves the following initial value problem:

\begin{align} \label{E:ODEFORTCOMPONENTOFMUULUNITINTEGRALCURVE}
\frac{\mathrm{d}}{\mathrm{d} \eik} \timeisafunctionofeikonalcauchyhorizon(\eik)
& = \frac{1}{2} \upmu\left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right),
\timeisafunctionofeikonalcauchyhorizon(0)
& = \blowuptime.
\end{align}

Then for an interval $I \subset [0,\infty)$ of non-negative[In Theorem <ref>, the
Cauchy horizon will correspond to non-negative values of $\eik$.]
we define the truncated Cauchy horizon to be:

\begin{align} \label{E:CAUCHYHORIZON}
\cauchyhor_I
& := \intcurvenewuL(I).
\end{align}

We now define $\classicaldev$, which is a subset of $\mathbb{R}_t \times \mathbb{R}_{\eik}$
that is tailored to the shape of the $\hfour$-MGHD.

Let $\singularcurve_{[-\interestingu,0]}$ and $\cauchyhor_{[0,\interestingu]}$ denote the truncated singular curve and Cauchy horizon from Defs. <ref>–<ref>,
where $0 < \interestingu < 1$ is the constant from Lemma <ref>.
Let $\timeisafunctionofeikonalinsingularcurve$ and $\timeisafunctionofeikonalcauchyhorizon$
be the functions of $\eik$ from (<ref>) and (<ref>).
We define the following subsets of geometric coordinate space $\mathbb{R}_t \times \mathbb{R}_{\eik}$:

\begin{align}
\classicaldevsingular
\left\{
(t,\eik) \in \mathbb{R}^2
\ | \
0 \leq t  < \timeisafunctionofeikonalinsingularcurve(\eik), \, \eik \in [-\interestingu,0]
\right\},
\label{E:SINGULARPORTIONOFCLASSICALDEVBOUNDARYININTERESTINGREGION}
\\
\classicaldevregular
& :=
\left\{
(t,\eik) \in \mathbb{R}^2
\ | \  0 \leq t \leq \timeisafunctionofeikonalcauchyhorizon(\eik), \, \eik \in (0, \interestingu]
\right\},
\label{E:REGULARPORTIONOFCLASSICALDEVBOUNDARYININTERESTINGREGION} \\
\classicaldev & :=\classicaldevsingular \cup\classicaldevregular. \label{E:INTERESTINGREGION}
\end{align}

The set $\classicaldev$ defined by (<ref>)
is the region of geometric coordinate space
lying between $\Sigma_0^{[-\interestingu,\interestingu]}$ and $\cauchyhor_{(0,\interestingu]} \cup \singularcurve_{[-\interestingu,0]}$,
where the boundary portions
$\nullhyparg{\interestingu}^{\left[0,\timeisafunctionofeikonalcauchyhorizon(\interestingu)\right]}$ are included,
but $\singularcurve_{[-\interestingu,0]}$ is not; see Fig. <ref>.
Theorem <ref> shows that $\Upsilon(\classicaldev) \subset \mathbb{R}_t \times \mathbb{R}_{x^1}$
is a subset of Minkowski-rectangular coordinate space where we can justify
the use of $(t,\eik)$ coordinates to reach conclusions about the behavior of the solution in $(t,x^1)$ coordinates.[With some additional effort, given the results of Theorem <ref>, we could extend the region of spacetime on which $\Upsilon(t,\eik) = (t,x^1)$ is a homeomorphism to include the infinite region $\classicaldev^{\textnormal{Large}}
\classicaldev
\bigcup
\left(
[0,\blowuptime] \times (-\infty,-\rightu) \cup (\leftu,\infty)
\right)$.
We refrain from
giving the proof because it would require some technical, but uninteresting, adjustments of the arguments presented in the proof of the theorem. The details are uninteresting in the sense that $\upmu$ is uniformly positive on
$\classicaldev^{\textnormal{Large}} \setminus \classicaldev$
(see (<ref>))
and thus no singularity occurs in that region.]

