# Supersonic flow past an airfoil

Yannis Angelopoulos  The Division of Physics, Mathematics and Astronomy,
Caltech, 1200 E California Blvd, Pasadena CA 91125, USA

###### Abstract

We consider a supersonic flow past an airfoil in the context of the steady,
isentropic and irrotational compressible Euler equations. We show that for
appropriate data, either a shock forms, in which case certain derivatives of
the solution blow up, or a sonic line forms, which translates to the fact that
the hyperbolic character of the equations degenerates.

##  1 Introduction

In this paper we make several remarks on the study of shock formation in the
case of supersonic flows around airfoils. This is a problem coming from
physics and engineering concerning the properties of supersonic flight.

Our setup is that of the 2-dimensional steady, isentropic and irrotational
Euler equations as an initial value problem, where the initial data are given
on a hypersurface, part of which models an airfoil. On the initial
hypersurface we assume that the Mach number is bigger than one, making our
equations hyperbolic. For data of appropriate size we show that either a shock
or a sonic line forms. We are also able to have a precise estimate for the
time where the shock forms and where the sonic line forms relative to the size
of the initial data. A rough version of the main result of the paper is the
following:

######  Theorem 1.1.

Consider a solution of the steady, isentropic and irrotational Euler equations
in two dimensions (denoted by variables $x$ and $y$) given in the form of a
potential flow $\phi$ emanating from data set on an initial smooth
hypersurface $\Sigma$ as in figure 1, where we assume that

$\sqrt{(\partial_{y}\phi)^{2}+(\partial_{x}\phi)^{2}}>q_{cr},$

a condition that makes the second order quasilinear equation satisfied by
$\phi$ initially hyperbolic.

Then in the upper half space relative to $\Sigma$ we construct the maximal
domain of existence and uniqueness of the potential flow $\phi$, and on the
boundary of that domain either a shock forms, i.e. as we approach the boundary
we have that $\partial^{2}\phi\rightarrow\infty$, or a sonic line forms, i.e.
as we approach the boundary we have that

$\sqrt{(\partial_{y}\phi)^{2}+(\partial_{x}\phi)^{2}}\rightarrow q_{cr}.$

Figure 1: The initial hypersurface $\Sigma$, and the dotted area in red where
the solution is constructed, and a shock and a sonic line form.

######  Remark 1.1.

Let us note that the boundary of the maximal domain of existence and
uniqueness of the solution constructed here can contain points where a shock
forms or points where a sonic line forms. The full maximal boundary is
expected to contain both, such a boundary is sometimes referred to in the
literature as a shock polar.

As far as we know, the problem of the two dimensional steady, isentropic and
irrotational flow has been studied extensively from a mathematical viewpoint,
but only in the subsonic and transonic cases. For the subsonic case (which
involves the study of a quasilinear elliptic equation) see the work of Bers
[6], [5], [7] and Shiffman [31], while for the transonic case (where the
equation changes character from hyperbolic to elliptic) we refer to the
important work of Morawetz [26], [27], [28] (see also the review article [30]
and the companion paper of the author [4])).

For the supersonic case, there have been numerical results and heuristical
arguments that point towards the formation of shocks and sonic lines, but no
rigorous mathematical proof. For a reference on numerical and heuristical
results one can consult the book of Courant and Friedrichs [15] (see also the
lecture notes of Morawetz [29]).

On the other hand, during the last 15 years, there has been significant
progress on the shock formation problem for hyperbolic equations. The main
reference is the monumental work of Christodoulou [10] where he constructs
shocks for three dimensional relativistic fluids. For the non-relativistic
case see [14]. Alinhac has also obtained weaker results on shock formation for
quasilinear wave equations, see [3], [2] (for even earlier results see the
pioneering work of John [17]). The methods of Christodoulou have been used to
show shock formation for quasilinear wave equations, see the book of Speck
[33]. Since then the same methods have been used to show shock formation in
several different scenarios, always for the time-dependent Euler equations,
see [35], [16], [22] to mention a few. Speck has also used Christodoulou’s
method to show degeneration of hyperbolicity for certain quasilinear wave
equations [34].

In this work we also use the methods of Christodoulou. A natural question is
in the case where a shoci forms, whether the solution can be continued beyond
the shock and reach the sonic line, hence constructing a shock polar. The
problem of shock development has been considered in the important works of
Christodoulou-Lisibach [13] and Christodoulou [12] (see also the work of Majda
[23], [24]), but the main issue in the present situation is the formation of
the sonic line. Note that we cannot expect to continue the solution past the
sonic line as such a problem is in general ill-posed (see for instance [25],
[20], [21], [4]).

##  2 The steady, isentropic and irrotational Euler equations in 2D

We want to study the 2-dimensional irrotational, isentropic and steady Euler
flow past a symmetric airfoil $\mathcal{P}$ (that will be defined later).

Consider first an isentropic and irrotational flow on
$\mathbb{R}\times\mathbb{R}^{2}$ for which we have the following sets of
equations:

$\begin{cases}\partial_{t}\rho+\partial_{x_{1}}(\rho
u_{1})+\partial_{x_{2}}(\rho u_{2})=0,\\\
\partial_{t}u_{i}+u_{j}\cdot\nabla_{x_{1},x_{2}}u_{i}-\frac{1}{\rho}\partial_{x_{i}}p=0\mbox{
for $i,j\in\\{1,2\\}$},\end{cases}$ (2.1)

where $\rho$ is the density, $(u_{1},u_{2})$ is the velocity vector, and
$p\doteq p(\rho)$ is the pressure. Let us denote by $q$ the speed, i.e.

$q=\sqrt{u_{1}^{2}+u_{2}^{2}},$

and by $c$ the speed of sound given by

$c^{2}=-\frac{\rho q}{\rho^{\prime}(q)},$

something that follows from Bernoulli’s law.

Now we make the assumption that the flow is steady, i.e. independent of time.
We note also that as $p$ depends only on $\rho$ it can be eliminated from
(2.1), and by the assumption that the flow is irrotational as we have that

$\partial_{x_{2}}u_{1}-\partial_{x_{1}}u_{2}=0,$

we can find a function $\phi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that

$u_{1}=\partial_{x_{1}}\phi,\quad u_{2}=\partial_{x_{2}}\phi.$

Setting $x_{1}=x$ and $x_{2}=y$, the potential function $\phi$ then satisfies
the following equation:

$\left(1-\frac{(\partial_{x}\phi)^{2}}{c^{2}}\right)\partial_{xx}^{2}\phi-\frac{2\partial_{x}\phi\partial_{y}\phi}{c^{2}}\partial_{xy}^{2}\phi+\left(1-\frac{(\partial_{y}\phi)^{2}}{c^{2}}\right)\partial_{yy}^{2}\phi=0.$
(2.2)

Letting

$M=\frac{q}{c}=\frac{\sqrt{(\partial_{x}\phi)^{2}+(\partial_{y}\phi)^{2}}}{c},$

we note that equation (2.2) is elliptic if $M<1$ and hyperbolic if $M>1$. We
call the flow subsonic if equation (2.2) is elliptic, and we call it
supersonic if equation (2.2) is hyperbolic.

######  Remark 2.1.

Another way to study the problem of the irrotational, isentropic and steady
flow in $1+2$ dimensions (or just two dimensions as time is eliminated) is to
use the equation

$\partial_{x}(\rho u_{1})+\partial_{y}(\rho u_{2})=0,$

in order to introduce a stream function $\psi$ such that

$\rho u_{1}=\partial_{y}\psi,\quad\rho u_{2}=-\partial_{x}\psi.$

For $\psi$ we can use then the so called hodograph transformation. Let

$\vartheta=\arctan\left(\frac{u_{2}}{u_{1}}\right),$

and let us consider the change of coordinates

$(x,y)\rightarrow(\vartheta,q).$

This is a dynamic change of variables, for which we note that

$d\phi=u_{1}dx+u_{2}dy,\quad\frac{1}{\rho}d\psi=-u_{2}dx+u_{1}dy,$

which implies that for $z=x+iy$ we have that

$d\phi+i\frac{1}{\rho}d\psi=qe^{-i\vartheta}dz\Rightarrow
dz=\frac{e^{i\vartheta}}{q}\left(d\phi+i\frac{1}{\rho}d\psi\right).$

We can now consider the partial derivatives of $z$ and we have that

$\frac{\partial
z}{\partial\vartheta}=\frac{e^{i\vartheta}}{q}\left(\frac{\partial\phi}{\partial
q}+i\frac{1}{\rho}\frac{\partial\psi}{\partial q}\right),\quad\frac{\partial
z}{\partial\vartheta}=\frac{e^{i\vartheta}}{q}\left(\frac{\partial\phi}{\partial\vartheta}+i\frac{1}{\rho}\frac{\partial\psi}{\partial\vartheta}\right).$

Since $\frac{\partial^{2}z}{\partial
q\partial\vartheta}=\frac{\partial^{2}z}{\partial\vartheta\partial q}$, we
have the following hodograph equations:

$\frac{\partial\phi}{\partial
q}=q\frac{\partial\psi}{\partial\vartheta}\frac{\partial}{\partial
q}\left(\frac{1}{q\rho}\right),\quad\frac{\partial\phi}{\partial\vartheta}=\frac{q}{\rho}\frac{\partial\psi}{\partial
q}.$

Using Bernoulli’s Theorem (that $\frac{dp}{dq}=-q\rho$) and that
$\frac{\partial^{2}\phi}{\partial
q\partial\vartheta}=\frac{\partial^{2}\phi}{\partial\vartheta\partial q}$ we
have the following second order equation for $\psi$:

$(1-M^{2})\frac{\partial^{2}\psi}{\partial\vartheta^{2}}+q^{2}\frac{\partial^{2}\psi}{\partial
q^{2}}+q(1+M^{2})\frac{\partial\psi}{\partial q}=0,$ (2.3)

for $M=\frac{q}{c}$ as before.