§.§.§ The main theorem

We now prove Theorem <ref>, which is the main result of
Sect. <ref>. On the region $\classicaldev$, the theorem provides a sharp description of the solution in the differential structure of the geometric coordinates $(t,\eik)$ as well as the behavior of the solution on
$\Upsilon(\classicaldev)$ in differential structure of the Minkowski-rectangular coordinates $(t,x^1)$,
where we recall that $\Upsilon(t,\eik) = (t,x^1)$ is the change of variables map.
In particular, the theorem reveals various diffeomorphism and homeomorphism properties of $\Upsilon$ and exhibits the blowup
of $|\partial_1 \RRiemannPS|$ as $\Upsilon(\singularcurve_{[-\interestingu,0]})$ is approached by points in $\Upsilon(\classicaldev)$.
In Fig. <ref>, we have depicted the region in geometric coordinate space
$\mathbb{R} \times \mathbb{R}_{\eik}$.
In Fig. <ref>, we have depicted the
region in Minkowski-rectangular coordinate space $\mathbb{R} \times \mathbb{R}_{x^1}$, i.e., the image
under $\Upsilon$ of the region from geometric coordinate space.

[scale=.7, grid = false, tics=5, trim=-.5cm -1cm -1cm -.5cm, clip]RealAndFakeSingBoundBurgersGeo.pdf
(52,11) $(0,0)$
(26,11) $(0,\interestingu)$
(77,11) $(0,-\interestingu)$
(60,51) $\crease$
(62,48) 90$=$
(58,45) $(\blowuptime,0)$
(28,46.5) $\cauchyhor_{[0,\interestingu]}$
(18,3.5) $\classicaldevregular$
(69,5) $\classicaldevsingular$
(86,47.5) $\singularcurve_{[-\interestingu,0]}$
(92,43) 90$=$
(86,40.5) $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu \leq 0 \rbrace$
(97.5,37.5) $\cap$
(90,34.5) $\lbrace \eik \in [-\interestingu,0] \rbrace$
(33,53) $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$
(4,21) $\eik$
(8,25.5) $t$
(95,13) $\Sigma_0$
(92,26.5) $\nullhyparg{-\interestingu}^{\left[0,\timeisafunctionofeikonalinsingularcurve(-\interestingu)\right)}$
(6,35.5) $\nullhyparg{\interestingu}^{\left[0,\timeisafunctionofeikonalcauchyhorizon(\interestingu)\right]}$

$\classicaldev$ in geometric coordinate space, with $\eik$ increasing from right to left

The dotted curve in Fig. <ref>
labeled “$\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$”
is not part of the $\hfour$-MGHD or its boundary. Formally, $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$
is the portion of the level set $\lbrace \upmu = 0 \rbrace$
that never has a chance to dynamically emerge because it is cut off by the Cauchy horizon portion
$\cauchyhor_{(0,\interestingu]}$, which lies below it
(as the proof of Theorem <ref> shows).
For this reason, we do not display $\Upsilon\left(\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace \right)$
in Fig. <ref>.
One can use Taylor expansions to show that formally,
$\Upsilon\left(\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace \right)$ would
be a curve in $\Upsilon(\classicaldevsingular)$ that lies below $\Upsilon(\singularcurve_{[-\interestingu,0]})$
and thus formally, $\Upsilon$ would not even be injective if its domain extended
up to $\lbrace \upmu = 0 \rbrace \cap \lbrace \muX \upmu > 0 \rbrace$.