##  3 The initial boundary problem for airfoils and some geometric assumptions

We will study equation (2.2) with data on

$\Sigma\doteq\mathcal{P}\cup\\{(x,y)\quad|\quad y=0,|x|\geq 1\\},$

where $\mathcal{P}$ is an airfoil that is defined as the graph of a smooth
concave function $f$ such that

$f(x)>0\mbox{ for $|x|<1$, }f(1)=f(-1)=0.$

Note that we have that

$\mathcal{P}\doteq\\{(x,y)\quad|\quad|x|<1,y=f(x)\\}.$

We will examine solutions of (2.2) only in the domain

$\mathcal{D}\doteq\\{(x,y)\quad|\quad y\geq 0\mbox{ for $|x|\geq 1$, and
}y\geq f(x)\mbox{ for $|x|<1$}\\}.$

On $\Sigma$ we let

$\left.\phi\right|_{\Sigma}\in
C^{\infty}(\mathbb{R}),\quad\left.\frac{\partial\phi}{\partial
n}\right|_{\Sigma}=0.$

We also let

$\lim_{|x|\rightarrow\infty}q(x)=K\mbox{ such that
}\lim_{|x|\rightarrow\infty}M(x)>1.$

Such a problem can be seen as an initial-boundary value problem, where we
assign a supersonic speed at infinity.

As noted before, if in $\mathcal{D}$ we have that $M>1$ then equation (2.2) is
a quasilinear hyperbolic equation in $\mathcal{D}$ in $1+1$ dimensions. If the
condition $M>1$ is satisfied by the initial data then we can treat (2.2) as a
hyperbolic equation, but this might not be the case.

For a given density function we note that there exists a number $q_{cr}$ (the
critical speed) such that

$M>1\mbox{ if $q>q_{cr}$, and }M<1\mbox{ if $q<q_{cr}$,}$

i.e. equation (2.2) is hyperbolic for $q>q_{cr}$ and elliptic for $q<q_{cr}$.

##  4 The purely hyperbolic regime

###  4.1 The equations

Since we will work with data that will guarantee that $M>1$ in the domain
where we will show existence of the solution, equation (2.2) can then be
treated as a quasilinear hyperbolic equation of the form:

$\bar{g}^{\alpha\beta}(\partial\phi)\partial^{2}_{\alpha\beta}\phi=0\mbox{ for
$\alpha,\beta\in\\{x,y\\}$},$

where

$\bar{g}^{yy}=1-\frac{(\partial_{y}\phi)^{2}}{c^{2}},\quad\bar{g}^{xy}=-\frac{(\partial_{x}\phi)(\partial_{y}\phi)}{c^{2}},\quad\bar{g}^{xx}=1-\frac{(\partial_{x}\phi)^{2}}{c^{2}}.$

Rewriting equation (2.2) as

$(c^{2}-(\partial_{x}\phi)^{2})\partial_{xx}^{2}\phi-2(\partial_{x}\phi)(\partial_{y}\phi)\partial_{xy}^{2}\phi+(c^{2}-(\partial_{y}\phi)^{2})\partial_{yy}^{2}\phi=0,$

we multiply the equation above with $-\frac{1}{c^{2}-(\partial_{y}\phi)^{2}}$
(as suggested in [33]), and after differentiating it by $\partial_{y}$ and
$\partial_{x}$ for

$\Phi_{1}\doteq\partial_{y}\phi,\quad\Phi_{2}\doteq\partial_{x}\phi,$

we get the following system for $\Phi=(\Phi_{1},\Phi_{2})$:

$\begin{cases}\Box_{g(\Phi)}\Phi_{1}=\mathcal{N}_{1}(\Phi_{1},\Phi_{2}),\\\
\Box_{g(\Phi)}\Phi_{2}=\mathcal{N}_{2}(\Phi_{1},\Phi_{2}),\end{cases}$ (4.1)

where

$g^{-1}(\Phi)=\begin{bmatrix}g^{00}&g^{01}\\\
g^{10}&g^{11}\end{bmatrix}=\begin{bmatrix}-1&\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\\\
\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}&-\frac{c^{2}-\Phi_{2}^{2}}{c^{2}-\Phi_{1}^{2}}\end{bmatrix},$

which implies that

$g(\Phi)=\begin{bmatrix}g_{00}&g_{01}\\\
g_{10}&g_{11}\end{bmatrix}=\begin{bmatrix}-\frac{(c^{2}-\Phi_{1}^{2})(c^{2}-\Phi_{2}^{2})}{c^{2}(c^{2}-\Phi_{1}^{2}-\Phi_{2}^{2})}&\frac{(c^{2}-\Phi_{1}^{2})\Phi_{1}\cdot\Phi_{2}}{c^{2}(c^{2}-\Phi_{1}^{2}-\Phi_{2}^{2})}\\\
\frac{(c^{2}-\Phi_{1}^{2})\Phi_{1}\cdot\Phi_{2}}{c^{2}(c^{2}-\Phi_{1}^{2}-\Phi_{2}^{2})}&-\frac{(c^{2}-\Phi_{1}^{2})^{2}}{c^{2}(c^{2}-\Phi_{1}^{2}-\Phi_{2}^{2})}\end{bmatrix},$

and where $\mathcal{N}_{1}$, $\mathcal{N}_{2}$ are the following nonlinear
terms:

$\displaystyle\mathcal{N}_{1}(\Phi_{1},\Phi_{2})$
$\displaystyle=\frac{\partial}{\partial
x}\left(\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{y}\Phi_{1}-\frac{\partial}{\partial
y}\left(\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{x}\Phi_{1}$
$\displaystyle+\frac{\partial}{\partial
y}\left(\frac{c^{2}-\Phi_{2}^{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{x}\Phi_{2}-\frac{\partial}{\partial
x}\left(\frac{c^{2}-\Phi_{2}^{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{x}\Phi_{1}$
$\displaystyle+\frac{1}{\sqrt{-\det(g)}}[\partial_{\alpha}(\sqrt{-\det(g)})]\cdot
g^{\alpha\beta}\partial_{\beta}\Phi_{1},$
$\displaystyle\mathcal{N}_{2}(\Phi_{1},\Phi_{2})$
$\displaystyle=\frac{\partial}{\partial
y}\left(\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{x}\Phi_{2}-\frac{\partial}{\partial
x}\left(\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\right)\partial_{y}\Phi_{2}$
$\displaystyle+\frac{1}{\sqrt{-\det(g)}}[\partial_{\alpha}(\sqrt{-\det(g)})]\cdot
g^{\alpha\beta}\partial_{\beta}\Phi_{2}.$

###  4.2 Coordinate systems

Let us define the eikonal function $u$ as the solution of the equation:

$\begin{split}&g^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u=0,\quad\partial_{y}u>0,\\\
&\left.u\right|_{\Sigma}=1-x\mbox{ for $|x|>1$ and extended smoothly in the
remaining part of $\Sigma$.}\end{split}$ (4.2)

Define the vector field $L$ with components:

$L^{\alpha}=-\mu g^{\alpha\beta}\partial_{\beta}u,$ (4.3)

where

$\mu=-\frac{1}{g^{0\alpha}\partial_{\alpha}u},$ (4.4)

is the inverse foliation density, which is the key quantity controlling the
shock formation phenomenon.

It can be checked that $L$ is null, i.e.

$g(L,L)=0.$

Define also the vector field $\underline{L}$ as one satisfying

$g(L,\underline{L})=-2,$

and also the vector field

$X=\mu\frac{\underline{L}-L}{2},$

with components

$X^{\alpha}=-L^{\alpha}-g^{0\alpha}\mbox{ for $\alpha\in\\{0,1\\}$.}$

Note that

$X^{0}=0,\quad X^{1}=-L^{1}+\frac{\Phi_{1}\Phi_{2}}{c^{2}-\Phi_{1}^{2}}.$

We also define the auxiliary vector field

$\check{X}=\frac{\underline{L}-L}{2}=\mu^{-1}X.$

The Christoffel symbols can be expressed as

$\Gamma_{\alpha\gamma\beta}=\frac{1}{2}(\partial_{\alpha}g_{\beta\gamma}+\partial_{\beta}g_{\alpha\gamma}-\partial_{\gamma}g_{\alpha\beta})\doteq\frac{1}{2}(H_{\beta\gamma}^{\lambda}\partial_{\alpha}\Phi_{\lambda}+H_{\alpha\gamma}^{\lambda}\partial_{\beta}\Phi_{\lambda}-H_{\alpha\beta}^{\lambda}\partial_{\gamma}\Phi_{\lambda}),$

for $\alpha,\beta,\gamma\in\\{0,1\\}$, where

$\partial_{0}=\partial_{y},\quad\partial_{1}=\partial_{x},$

and where the functions $H_{\alpha\beta}^{\lambda}$,
$\alpha,\beta\in\\{0,1\\}$, $\lambda\\{1,2\\}$ are defined from the
aforementioned inequality. Finally, for $y=t$, and $u$ as defined before, we
consider the coordinate system $(t,u)$ that we will call dynamical coordinates
due to their dependence on the solution itself. For the change of variables
map

$\mathcal{J}(t,u)=(y,x),$

we have the following computation for the Jacobian:

$\dfrac{\partial\mathcal{J}}{\partial(t,u)}\doteq\dfrac{\partial(y,x)}{\partial(t,u)}=\begin{pmatrix}1&0\\\
L^{1}&X^{1}\end{pmatrix}.$ (4.5)