[scale=.7, grid = false, tics=5, trim=-.5cm -1cm -1cm -.5cm, clip]Schematic1DFluidCauchyHorizon.pdf
(87,20) $x^1$
(83,24.5) $t$
(53,9) $\mbox{Shock-forming}$
(50,5) $\mbox{characteristic family}$
(62,1) 90$=$
(49,-2) $\mbox{Integral curves of } \Lunit$
(9,39.5) $\Upsilon(\cauchyhor_{[0,\interestingu]})$
(17,35.5) 90$=$
(8,32.5) $\mbox{Cauchy horizon}$
(17,28.5) 90$=$
(7,25.5) $\mbox{An integral curve}$
(14,21.5) $\mbox{of } \newuL$
(36.5,56.5) $\Upsilon(\crease)$
(48,52) $\Upsilon(\singularcurve_{[-\interestingu,0]})$
(-8,4)  $\mbox{Characteristic family}$
(-8,0)  $\mbox{``blocked'' by } \Upsilon(\cauchyhor_{[0,\interestingu]})$
(94.5,14) $\Sigma_0$
(18,8.5) $\Upsilon(\classicaldevregular)$
(33,7.5) $\Upsilon(\classicaldevsingular)$

$\Upsilon(\classicaldev)$ in Minkowski-rectangular coordinate space

Let $\dataRRiemannPS$ be initial data satisfying the assumptions of Sect. <ref>,
and let $\RRiemannPS$ be the corresponding solution of the quasilinear initial value problem (<ref>)
(with $\LRiemannPSdata \equiv 0$).
Under the assumptions and conclusions of Lemma <ref>,
perhaps shrinking the constant $\interestingu > 0$ if necessary, the following results hold,
where the notation $\positivebigO(1)$ is defined in the statement of
Lemma <ref>.

Classical existence with respect to the geometric coordinates.
With respect to the geometric coordinates $(t,\eik)$, the following results hold.

* $\RRiemannPS$ is a smooth function of $\eik$ alone, i.e., $\RRiemannPS(t,\eik) = \dataRRiemannPS(\eik)$.

* $\RRiemannPS$, the null vectorfields $\Lunit$ and $\newuL$,
the vectorfield $\muX$,
and the inverse foliation density $\upmu$ exist classically on
$\mathbb{R}_t \times \mathbb{R}_{\eik}$.

* On $\{(t,\eik) \in \R^2 \ | \ \eik \notin [-\rightu,\leftu]\}$,
$\RRiemannPS$ vanishes,
$\Lunit^1 := \Lunit x^1 = \overline{c}$,
$\newuL^1 := \newuL x^1 = - \overline{c}$,
and $\upmu = 1/\overline{c}$,
where the constant $\overline{c} > 0$ denotes the speed of sound when the enthalpy is constantly $\overline{\Enth}$,
where $\overline{\Enth} > 0$ is the constant fixed in (<ref>).

* The following estimate holds:

\begin{align} \label{E:QUANTITATIVENEGATIVITYOFSPEEDTIMESLMU}
\Lunit \upmu(t,\eik)
& = - \positivebigO(1)\blowupdelta,
& \mbox{for } (t,\eik) \in \R^2 \times [-\interestingu,\interestingu].
\end{align}

In particular, with $\overline{\classicaldev}$ denoting the closure of
the region $\classicaldev$ defined in (<ref>),
since Def. <ref> implies that
$\overline{\classicaldev} \subset \R^2 \times [-\interestingu,\interestingu]$,
it follows that the estimate (<ref>) holds on

Description of the $\hfour$-MGHD in geometric coordinates.

* Let $\singularcurve_{[-\interestingu,0]}$ be the singular curve portion
from (<ref>),
and let $\classicaldev$ be the subset of geometric coordinate space defined in (<ref>).
Then the following disjoint union holds,
where $\overline{\classicaldev}$ is the closure of $\classicaldev$:

\begin{align} \label{E:CLOSUREOFCLASSICALDEVREGION}
\overline{\classicaldev}
& = \classicaldev \sqcup \singularcurve_{[-\interestingu,0]}.
\end{align}

* The crease, which is defined in (<ref>),
is a single point:

\begin{align} \label{E:MAINTHMCREASEISSINGLEPOINT1D}
\crease
& = \{ \left(\blowuptime, 0 \right) \}.
\end{align}

* The singular curve function $\timeisafunctionofeikonalinsingularcurve = \timeisafunctionofeikonalinsingularcurve(\eik)$
from (<ref>)
satisfies the following estimates:

\begin{align}  \label{E:TAYLOREXPANSIONSUDERIVATIVEOFSINGULARCURVE}
\frac{\mathrm{d} }{\mathrm{d} \eik} \timeisafunctionofeikonalinsingularcurve(\eik)
& =
\positivebigO(1) \eik,
& & \eik \in [-\interestingu,0],
\\
\timeisafunctionofeikonalinsingularcurve(\eik)
& =
\blowuptime
\positivebigO(1) \eik^2,
& & \eik \in [-\interestingu,0].
\label{E:TAYLOREXPANSIONSOFSINGULARCURVE}
\end{align}

* The rectangular spatial coordinate $x^1$
satisfies the following estimate on $\singularcurve_{[-\interestingu,0]}$:

\begin{align} \label{E:TAYLOREXPANSIONSOFDERIVATIVERECTANGULARX1ALONGSINGULARCURVE}
\frac{\mathrm{d} }{\mathrm{d} \eik} x^1\left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right)
\positivebigO(1) \eik,
& & \eik \in [-\interestingu,0].
\end{align}

* The integral curve initial value problem for $\intcurvenewuL(\eik)$ given by
has a unique solution on the interval $\eik \in [0,\interestingu]$.
Hence, the truncated Cauchy horizon $\cauchyhor_{[0,\interestingu]}$,
defined in (<ref>),
can be parameterized as follows:

\begin{align} \label{E:MAIN1DTHEOREMPARAMETERIZATIONOFTRUNCATEDCAUCHYHORIZON}
\cauchyhor_{[0,\interestingu]}
& = \{ \left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right) \ | \ \eik \in [0,\interestingu]\},
\end{align}

as in (<ref>).
Moreover, $\timeisafunctionofeikonalcauchyhorizon(\eik)$ satisfies the following estimates:

\begin{align} \label{E:TAYLOREXPANSIONUDERIVATIVEOFCAUCHYHORIZON}
\frac{\mathrm{d} }{\mathrm{d} \eik} \timeisafunctionofeikonalcauchyhorizon(\eik)
\positivebigO(1) \eik^2,
& & \eik \in [0,\interestingu],
\\
\timeisafunctionofeikonalcauchyhorizon(\eik)
\blowuptime
\positivebigO(1) \eik^3,
& & \eik \in [0,\interestingu].
\label{E:TAYLOREXPANSIONOFCAUCHYHORIZON}
\end{align}

* The inverse foliation density $\upmu$,
defined in (<ref>),
satisfies the following estimate along the Cauchy horizon portion $\cauchyhor_{[0,\interestingu]}$:

\begin{align} \label{E:TAYLOREXPANSIONOFMUALONGCAUCHYHORIZON}
\upmu\left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right)
\positivebigO(1) \eik^2,
& & \eik \in [0,\interestingu].
\end{align}

* The rectangular spatial coordinate $x^1$
satisfies the following estimate along the Cauchy horizon portion $\cauchyhor_{[0,\interestingu]}$:

\begin{align} \label{E:TAYLOREXPANSIONDERIVATIVEOFRECTANGULARX1ALONGCAUCHYHORIZON}
\frac{\mathrm{d} }{\mathrm{d} \eik} x^1\left(\timeisafunctionofeikonalcauchyhorizon(\eik),\eik \right)
\positivebigO(1) \eik^2,
& & \eik \in [0,\interestingu].
\end{align}

* $\upmu$ is positive on $\classicaldev$ but vanishes on $\singularcurve_{[-\interestingu,0]}$.
In particular, (<ref>) implies
that $\upmu$ is positive on $\cauchyhor_{(0,\interestingu]}$
but vanishes on the crease $\crease \subset \cauchyhor_{[0,\interestingu]}$, which corresponds to $\eik = 0$ in

Description of the singularity formation and the $\hfour$-MGHD in Minkowski-rectangular coordinates.