From this we are able to compute the derivatives $\partial_{y}$ and
$\partial_{x}$ in terms of $L$ and $\underline{\check{L}}$ as follows:

$\frac{\partial}{\partial y}=\frac{\partial}{\partial
t}-\frac{L^{1}}{X^{1}}\frac{\partial}{\partial
u},\quad\quad\frac{\partial}{\partial
x}=\frac{1}{X^{1}}\frac{\partial}{\partial u},$

and since

$L=\frac{\partial}{\partial t},\quad X=\frac{\partial}{\partial u},$

we finally have that

$\frac{\partial}{\partial y}=\left(1+\frac{\mu
L^{1}}{2X^{1}}\right)L-\frac{L^{1}}{2X^{1}}\underline{\check{L}},\quad\quad\frac{\partial}{\partial
x}=-\frac{\mu}{2X^{1}}L+\frac{1}{2X^{1}}\underline{\check{L}}.$ (4.6)

###  4.3 The equations in dynamical coordinates

We want to express the system of equations (4.1) with respect to the
$(L,\underline{\check{L}})$ vector fields, where

$\underline{\check{L}}\doteq\mu\underline{L},$

and we note that

$\frac{\partial}{\partial y}=L,\quad\frac{\partial}{\partial
x}=-\left(\frac{2L^{1}+\mu}{2X^{1}}\right)L+\frac{1}{2X^{1}}\underline{\check{L}}.$

For the nonlinearities we not that we have that:

$\mathcal{N}_{1}(\Phi_{1},\Phi_{2})=H^{\nu\alpha\beta}[(\partial_{\beta}\Phi_{\alpha+1})(\partial_{\nu}\Phi_{1})-(\partial_{\nu}\Phi_{\alpha+1})(\partial_{\beta}\Phi_{1})]-g^{\alpha\beta}\Omega^{\lambda}(\partial_{\alpha}\Phi_{\lambda})(\partial_{\beta}\Phi_{1}),$
$\mathcal{N}_{2}(\Phi_{1},\Phi_{2})=H^{\nu\alpha\beta}[(\partial_{\beta}\Phi_{\alpha+1})(\partial_{\nu}\Phi_{1})-(\partial_{\nu}\Phi_{\alpha+1})(\partial_{\beta}\Phi_{2})]-g^{\alpha\beta}\Omega^{\lambda}(\partial_{\alpha}\Phi_{\lambda})(\partial_{\beta}\Phi_{2}),$

where

$H^{\nu\alpha\beta}\doteq
g^{\alpha\alpha_{0}}g^{\beta\beta_{0}}H^{\nu}_{\alpha_{0}\beta_{0}},$
$\Omega^{\lambda}=\frac{1}{\sqrt{-\det(g)}}\frac{\partial(\sqrt{-\det(g)})}{\partial\Phi_{\lambda}},$

for $\nu,\alpha,\alpha_{0},\beta,\beta_{0}\in\\{0,1\\}$ and
$\lambda\in\\{1,2\\}$.

Let

$\omega_{1}=-\frac{1}{2}H^{\gamma}_{\alpha\beta}L^{\alpha}L^{\beta}L_{\gamma}\check{X}^{\lambda}(X\Phi_{\lambda}),$
(4.7)

and

$\omega_{2}=-\frac{1}{2}H^{\gamma}_{\alpha\beta}L^{\alpha}L^{\beta}L_{\gamma}\check{X}^{\lambda}(L\Phi_{\lambda})-\frac{1}{2}H^{\gamma}_{\alpha\beta}L^{\alpha}L^{\beta}\check{X}_{\gamma}\check{X}^{\lambda}(L\Phi_{\lambda}).$
(4.8)

According to Proposition 4.3.1 of [33], we have that

$\displaystyle\mu\Box_{g(\Phi)}\Phi_{\kappa}=-L\check{\underline{L}}\Phi_{\kappa},$
$\displaystyle\mu\Box_{g(\Phi)}\Phi_{\kappa}=-\check{\underline{L}}L\Phi_{\kappa}-(\omega_{1}+\mu\omega_{2})L\Phi_{\kappa},$

for $\kappa\in\\{1,2\\}$. Hence, we will work with the following system of
equations (derived from (4.1)):

$\begin{cases}L\check{\underline{L}}\Phi_{1}=\check{\mathcal{N}}_{1}(\Phi,L\Phi,\check{\underline{L}}\Phi),\\\
\check{\underline{L}}L\Phi_{1}+(\omega_{1}+\mu\omega_{2})L\Phi_{1}=\check{\mathcal{N}}_{1}(\Phi,L\Phi,\check{\underline{L}}\Phi),\\\
L\check{\underline{L}}\Phi_{2}=\check{\mathcal{N}}_{2}(\Phi,L\Phi,\check{\underline{L}}\Phi),\\\
\check{\underline{L}}L\Phi_{2}+(\omega_{1}+\mu\omega_{2})L\Phi_{2}=\check{\mathcal{N}}_{2}(\Phi,L\Phi,\check{\underline{L}}\Phi),\end{cases}$
(4.9)

where

$\check{\mathcal{N}}_{\kappa}(\Phi,L\Phi,\check{\underline{L}}\Phi)=-\mu\mathcal{N}_{\kappa}(\Phi_{1},\Phi_{2})\mbox{
for $\kappa\in\\{1,2\\}$,}$

and we note that both nonlinearities can be written only in terms of the $L$
and $\check{\underline{L}}$ of $\Phi_{1}$ and $\Phi_{2}$ by our previous
computations.

Moreover, according to Lemma 3.2.1 from [33] we have the following equation
for the inverse foliation density:

$L\mu=\omega_{1}+\mu\omega_{2}.$ (4.10)

The above equation involves not only the $H$ terms (that have been computed
previously) and $L^{0}$ and $L^{1}$. By definition $L^{0}=1$, while for
$L^{1}$ we have the following equation from Proposition 3.4.1 of [33]:

$LL^{1}=\frac{1}{2}H_{\alpha\beta}^{\lambda}L^{\alpha}L^{\beta}(L\Phi_{\lambda})\cdot\left(-L^{1}-\frac{\Phi_{1}\cdot\Phi_{2}}{c^{2}-\Phi_{1}^{2}}\right).$
(4.11)

##  5 The main theorem

In this section we state the main results that we are going to prove in this
paper, and we also describe the method of proof.

######  Definition 5.1.

Let $T^{*}_{U}$ be the classical lifespan of a solution of the system of
equations (4.1) with data corresponding to the ones described in section 3
which is the supremum of all $t>0$ where $\Phi$ remains in $C^{2}\times C^{2}$
in

$\mathcal{R}_{T,U}\doteq\\{(t,u)\quad|\quad t\in[0,T),u\in(-\infty,U]\\}.$

Our main result gives an estimate on the size of $T*_{U}$ in terms of the
sizes of $\Phi$ and $\check{\underline{L}}\Phi$ initially, and also gives an
estimate on the behaviour of $\mu$ in $\mathcal{R}_{T,U}$.

######  Theorem 5.1.

Let $\left(\left.(\Phi_{1},\frac{\partial\Phi_{1}}{\partial
n})\right|_{\Sigma},\left.(\Phi_{2},\frac{\partial\Phi_{2}}{\partial
n})\right|_{\Sigma}\right)$ be an initial data set for the system of equations
(4.9) such that

$\|\Phi_{1}\|_{L^{\infty}}\leq\epsilon,\quad\|L\Phi_{1}\|_{L^{\infty}}\leq\epsilon,\quad\|\underline{\check{L}}\Phi_{1}\|_{L^{\infty}}\leq\delta,$
(5.1)
$\|L\Phi_{2}\|_{L^{\infty}}\leq\epsilon,\quad\|\underline{\check{L}}\Phi_{2}\|_{L^{\infty}}\leq\delta,$
(5.2) $\sup_{\Sigma}|\Phi_{2}|=K,\quad\inf_{\Sigma}|\Phi_{2}|\geq K-\epsilon,$
(5.3) $\|\Phi_{1}\|_{L^{2}(\Sigma\cap\\{u\leq
U_{c}\\})}\leq\epsilon,\quad\|\Phi_{2}\|_{L^{2}(\Sigma\cap\\{u\leq
U_{c}\\})}\leq\epsilon,$ (5.4) $\|L\Phi_{1}\|_{L^{2}(\Sigma\cap\\{u\leq
U_{c}\\})}\leq\epsilon,\quad\|L\Phi_{2}\|_{L^{2}(\Sigma\cap\\{u\leq
U_{c}\\})}\leq\epsilon,$ (5.5)

for $\epsilon,\delta>0$, $\epsilon$ small enough with respect to $\delta$, for
some $U_{c}<\infty$, and $K$ and $\epsilon$ additionally satisfying the
following:

$\mbox{for }\left.q\right|_{\Sigma}\doteq q_{in}\mbox{ we have that
}q_{in}>q_{cr}.$

Then the classical lifespan time which is the maximal time (in $(t,u)$
coordinates) where the solution exists in
$\cup_{t^{\prime}\in[0,t)}\Sigma_{t^{\prime}}$ and is $C^{2}$, is given by

$T^{*}=\sup\left\\{t>0\quad|\quad\mu>0\mbox{ and }\frac{q}{c}>1\right\\},$

and as $\frac{q}{c}>1$ is equivalent to $q>q_{cr}$, $T^{*}$ is also given by

$T^{*}=\sup\left\\{t>0\quad|\quad\mu>0\mbox{ and }q>q_{cr}\right\\}.$

For

$\delta_{1}\doteq\sup_{\Sigma}[\omega_{1}]_{-}\mbox{ and
}\delta_{2}\doteq\sup_{\Sigma}[\Phi_{2}\cdot L\Phi_{2}]_{-},$

where $\omega_{1}$ was defined in (4.7) as

$\omega_{1}=-\frac{1}{2}H^{\gamma}_{\alpha\beta}L^{\alpha}L^{\beta}L_{\gamma}X^{\lambda}(\check{X}\Phi_{\lambda}),$

and where for a number $p$ we have that $[p]_{-}\doteq|p|$ if $p<0$ and
$[p]_{-}=0$ otherwise, we have that

$0<T^{*}\leq\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right),$

and in particular we have that

$\begin{split}&T^{*}=\left(1+O(\epsilon)\right)\cdot\frac{1}{\delta_{1}}\quad\mbox{
if $\frac{1}{\delta_{1}}<\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{4\delta_{2}}$,
or }\\\
&T^{*}=\left(1+O(\epsilon^{2})\right)\cdot\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{4\delta_{2}}\quad\mbox{
if
$\frac{1}{\delta_{1}}>\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{4\delta_{2}}$}.\end{split}$
(5.6)

For $(t,u)\in[0,T^{*})$ we have the following estimates:

$\|L\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|L\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\|L\Phi_{1}\|_{L^{\infty}(\Sigma)}+\|L\Phi_{2}\|_{L^{\infty}(\Sigma)}+O(\epsilon),$
(5.7)
$\|\underline{\check{L}}\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|\underline{\check{L}}\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\|\underline{\check{L}}\Phi_{1}\|_{L^{\infty}(\Sigma)}+\|\underline{\check{L}}\Phi_{2}\|_{L^{\infty}(\Sigma)}+O(\epsilon),$
(5.8)
$\mu(t,u)=c+CK(\underline{\check{L}}\Phi_{1}+\underline{\check{L}}\Phi_{2})t+O(\epsilon),$
(5.9)

for some constants $c$, $C.$. Note that the above estimates imply that

$\|\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}\leq\bar{C}\epsilon,$
$\|L\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|L\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\bar{C}\epsilon,$
$\|\underline{\check{L}}\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|\underline{\check{L}}\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\bar{C}\delta,$

and

$\|\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\bar{C}K,$

for some constant $\bar{C}$.

In particular, $\underline{L}\Phi$ blows up along

$\\{(T^{*},u)\quad|\quad\mu(T^{*},u)=0\\}\subseteq\Sigma_{T^{*}}\mbox{ if
$\frac{1}{\delta_{1}}<\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{4\delta_{2}}$,}$

while a sonic line forms for

$\left\\{(T^{*},u)\quad|\quad\frac{q(T^{*},u)}{c(T^{*},u)}=1\right\\}\subseteq\Sigma_{T^{*}}\mbox{
or }\left\\{(T^{*},u)\quad|\quad
q(T^{*},u)=q_{cr}\right\\}\subseteq\Sigma_{T^{*}}\mbox{ if
$\frac{1}{\delta_{1}}>\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{4\delta_{2}}$.}$

Finally for all $t$ such that $0\leq t<T^{*}$ we have that

$\lim_{u\rightarrow-\infty}\left.\Phi_{1}(x)\right|_{\Sigma_{t}}=0,\quad\lim_{u\rightarrow-\infty}\left.\Phi_{2}(x)\right|_{\Sigma_{t}}=K.$

It is worth noticing that the boundary of the spacetime region of existence of
the solution can consist not only of points where a shock forms, i.e. where
$\mu=0$, but also of points where a sonic line forms, i.e. where
$\frac{q}{c}=1$. Moreover, the last part of the Theorem essentially states
that the Mach number at infinity is fixed throughout the region of existence.

Note that for data

$\left.\Phi_{1}\right|_{\Sigma}=0,\quad\left.\frac{\partial\Phi_{1}}{\partial
n}\right|_{\Sigma}\sim\epsilon,\quad\left.\Phi_{2}\right|_{\Sigma}\sim
q_{cr},\quad
L\Phi_{2}\sim\epsilon,\quad\underline{\check{L}}\Phi_{2}\sim\delta,$

we expect that in the region where the solution exists we have that

$c^{2}\sim q_{cr}^{2},\quad c^{2}-\Phi_{1}^{2}\sim q_{cr}^{2},\quad
c^{2}-\Phi_{1}^{2}-\Phi_{2}^{2}\sim-\epsilon^{2}.$

Note then that for the nonlinearities $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$
– taking into consideration the size of $K$ and $\epsilon$ – we have
schematically that

$\mathcal{N}_{1}\sim\mathcal{N}_{2}\sim\sum_{i,j,k\in\\{1,2\\}}(1+\Phi_{1}\Phi_{2})\left[\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{L}\Phi_{k})\right],$

and as $\epsilon$ is taken to be small, we essentially have – once again
schematically – that

$\mathcal{N}_{1}\sim\mathcal{N}_{2}\sim\sum_{i,j,k\in\\{1,2\\}}\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{L}\Phi_{k}).$

Writing the system of equation (4.1) using the vector fields
$(L,\underline{\check{L}})$ schematically we obtain the following system

$\begin{cases}L\underline{\check{L}}\Phi_{1}\sim\sum_{i,j,k\in\\{1,2\\}}\left[\mu\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{\check{L}}\Phi_{k})\right],\\\
\underline{\check{L}}L\Phi_{1}\sim\sum_{i,j,k\in\\{1,2\\}}\left[\mu\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{\check{L}}\Phi_{k})\right],\\\
L\underline{\check{L}}\Phi_{2}\sim\sum_{i,j,k\in\\{1,2\\}}\left[\mu\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{\check{L}}\Phi_{k})\right],\\\
\underline{\check{L}}L\Phi_{2}\sim\sum_{i,j,k\in\\{1,2\\}}\left[\mu\Phi_{i}(L\Phi_{j})(L\Phi_{k})+\Phi_{i}(L\Phi_{j})(\underline{\check{L}}\Phi_{k})\right].\end{cases}$
(5.10)

Note that the inverse foliation density $\mu$ satisfies a schematic equation
of the following form (see Lemma A.2.1 of [33]):

$L\mu\sim\sum_{i,j\in\\{1,2\\}}\Phi_{i}(\mu
L\Phi_{j}+\underline{\check{L}}\Phi_{j}).$ (5.11)

We will argue through a bootstrap argument. We will assume that:

$\|\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}\leq
c_{1}\epsilon,\quad\|L\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}\leq
c_{2}\epsilon,\quad\|L\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq c_{3}\epsilon,$

for some $c_{1},c_{2},c_{3}>0$,

$\|\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq
d_{1}K,\quad\|\underline{\check{L}}\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}\leq
d_{2}\delta,\quad\|\underline{\check{L}}\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq
d_{3}\delta,$

for some $d_{1},d_{2},d_{3}>0$, and

$\begin{split}&\|\Phi_{1}\|_{L^{2}(\Sigma\cap\\{u\leq U_{c}\\})}\leq
e_{1}\epsilon,\quad\|\Phi_{2}-K\|_{L^{2}(\Sigma\cap\\{u\leq U_{c}\\})}\leq
e_{2}\epsilon,\\\ &\|L\Phi_{1}\|_{L^{2}(\Sigma\cap\\{u\leq U_{c}\\})}\leq
e_{3}\epsilon,\quad\|L\Phi_{2}\|_{L^{2}(\Sigma\cap\\{u\leq U_{c}\\})}\leq
e_{4}\epsilon,\end{split}$

for some $e_{1},e_{2},e_{3},e_{4}>0$. The goal is to improve the $c_{i}$’s,
the $d_{i}$’s, and the $e_{i}$’s.

We start by constructing the solution close to spacelike infinity. Note that
an outcome of the estimates we will establish is that the Mach number at
infinity remains fixed (which can be seen as a boundary condition). We will
use a basic energy estimate. We will show the following estimate:

$\displaystyle\int_{\Sigma_{t}}$
$\displaystyle\Big{[}(1+\mu)\mu(L\Phi_{1})^{2}+\mu(L\Phi_{1})(\check{X}\Phi_{1})+(\check{X}\Phi_{1})^{2}\Big{]}\,du$
$\displaystyle+\int_{\Sigma_{t}}\Big{[}(1+\mu)\mu(L(\Phi_{2}-K))^{2}+\mu(L(\Phi_{2}-K))(\check{X}(\Phi_{2}-K))+(\check{X}(\Phi_{2}-K))^{2}\Big{]}\,du$
$\displaystyle\leq$
$\displaystyle\int_{\Sigma_{0}}\Big{[}(1+\mu)\mu(L\Phi_{1})^{2}+\mu(L\Phi_{1})(\check{X}\Phi_{1})+(\check{X}\Phi_{1})^{2}\Big{]}\,du$
$\displaystyle+\int_{\Sigma_{0}}\Big{[}(1+\mu)\mu(L(\Phi_{2}-K))^{2}+\mu(L(\Phi_{2}-K))(\check{X}(\Phi_{2}-K))+(\check{X}(\Phi_{2}-K))^{2}\Big{]}\,du$
$\displaystyle+O(\epsilon^{3}),$

using the last set of the aforementioned bootstrap assumptions, which will
allows us to conclude that

$\displaystyle\int_{\Sigma_{t}}$
$\displaystyle\Big{[}(1+\mu)\mu(L\Phi_{1})^{2}+\mu(L\Phi_{1})(\check{X}\Phi_{1})+(\check{X}\Phi_{1})^{2}\Big{]}\,du$
$\displaystyle+\int_{\Sigma_{t}}\Big{[}(1+\mu)\mu(L(\Phi_{2}-K))^{2}+\mu(L(\Phi_{2}-K))(\check{X}(\Phi_{2}-K))+(\check{X}(\Phi_{2}-K))^{2}\Big{]}\,du<\infty,$

for all $t\in[0,T^{*})$. The result will follow by the coercive estimate

$\displaystyle\int_{\Sigma_{t}}\Big{[}$
$\displaystyle\Phi_{1}^{2}+(\Phi_{2}-K)^{2}\Big{]}\,du\leq\int_{\Sigma_{t}}\Big{[}(1+\mu)\mu(L\Phi_{1})^{2}+\mu(L\Phi_{1})(\check{X}\Phi_{1})+(\check{X}\Phi_{1})^{2}\Big{]}\,du$
$\displaystyle+\int_{\Sigma_{t}}\Big{[}(1+\mu)\mu(L(\Phi_{2}-K))^{2}+\mu(L(\Phi_{2}-K))(\check{X}(\Phi_{2}-K))+(\check{X}(\Phi_{2}-K))^{2}\Big{]}\,du,$

again for $t\in[0,T^{*})$. We note that we need to work with a slightly
different system compared to (4.1), one satisfied by $\Phi_{1}$ and
$\Phi_{2}-K$.