* Let $\Upsilon(t,\eik) := (t,x^1)$ be the change of variables map from geometric coordinates to Minkowski-rectangular coordinates.
Then $\Upsilon$ is a diffeomorphism from $\classicaldev$ onto its image. Moreover,
$\Upsilon$ is a homeomorphism from $\overline{\classicaldev}$ onto its image.

* The solution $\RRiemannPS$ exists classically relative to the Minkowski-rectangular coordinates $(t,x^1)$
on the subset $\Upsilon(\classicaldev)$ of $\mathbb{R}_t \times \mathbb{R}_{x^1}$.

* The following estimate holds on $\Upsilon\left(\classicaldev \right)$:

\begin{align}\label{E:BLOWUPOFCARTESIANCOORDINATES}
\left| [\p_1 \RRiemannPS] \circ \Upsilon^{-1}(t,x^1) \right|
& =
\frac{1}{\upmu}
\left|
\frac{\normalizer}{\speed}  \muX \RRiemannPS
\right| \circ \Upsilon^{-1}(t,x^1)
\frac{1}{\upmu} \positivebigO(1), & & (t,x^1) \in \Upsilon\left(\classicaldev \right).
\end{align}

In particular, since $\upmu$ vanishes along $\Upsilon(\singularcurve_{[0,\interestingu]})$,
given any point $(t_*,x_*^1) \in \Upsilon(\singularcurve_{[0,\interestingu]})$
and any sequence of points $\{ (t_m, x_m^1) \}_{m \in \mathbb{N}} \subset \Upsilon\left(\classicaldev \right)$
with $\lim_{m \to \infty} (t_m, x_m^1) = (t_*,x_*^1)$,
we have that $|\p_1 \RRiemannPS| \circ \Upsilon^{-1} \left((t_m, x_m^1) \right) \to \infty$ as $m \to \infty$.

* Recall that the vectorfield $\Lunit$ is defined relative to the $(t,x^1)$-coordinates in (<ref>)
(see also (<ref>)).
With respect to the $(t,x^1)$-coordinates,
the solution $\RRiemannPS$ and all of its $\Lunit$-derivatives of any order, i.e., $\Lunit^M \RRiemannPS$ for any $M$,
are continuous on $\Upsilon(\overline{\classicaldev})$,
including the singular curve portion $\Upsilon(\singularcurve_{[0,\interestingu]})$.
on $\Upsilon(\overline{\classicaldev})$,
relative to the $(t,x^1)$-coordinates,
the $1D$ acoustical metric $\hfour_{\alpha \beta}$ from (<ref>)
is a continuous, non-degenerate, $2 \times 2$ matrix of signature $(-,+)$.

The causal structure of the Cauchy horizon.

* $\cauchyhor_{[0,\interestingu]}$
is a $\hfour$-null curve portion, i.e., its tangent vector
is null with respect to $\hfour$.

The causal structure of the singular curve.

* Relative to the geometric coordinates $(t,\eik)$,
the vectorfield $\singcurvevectorfield$ defined by:

\begin{align} \label{E:SINGCURVEVECTORFIELD}
\singcurvevectorfield
& : = \Lunit -  \frac{\Lunit(\frac{\speed}{\normalizer} \upmu)}{\muX (\frac{\speed}{\normalizer} \upmu)} \muX
\end{align}

is well-defined on the singular curve portion $\singularcurve_{[-\interestingu,0)}$, and $\singcurvevectorfield$
is tangent to $\singularcurve_{[-\interestingu,0)}$.