Next we will construct the solution in the interior. We will use equations
(5.10) and show first the estimates (5.7), (5.8), and (5.9). We start with
$L\Phi$, and use the second and fourth equations of (5.10) and we have that
for $k\in\\{1,2\\}$:

$|\underline{\check{L}}L\Phi_{k}|(t,u)\leq\sum_{i=1}^{2}K\delta|L\Phi_{i}|(t,u),$

and considering $u$ in a bounded (potentially small) interval, we can apply a
Grönwall estimate using the previous inequality (integrating along the
integral curves of $\underline{\check{L}}$), and we have that for
$k\in\\{1,2\\}$:

$|L\Phi_{k}|(t,u)\leq|L\Phi_{k}|(t,u_{0})+C_{1}K\delta\epsilon+O(\epsilon^{2}).$

Integrating then along the integral curves of $L$ we have that

$|\Phi_{1}|(t)\leq|\Phi_{1}|(0)+C_{2}K\delta\epsilon+O(\epsilon^{2})\leq
K+C_{2}K\delta\epsilon+O(\epsilon^{2}),$
$|\Phi_{2}|(t)\leq|\Phi_{2}|(0)+C_{3}K\delta\epsilon+O(\epsilon^{2})\leq\epsilon+C_{3}K\delta\epsilon+O(\epsilon^{2}).$

Having established the previous estimates we can now turn to
$\underline{\check{L}}$ using the first and third equations of (5.10), and we
have that for $k\in\\{1,2\\}$:

$|\underline{\check{L}}\Phi_{k}|(t)\leq|\underline{\check{L}}\Phi_{k}|(0)+C_{4}K\delta\epsilon+O(\epsilon^{2})\leq\delta+C_{4}K\delta\epsilon+O(\epsilon^{2}).$

Concerning shock formation, for $\mu$ we use equation (4.10) and we have that

$L\mu\sim
K(\underline{\check{L}}\Phi_{1}+\underline{\check{L}}\Phi_{2})+O(\epsilon).$

After integrating along the integral curves of $L$ we have that

$\mu(t)\sim\mu(0)+K(\underline{\check{L}}\Phi_{1}+\underline{\check{L}}\Phi_{2})t+O(\epsilon),$

and the definition of $\delta_{0}$ as the quantity controlling the blow up
time follows from the previous calculation. Note that in the area close to
spacelike infinity it can be shown through the same method that $\mu$ remains
positive and bounded away from zero, so a shock can form only in the interior
region.

Finally, for the formation of the sonic line we consider the quantity
$L(\Phi_{1}^{2}+\Phi_{2}^{2})$ that we integrate in $t$. We use again the
established estimates for $\Phi_{i}$, $L\Phi_{i}$, $i\in\\{1,2\\}$ in order to
compute the estimated time where $\Phi_{1}^{2}+\Phi_{2}^{2}=q_{cr}^{2}$. Note
again that the previously established estimates for $\Phi_{i}$, $L\Phi_{i}$,
$i\in\\{1,2\\}$ guarantee that close to spacelike infinity we have that
$\Phi_{1}^{2}+\Phi_{2}^{2}$ remains bounded away from $q_{cr}^{2}$, so a sonic
line can form only in the interior region.

######  Remark 5.1.

We note that $\epsilon$ is chosen to be small enough not only in order to be
able to close the bootstrap assumptions, but also in order to ensure that
equation (2.2) remains hyperbolic throughout the region where the solution
exists, i.e. that

$q>q_{cr}\mbox{ for all $0\leq t<T^{*}$.}$

##  6 Proof of the main theorem

In this section we prove the main Theorem 5.1. We start by constructing the
set of initial data that we consider. The proof follows by a standard
bootstrap argument that occupies the largest part of this section. We will
state certain bootstrap assumptions that will imply certain estimates on the
nonlinearities. Using these we will be able to close the bootstrap argument on
an energy and pointwise level.

As we need to work with energy spaces we define the following renormalized
quantities:

$\Psi_{1}\doteq\Phi_{1},\quad\quad\Psi_{2}\doteq\Phi_{2}-K.$

The new quantity $\Psi=(\Psi_{1},\Psi_{2})$ satisfies the following system of
equations

$\begin{cases}\Box_{\mathbf{g}(\Psi)}\Psi_{1}=\mathbf{N}_{1}(\Psi_{1},\Psi_{2}),\\\
\Box_{\mathbf{g}(\Psi)}\Psi_{2}=\mathbf{N}_{2}(\Psi_{1},\Psi_{2}),\end{cases}$
(6.1)

where

$\mathbf{g}^{-1}(\Psi)=\begin{bmatrix}-1&-\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}\\\
-\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}&-\frac{\mathbf{c}^{2}-(\Psi_{2}+K)^{2}}{\mathbf{c}^{2}-\Psi_{1}^{2}}\end{bmatrix},$

where

$\mathbf{c}(\Psi)=c(\Psi_{1}^{2}+(\Psi_{2}+q_{cr})^{2}),$

and where

$\displaystyle\mathbf{N}_{1}(\Psi_{1},\Psi_{2})$
$\displaystyle=\frac{\partial}{\partial
x}\left(\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{y}\Psi_{1}-\frac{\partial}{\partial
y}\left(\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{x}\Psi_{1}$
$\displaystyle+\frac{\partial}{\partial
y}\left(\frac{\mathbf{c}^{2}-(\Psi_{2}+K)^{2}}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{x}\Psi_{2}-\frac{\partial}{\partial
x}\left(\frac{\mathbf{c}^{2}-(\Psi_{2}+K)^{2}}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{x}\Psi_{1}$
$\displaystyle+\frac{1}{\sqrt{-\det(\mathbf{g})}}[\partial_{\alpha}(\sqrt{-\det(\mathbf{g})})]\cdot\mathbf{g}^{\alpha\beta}\partial_{\beta}\Psi_{1},$
$\displaystyle\mathbf{N}_{2}(\Psi_{1},\Psi_{2})$
$\displaystyle=\frac{\partial}{\partial
y}\left(\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{x}\Psi_{2}-\frac{\partial}{\partial
x}\left(\frac{\Psi_{1}\cdot(\Psi_{2}+K)}{\mathbf{c}^{2}-\Psi_{1}^{2}}\right)\partial_{y}\Psi_{2}$
$\displaystyle+\frac{1}{\sqrt{-\det(\mathbf{g})}}[\partial_{\alpha}(\sqrt{-\det(\mathbf{g})})]\cdot\mathbf{g}^{\alpha\beta}\partial_{\beta}\Psi_{2}.$

###  6.1 Construction of the initial data set

We can choose any smooth functions $(f_{1},f_{2})$ in one variable for

$f_{1}=\left.\Phi_{1}\right|_{\Sigma},\quad\quad
f_{2}=\left.\Phi_{2}\right|_{\Sigma},$

with the appropriate size restrictions in $L^{\infty}$ and $L^{2}$. If the
size restrictions of the derivatives of $(f_{1},f_{2})$ are not satisfied,
then we can argue by rescaling in order to get the desired bounds.

On the other hand, a consequence of (4.3) is that

$\left.\mu\right|_{\Sigma}=\frac{1}{\sqrt{(\underline{g}^{-1})^{00}}},$

for $\underline{g}=\Pi g$, $\Pi$ the projection to $\Sigma$ (for a proof see
Lemma 7.2 of [35]). Initially at $\Sigma$ this gives us that
$\inf_{\Sigma}\mu=1$.

###  6.2 The bootstrap assumptions

Let us make the following bootstrap assumptions: for some constant
$\mathcal{C}$ we have that

$\|\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}\leq\mathcal{C}\epsilon,$ (6.2)
$\|L\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|L\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\mathcal{C}\epsilon,$
(6.3)
$\|\check{\underline{L}}\Phi_{1}\|_{L^{\infty}(\Sigma_{t})}+\|\check{\underline{L}}\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\mathcal{C}\delta,$
(6.4) $\|\Phi_{2}\|_{L^{\infty}(\Sigma_{t})}\leq\mathcal{C}K,$ (6.5)

for $\epsilon$, $\delta$ and $K$ as in the assumptions and for $t\in[0,T^{0})$
for $T^{0}$ to be a bootstrap time such that

$0<T^{0}<\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right).$

Moreover we also assume that again for $t\in[0,T^{0})$ we have that

$\|\Phi_{1}\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}+\|\Phi_{2}-K\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}\leq\underline{\mathcal{C}}\epsilon,$ (6.6)
$\|L\Phi_{1}\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}+\|L\Phi_{2}\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}\leq\underline{\mathcal{C}}\epsilon,$ (6.7)
$\|\check{\underline{L}}\Phi_{1}\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}+\|\check{\underline{L}}\Phi_{2}\|_{L^{\infty}(\Sigma_{t}\cap\\{u\leq
U_{c}\\})}\leq\underline{\mathcal{C}}\epsilon.$ (6.8)

Under the bootstrap assumptions (6.2), (6.3), (6.4) and (6.5) we will show
that:

$|\check{\mathcal{N}}_{1}(\Phi,L\Phi,\check{\underline{L}}\Phi)|+|\check{\mathcal{N}}_{2}(\Phi,L\Phi,\check{\underline{L}}\Phi)|+(|\omega_{1}|+\mu|\omega_{2}|)|L\Phi_{\kappa}|\leq\sum_{i=1}^{2}\mathcal{C}^{\prime}K\delta|L\Phi_{i}|,$
(6.9)

and for $\kappa\in\\{1,2\\}$ and for all $(t,u)\in[0,T^{0})\times(-\infty,U]$.
Note that for $(t,u)\in[0,T^{0})\times(-\infty,U_{0}]$ we do not need the
bootstrap assumptions but we can just use the estimates of the previous
section. This also implies that:

$|\omega_{1}|+\mu|\omega_{2}|\leq\mathcal{C}^{\prime\prime}K\delta,$ (6.10)
$|\check{\mathcal{N}}_{1}(\Phi,L\Phi,\check{\underline{L}}\Phi)|+|\check{\mathcal{N}}_{2}(\Phi,L\Phi,\check{\underline{L}}\Phi)|+(|\omega_{1}|+\mu|\omega_{2}|)|L\Phi_{\kappa}|\leq\mathcal{C}^{\prime\prime\prime}K\delta\epsilon,$
(6.11)

again for $\kappa\in\\{1,2\\}$ and for all
$(t,u)\in[0,T^{0})\times(-\infty,U]$, and it also implies that:

$|\mathbf{N}_{1}(\Psi_{1},\Psi_{2})|+|\mathbf{N}_{2}(\Psi_{1},\Psi_{2})|\leq\sum_{i=1}^{2}\mathcal{C}^{\prime\prime\prime\prime}\epsilon^{2}|L\Psi_{i}|,$
(6.12)

for all $(t,u)\in[0,T^{0})\times(-\infty,U_{c}]$.

We start by showing (6.9), examining first the terms $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$. From Lemma A.3.1 of [33] we have that

$\mathcal{N}_{1}=f_{11}\cdot(L\Phi_{1})(L\Phi_{2})+f_{12}(L\Phi_{1})(X\Phi_{2})+f_{13}(X\Phi_{1})(L\Phi_{2}),$
$\mathcal{N}_{2}=f_{21}\cdot(L\Phi_{1})(L\Phi_{2})+f_{22}(L\Phi_{1})(X\Phi_{2})+f_{23}(X\Phi_{1})(L\Phi_{2}),$

where the $f_{i}$’s, $i\in\\{1,2,3\\}$, are smooth functions of the
$\Phi_{j}$’s, $j\in\\{1,2\\}$ and the components $L^{j}$’s and $X^{j}$’s,
$j\in\\{1,2\\}$. From the above estimate (6.9) for $\mathcal{N}_{1}$,
$\mathcal{N}_{2}$, follows from the bootstrap assumptions (6.2), (6.3), (6.4)
and (6.5) and the form of the nonlinearities, after showing again through a
bootstrap argument that

$|L^{i}|\leq c_{L},\quad|X^{i}|\leq c_{X}\mbox{ for $i\in\\{1,2\\}$},$ (6.13)

for some constants $c_{L}>0$, $c_{X}>0$. This can be done similarly to Lemma
11.9.1 from [33], with obvious modifications for $u\in(-\infty,U_{c}]$.

From the form of the $\mathcal{N}_{i}$’s, $i\in\\{1,2\\}$, we have that

$|f_{ij}|\leq\sum_{k=1}^{2}c_{f}|\Phi_{k}|\mbox{ for $i\in\\{1,2,3\\}$,
$j\in\\{1,2\\}$, and a constant $c_{f}>0$.}$

This completes the proof of estimate (6.9) for $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$. Estimate (6.9) for the $\omega$’s follows from the form of
the $H^{\nu}_{\alpha\beta}$’s, while estimate (6.11) and (6.12) are
consequences of estimate (6.9).

###  6.3 Construction of the solution near spacelike infinity

First we construct the solution close to spacelike infinity, that is for
$|x|\rightarrow\infty$ along $\Sigma_{t}$ for any $t\in[0,T^{0})$ (the fact
that this remains fixed will follow by the construction as the $\Sigma_{t}$
hypersurfaces depend on the solution $\Phi$), also showing that throughout the
region of existence and uniqueness of the solution $\Phi$, the Mach speed at
infinity will also remain fixed (so there is no need imposing a boundary
condition). Eventually after closing the bootstrap argument we will have the
same result for $t\in[0,T^{*})$.

The estimates that we will establish will imply that

$\lim_{u\rightarrow-\infty}\sqrt{\Psi_{1}^{2}(t,u)+\Psi_{2}^{2}(t,u)}=0,$

ifor all $t\in[0,T^{*})$, hence ensuring that the Mach speed at infinity
remains fixed.

Recall that the energy-momentum tensor for the wave equation is given by:

$\mathbb{T}_{\alpha\beta}[\psi]\doteq\partial_{\alpha}\psi\cdot\partial_{\beta}\psi-\frac{1}{2}\mathbf{g}_{\alpha\beta}\mathbf{g}^{\sigma\tau}\partial_{\sigma}\psi\cdot\partial_{\tau}\psi\mbox{
for $\alpha,\beta,\sigma,\tau\in\\{0,1\\}$}.$

For the system (6.1) we have the following well-known computations:

$\partial_{\alpha}\mathbb{T}^{\alpha\beta}[\Psi_{1}]=\mathbf{N}_{1}(\Psi_{1},\Psi_{2})\cdot\partial^{\beta}\Psi_{1}\mbox{
and
}\partial_{\alpha}\mathbb{T}^{\alpha\beta}[\Psi_{2}]=\mathbf{N}_{2}(\Psi_{1},\Psi_{2})\cdot\partial^{\beta}\Psi_{2}.$

For a vector field $V$ we define its energy current as:

$\mathbb{J}^{V}_{\alpha}[\psi]\doteq\mathbb{T}_{\alpha\beta}[\psi]\cdot
V^{\beta}.$

For the system (6.1) we have another well-known computation:

$\mathrm{Div}(J^{V}[\Psi_{1}])=\mathbf{N}_{1}(\Psi_{1},\Psi_{2})\cdot
V\Psi_{1}+\mathbb{T}_{\alpha\beta}[\Psi_{1}]\cdot\pi_{V}^{\alpha\beta}\mbox{
and }$ $\mathrm{Div}(J^{V}[\Psi_{2}])=\mathbf{N}_{2}(\Psi_{1},\Psi_{2})\cdot
V\Psi_{1}+\mathbb{T}_{\alpha\beta}[\Psi_{2}]\cdot\pi_{V}^{\alpha\beta},$

where $\pi_{V}$ is the deformation tensor with respect to $V$ defined as:

$(\pi_{V})_{\alpha\beta}\doteq\partial_{\alpha}V_{\beta}+\partial_{\beta}V_{\alpha}.$

Let us choose some $-\infty<U_{0}<U_{c}<U$. An application of the divergence
theorem gives the following identity (for a detailed proof see Lemma 3.1 of
[32]):

$\begin{split}\int_{\Sigma_{T}^{U_{0}}}\mu J^{V}_{\alpha}[\psi]\cdot
n_{\Sigma_{T}}^{\alpha}\,du+&\int_{\mathcal{N}_{U_{0}}^{T}}J^{V}_{\alpha}[\psi]\cdot
L^{\alpha}\,du=\int_{\Sigma_{0}^{U_{0}}}\mu J^{V}_{\alpha}[\psi]\cdot
n_{\Sigma}\,du\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\psi]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\Box_{\mathbf{g}}\psi\cdot
V\psi\,dtdu,\end{split}$ (6.14)

for

$\Sigma_{T}^{U_{0}}\doteq\\{t=T,-\infty<u\leq
U_{0}\\},\quad\mathcal{N}_{U_{0}}^{T}=\\{u=U_{0},0\leq t<T\\},$

for $\mathcal{R}_{T,U_{0}}$ as defined before, and for $n_{\Sigma}$ the normal
to the surface $\Sigma$. We choose $T\leq T^{0}$.