* On $\singularcurve_{[-\interestingu,0)}$, we have
$\singcurvevectorfield t > 0$ and $\singcurvevectorfield \eik < 0$. In particular,
$\singcurvevectorfield$ is future-directed and transversal to the characteristics

* For every $q \in \singularcurve_{[-\interestingu,0)}$,
the pushforward vectorfield $[\mathrm{d} \Upsilon(q)] \cdot \singcurvevectorfield(q)$, which is tangent to the curve
$\Upsilon(\singularcurve_{[-\interestingu,0)}) \subset \mathbb{R}_t \times \mathbb{R}_{x^1}$,
is equal to $\Lunit|_{\Upsilon(q)} = [\Lunit^{\kappa} \partial_{\kappa}]_{\Upsilon(q)}$.

* For every $q \in \singularcurve_{[-\interestingu,0)}$, $\Lunit|_{\Upsilon(q)}$ is $\hfour$-orthogonal to the tangent space of

Since there exist integral curves of the $\hfour$-null vectorfield $\Lunit^{\kappa} \partial_{\kappa}$ that foliate and are $\hfour$-orthogonal to $\Upsilon(\singularcurve_{[-\interestingu,0)})$, it follows that $\Upsilon(\singularcurve_{[-\interestingu,0)})$ is a $\hfour$-null curve in the Minkowskian coordinate differential structure.

Before proving the theorem, we make some remarks.

* All the important results of the theorem are stable under perturbations of the initial data of
$\RRiemannPS$. That is, even though the theorem was stated only for initial data that satisfy
the assumptions of Sect. <ref>, the proof shows that
if one perturbs any of those data by an arbitrary (small) $C^4$ function, then the corresponding
perturbed simple isentropic plane-wave solution has an $\hfour$-MGHD/exhibits singularity formation
that is quantitatively and qualitatively close to the un-perturbed solution.

* In a similar vein as the previous point, with only modest additional effort, the results of
Theorem <ref> could be extended to
a large, open set of (non-simple) isentropic plane-symmetric solutions
in which $\LRiemannPS$ is sufficiently small compared to $\RRiemannPS$
(or, alternatively, in which $\RRiemannPS$ is sufficiently small compared to $\LRiemannPS$).

* The emergence of the Cauchy horizon from the crease is fundamentally tied to the
fact that in $1D$, the isentropic compressible Euler equations have two characteristic directions,
$\Lunit$ and $\uLunit$, as is evident from equations (<ref>)–(<ref>).
In particular, even within the class of simple isentropic plane-symmetric solutions
in which $\LRiemannPS$ (which is transported by $\uLunit$) vanishes,
when investigating the structure of the $\hfour$-MGHD,
one cannot “ignore” the $\uLunit$ direction.
This is in contrast to Burgers' equation, in which there is only one characteristic direction.
Relatedly, for Burgers' equation, there does not exist a notion of “globally hyperbolic”
that enjoys the same fundamental significance as the notion of “globally hyperbolic”
in the context of Lorentzian geometry, relativistic Euler flow, or wave equations.

* From the point of view of determinism, the Cauchy horizon is an ally in that it “blocks” the non-uniqueness
of classical solutions (in $\hfour$-globally hyperbolic regions).
In particular, in Fig. <ref>,
one can see that the Cauchy horizon prevents the characteristics in the left of the figure from entering into the region on the right,
where the fluid's gradient blows up. That is, the Cauchy horizon prevents crossing of the characteristics
$\Upsilon(\classicaldev)$, which in turn prevents multi-valuedness/non-uniqueness of classical solutions in this region.
This should be contrasted against Burgers' equation,
which exhibits non-uniqueness of classical solutions due to multi-valuedness stemming from the crossing of
characteristics emanating from widely separated regions in the initial data hypersurface.

* In Fig. <ref>,
along $\Upsilon(\singularcurve_{[-\interestingu,0]})$,
one sees the interesting phenomenon of
non-uniqueness of integral curves of $\Lunit$. In particular, $\Lunit$ is tangent to
and the characteristics, which are the straight lines in the figure
that intersect $\Upsilon(\singularcurve_{[-\interestingu,0]})$ tangentially.
The non-uniqueness is tied to the blowup of $|\partial_1 \Lunit^1|$
along $\Upsilon(\singularcurve_{[-\interestingu,0]})$, which in particular