Choosing

$V=(1+\mu)L+\check{\underline{L}}=(1+2\mu)L+2\check{X},$

and using the aforementioned (6.14) we have that

$\begin{split}\int_{\Sigma_{T}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)&\mu(L\psi)^{2}+2\mu(L\psi)(\check{X}\psi)+2(\check{X}\psi)^{2}\Big{]}\,du+\int_{\mathcal{N}_{U_{0}}^{T}}(1+\mu)(L\psi)^{2}\,du\\\
=&\int_{\Sigma_{0}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)\mu(L\psi)^{2}+2\mu(L\psi)(\check{X}\psi)+2(\check{X}\psi)^{2}\Big{]}\,du\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\psi]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\Box_{\mathbf{g}}\psi\cdot[(1+2\mu)(L\psi)+2(\check{X}\psi)]\,dtdu.\end{split}$
(6.15)

For a proof see Lemma 3.2 and Proposition 3.4 of [32]. Applying the identity
(6.15) to the solutions $\Psi_{1}$ and $\Psi_{2}$ of the system (6.1) we have
that:

$\begin{split}\int_{\Sigma_{T}^{U_{0}}}\Big{[}\frac{1}{2}&(1+2\mu)\mu(L\Psi_{1})^{2}+2\mu(L\Psi_{1})(\check{X}\Psi_{1})+2(\check{X}\Psi_{1})^{2}\Big{]}\,du+\int_{\mathcal{N}_{U_{0}}^{T}}(1+\mu)(L\Psi_{1})^{2}\,du\\\
&+\int_{\Sigma_{T}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)\mu(L\Psi_{2})^{2}+2\mu(L\Psi_{2})(\check{X}\Psi_{2})+2(\check{X}\Psi_{2})^{2}\Big{]}\,du+\int_{\mathcal{N}_{U_{0}}^{T}}(1+\mu)(L\Psi_{2})^{2}\,du\\\
=&\int_{\Sigma_{0}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)\mu(L\Psi_{1})^{2}+2\mu(L\Psi_{1})(\check{X}\Psi_{1})+2(\check{X}\Psi_{1})^{2}\Big{]}\,du\\\
&+\int_{\Sigma_{0}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)\mu(L\Psi_{2})^{2}+2\mu(L\Psi_{2})(\check{X}\Psi_{2})+2(\check{X}\Psi_{2})^{2}\Big{]}\,du\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\Psi_{1}]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{N}_{1}(\Psi_{1},\Psi_{2})\cdot[(1+2\mu)(L\Psi_{1})+2(\check{X}\Psi_{1})]\,dtdu\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\Psi_{2}]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{N}_{2}(\Psi_{1},\Psi_{2})\cdot[(1+2\mu)(L\Psi_{2})+2(\check{X}\Psi_{2})]\,dtdu.\end{split}$
(6.16)

We note that according to Proposition 3.4 of [32] we have that

$\mu\mathbb{T}_{\alpha\beta}[\psi]\cdot\pi_{V}^{\alpha\beta}=(L\psi)^{2}\Big{[}-\frac{1}{2}(L\mu)+\mu(\check{X}\mu)\Big{]}.$
(6.17)

By Proposition 3.3 of [32] and an application of the Cauchy-Schwarz inequality
in the estimate (6.16) we have that

$\begin{split}\int_{\Sigma_{T}^{U_{0}}}\Big{[}\frac{1}{2}&(1+2\mu)\mu(L\Psi_{1})^{2}+2(\check{X}\Psi_{1})^{2}\Big{]}\,du+\int_{\mathcal{N}_{U_{0}}^{T}}(1+\mu)(L\Psi_{1})^{2}\,du\\\
&+\int_{\Sigma_{T}^{U_{0}}}\Big{[}\frac{1}{2}(1+2\mu)\mu(L\Psi_{2})^{2}+2(\check{X}\Psi_{2})^{2}\Big{]}\,du+\int_{\mathcal{N}_{U_{0}}^{T}}(1+\mu)(L\Psi_{2})^{2}\,du\\\
\leq&\int_{\Sigma_{0}^{U_{0}}}\Big{[}2(1+2\mu)\mu(L\Psi_{1})^{2}+4(\check{X}\Psi_{1})^{2}\Big{]}\,du\\\
&+\int_{\Sigma_{0}^{U_{0}}}\Big{[}2(1+2\mu)\mu(L\Psi_{2})^{2}+4(\check{X}\Psi_{2})^{2}\Big{]}\,du\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\Psi_{1}]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{N}_{1}(\Psi_{1},\Psi_{2})\cdot[(1+2\mu)(L\Psi_{1})+2(\check{X}\Psi_{1})]\,dtdu\\\
&-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{T}_{\alpha\beta}[\Psi_{2}]\cdot\pi_{V}^{\alpha\beta}\,dtdu-\int_{\mathcal{R}_{T,U_{0}}}\mu\mathbb{N}_{2}(\Psi_{1},\Psi_{2})\cdot[(1+2\mu)(L\Psi_{2})+2(\check{X}\Psi_{2})]\,dtdu\\\
\doteq&I+II+III+IV.\end{split}$ (6.18)

Note that terms $I$ and $III$ are similar, while the same is true for terms
$II$ and $IV$. We start with the last two, and by bootstrap assumption (6.12)
and Cauchy-Schwarz we have that

$|II|+|IV|\leq
2\mathcal{C}^{\prime\prime\prime\prime}\epsilon^{2}\int_{\mathcal{R}_{T,U_{0}}}(1+2\mu)\mu[(L\Psi_{1})^{2}+(L\Psi_{2})^{2}]+2[(\check{X}\Psi_{1})^{2}+(\check{X}\Psi_{2})^{2}]\,dtdu.$

Using the fact that
$T\leq\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right)$
and that neither $\delta_{1}$ nor $\delta_{2}$ are taken to be small (while
$\epsilon$ is), we have that

$|II|+|IV|\leq\bar{\mathcal{C}}^{\prime\prime\prime\prime}\epsilon^{2}\sup_{t\in[0,T]}\int_{\Sigma_{t}\cap\\{u\leq
U_{0}\\}}(1+2\mu)\mu[(L\Psi_{1})^{2}+(L\Psi_{2})^{2}]+2[(\check{X}\Psi_{1})^{2}+(\check{X}\Psi_{2})^{2}]\,du,$

and by choosing $\epsilon$ sufficiently small we can absorb the last term on
the left hand side of (6.18) (by taking the supremum of $t$ on the left hand
side as well).

For terms $I$ and $III$ we have by (6.17) that:

$|I|+|III|\leq\int_{\mathcal{R}_{T,U_{0}}}(|L\mu|+|X\mu|)[(L\Psi_{1})^{2}+(L\Psi_{2})^{2}]\,dtdu.$

Under the bootstrap assumptions by using equation (4.10) we get that

$|L\mu|\leq\underline{c}\epsilon^{2},\quad|X\mu|\leq\underline{c}^{\prime}\epsilon^{2},\quad\mu\geq\underline{c}^{\prime\prime}>0,$
(6.19)

for all $(t,u)\in[0,T)\times(-\infty,U_{0}]$, and using the above and that
$T\leq\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right)$
and that neither $\delta_{1}$ nor $\delta_{2}$ are taken to be small (while
$\epsilon$ is) as before, we get that

$|I|+|III|\leq\underline{c}^{\prime\prime\prime}\epsilon^{2}\sup_{t\in[0,T]}\int_{\Sigma_{t}\cap\\{u\leq
U_{0}\\}}(1+2\mu)\mu[(L\Psi_{1})^{2}+(L\Psi_{2})^{2}]\,du,$

and using again the smallness of $\epsilon$ we can absorb the last term on the
left hand side of (6.18) (by taking the supremum of $t$ on the left hand side
as well). So in the end it remains to show the estimates (6.19).

##  7 Construction of the solution in the interior region

We work in the region

$\mathrm{Int}\doteq[0,T^{*})\times[U_{0},U].$

Using the bootstrap assumption (6.9) and using the second and fourth equation
from (4.9) we get that:

$|\check{\underline{L}}L\Phi_{1}|(t,u)\leq\sum_{i=1}^{2}\mathcal{C}^{\prime}K\delta|L\Phi_{i}|(t,u),$

and

$|\check{\underline{L}}L\Phi_{2}|(t,u)\leq\sum_{i=1}^{2}\mathcal{C}^{\prime}K\delta|L\Phi_{i}|(t,u),$

for $(t,u)\in\mathrm{Int}$. By a matrix version of the Grönwall inequality and
using the finiteness of the $u$-interval we get that:

$|L\Phi_{1}|(t,u)+|L\Phi_{2}|(t,u)\leq\mathcal{C}(U_{0},U)(|L\Phi_{1}|(t,U_{0})+|L\Phi_{2}|(t,U_{0}),$

for $\mathcal{C}(U_{0},U)<\infty$ a quantity depending on $U_{0}$ and $U$
(that can be considered fixed as both $U_{0}$ and $U$ are). Using the
estimates of the previous section we then get that:

$|L\Phi_{1}|(t,u)+|L\Phi_{2}|(t,u)\leq\mathcal{C}^{\prime}(U_{0},U)\epsilon,$

for all $(t,u)\in\mathrm{Int}$. Integrating in $t$ the last inequality we have
that:

$|\Phi_{1}|(t,u)\leq|\Phi_{1}|(0,u)+\mathcal{C}^{\prime}(U_{0},U)\epsilon\cdot
t,$

and

$|\Phi_{2}|(t,u)\leq|\Phi_{2}|(0,u)+\mathcal{C}^{\prime}(U_{0},U)\epsilon\cdot
t.$

Note that as
$t\leq\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{\delta_{2}}\right)$
and that neither $\delta_{1}$ nor $\delta_{2}$ are taken to be small, we have
for some constants $\bar{c}_{1}$, $\bar{c}_{2}$, that

$|\Phi_{1}|(t,u)\leq\bar{c}_{1}\epsilon,$

and

$|\Phi_{2}|(t,u)\leq\bar{c}_{2}K,$

for all $(t,u)\in\mathrm{Int}$.

For the $\underline{\check{L}}$ derivatives, we use the bootstrap assumption
(6.11) and we have that

$|L\underline{\check{L}}\Phi_{1}|(t,u)+|L\underline{\check{L}}\Phi_{2}|(t,u)\leq
2\mathcal{C}^{\prime\prime\prime}K\delta\epsilon,$

for all $(t,u)\in[0,T^{*})\times(-\infty,U]$. Integrating in $t$ the last
inequality we get that

$|\underline{\check{L}}\Phi_{1}|(t,u)\leq|\underline{\check{L}}\Phi_{1}|(0,u)+\mathrm{c}\epsilon,$

and

$|\underline{\check{L}}\Phi_{2}|(t,u)\leq|\underline{\check{L}}\Phi_{2}|(0,u)+\mathrm{c}\epsilon,$

for some $\mathrm{c}>0$ that does not depend on $K$ or $\delta$ (using that
$t\leq\min\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right)$
and the relationship between $\delta_{1}$ and $K$ and $\delta$). Since
$\epsilon$ is taken to be small compared to $\delta$, we get in the end that

$|\underline{\check{L}}\Phi_{1}|(t,u)\leq\mathrm{c}_{1}\delta,$

and

$|\underline{\check{L}}\Phi_{1}|(t,u)\leq\mathrm{c}_{1}\delta,$

for all $(t,u)\in[0,T^{*})\times(-\infty,U]$. Note that we can improve the
estimates above in the region $(t,u)\in[0,T^{*})\times(-\infty,U_{c}]$ to the
following:

$|\underline{\check{L}}\Phi_{1}|(t,u)\leq\mathrm{c}_{2}\epsilon,$

and

$|\underline{\check{L}}\Phi_{1}|(t,u)\leq\mathrm{c}_{2}\epsilon.$

####  7.0.1 Formation of a shock

Let us examine the equation for $\mu$, that is (4.10). We will use the already
established bounds on $\Phi_{1}$, $\Phi_{2}$, $L\Phi_{1}$, $L\Phi_{2}$,
$\underline{\check{L}}\Phi_{1}$ and $\underline{\check{L}}\Phi_{2}$ to
estimate the time $T^{1}$ where $\mu$ can potentially vanish. We will consider
the case of $u\in[U_{0},U]$ as we showed earlier that $\mu$ remains positive
and bounded away from zero for $u\in(-\infty,U_{0}]$.

Note that the estimates of the previous section can be stated as equalities in
the following way:

$\Phi_{\kappa}(t,u)=\Phi(0,u)+\mathcal{O}(\epsilon),$ (7.1)
$\underline{\check{L}}\Phi_{\kappa}(t,u)=\underline{\check{L}}\Phi_{\kappa}(0,u)+\mathcal{O}(\epsilon),$
(7.2)

and

$L\Phi_{\kappa}(t,u)=L\Phi_{\kappa}(0,u)+\mathcal{O}(\epsilon),$ (7.3)

for $\kappa\in\\{1,2\\}$, and for $(t,u)\in[0,T^{*})\times[U_{0},U]$.
Multiplying these two identities we get the following using the estimates
we’ve already established:

$\sum_{\kappa_{1},\kappa_{2}\\{1,2\\}}\Phi_{\kappa_{1}}(t,u)\underline{\check{L}}\Phi_{\kappa_{2}}(t,u)=2\sum_{\kappa_{1},\kappa_{2}\\{1,2\\}}\Phi_{\kappa_{1}}(0,u)\underline{\check{L}}\Phi_{\kappa_{2}}(0,u)+\mathcal{O}(K\cdot\epsilon)+\mathcal{O}(\delta\cdot\epsilon)+\mathcal{O}(\epsilon^{2}),$

again for $(t,u)\in[0,T^{*})\times[U_{0},U]$. We note now that the first term
of the last estimate is of size $\sim K\delta$, so all the $\mathcal{O}$ terms
can be considered as errors.

We note that

$\omega_{1}\sim\sum_{\kappa_{1},\kappa_{2}\\{1,2\\}}\Phi_{\kappa_{1}}\underline{\check{L}}\Phi_{\kappa_{2}}\mbox{
and
}\omega_{2}\sim\sum_{\kappa_{1},\kappa_{2}\\{1,2\\}}\Phi_{\kappa_{1}}L\Phi_{\kappa_{2}},$

so using equation (4.10), we have that:

$L\mu(t,u)=\omega_{1}(0,u)+\mathcal{O}(K\cdot\epsilon)+\mathcal{O}(\delta\cdot\epsilon)+\mathcal{O}(\epsilon^{2}).$

We integrate this last equation in $t$ and we get that:

$\mu(t,u)=\mu(0,u)+\omega_{1}(0,u)\cdot t+\mathcal{O}(\epsilon),$

using for the last terms that
$t\leq\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right)$.

We take the infimum in $u$ in the last equality and we have that:

$\inf_{u\in[U_{0},U]}\mu(t,u)=\inf_{u\in[U_{0},U]}\mu(0,u)-\delta_{1}\cdot
t+\mathcal{O}(\epsilon).$

Since we assume that $\inf_{u\in[U_{0},U]}\mu(0,u)=1$, for
$\frac{2}{\delta_{1}}<\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}$, we
can verify estimate (5.6).

####  7.0.2 Formation of the sonic line

We will use again the established bounds on $\Phi_{1}$, $\Phi_{2}$,
$L\Phi_{1}$, $L\Phi_{2}$, $\underline{\check{L}}\Phi_{1}$ and
$\underline{\check{L}}\Phi_{2}$ to estimate the time $T^{2}$ where we can
potentially have that

$\frac{q}{c}=1.$

Recall that

$\frac{q}{c}>1\mbox{ for $q>q_{cr}$ and }\frac{q}{c}<1\mbox{ for $q<q_{cr}$.
}$

We use the basic calculation:

$Lq^{2}=2\Phi_{1}\cdot L\Phi_{1}+2\Phi_{2}\cdot L\Phi_{2},$

and we integrate it along the integral curves of $L$, which gives us that:

$q^{2}(t,u)=q^{2}(0,u)+2\int_{0}^{t}[\Phi_{1}(t^{\prime},u)\cdot
L\Phi_{1}(t^{\prime},u)+\Phi_{2}(t^{\prime},u)\cdot
L\Phi_{2}(t^{\prime},u)]\,dt^{\prime}.$

Using equations (7.1) and (7.3) repeatedly, we get that

$q^{2}(t,u)=q^{2}(0,u)+4(\Phi_{1}(0,u)\cdot L\Phi_{1}(0,u)+\Phi_{2}(0,u)\cdot
L\Phi_{2}(0,u))\cdot t+\mathcal{O}(\epsilon^{2}),$

where we used also that
$t\leq\left(\frac{2}{\delta_{1}},\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}\right)$.

Using that

$\Phi_{2}\cdot L\Phi_{2}=\mathcal{O}(K\cdot\epsilon)\gg\Phi_{1}\cdot
L\Phi_{1}=\mathcal{O}(\epsilon^{2}),$

from the previously established bounds on $\Phi_{1}$, $\Phi_{2}$, $L\Phi_{1}$
and $L\Phi_{2}$, we have that after taking the infimum of $q^{2}(t,u)$ in $u$
the following emerges:

$\inf_{u\in[U_{0},U]}q^{2}(t,u)=\inf_{u\in[U_{0},U]}q^{2}(0,u)-4\delta_{2}\cdot
t+\mathcal{O}(\epsilon^{2}).$

Setting $\inf_{u\in[U_{0},U]}q^{2}(t,u)=q_{cr}^{2}$, for
$\frac{2}{\delta_{1}}>\frac{\min_{u}q^{2}(0,u)-q_{cr}^{2}}{2\delta_{2}}$, we
can verify estimate (5.6).

##  8 The development problem: discussion

At this point, an interesting situation arises: according to the choice of the
data, a shock might form first or a sonic line. In the later case, we cannot
expect to be able to extend the solution past such a boundary, but in the
former, one can try extending the solution past a shock surface. This is known
as the development problem and has a long history in the area of fluid
mechanics (see [12] for an impressive result in the case of the time-dependent
Euler equations, and also [8] and references therein).

In the case of hyperbolic systems in $\mathbb{R}^{+}\times\mathbb{R}$ (i.e. in
$1\times 1$ dimensions), there are several results on both the formation and
development of shocks. Let us mention here the work of Lebaud [19] who showed
shock formation and development for hyperbolic $p$-systems on
$\mathbb{R}^{+}\times\mathbb{R}$, the follow-up work of Chen and Dong [9] (for
the same type of equations, showing the same result for more generic data),
the work of Yin [36] on shock formation and development for the compressible
Euler equations with spherically symmetric data, and finally and more
relevantly to our case the work of Kong [18] on shock formation and
development for general quasilinear $2\times 2$ hyperbolic systems on
$\mathbb{R}^{+}\times\mathbb{R}$. Finally, shock formation for quasilinear
$N\times N$ strictly hyperbolic and genuinely nonlinear systems (see below for
a definition) on $\mathbb{R}^{+}\times\mathbb{R}$ has been shown in [11] with
techniques similar to the ones used in the present paper.

The results of [19], [9] and [18] rely on the fact that a quasilinear $2\times
2$ system on $\mathbb{R}^{+}\times\mathbb{R}$ of the form

$\partial_{t}u+A(u)\partial_{x}u=0,$ (8.1)

for $u(t,x)=(u_{1}(t,x),u_{2}(t,x))$, where $t\in\mathbb{R}^{+}$,
$x\in\mathbb{R}$, can be transformed into

$\begin{cases}\partial_{t}v_{1}+\lambda_{1}(v_{1},v_{2})\partial_{x}v_{1}=0,\\\
\partial_{t}v_{2}+\lambda_{2}(v_{1},v_{2})\partial_{x}v_{2}=0,\end{cases}$
(8.2)

for $\lambda_{i}$, $i\in\\{1,2\\}$, the eigenvalues of the matrix $A$. The
functions $v_{1}$ and $v_{2}$ are called the Riemann invariants.

The main result of [18] is the following:

######  Theorem 8.1.

Consider the system (8.2) in the case that it is strictly hyperbolic, that
means we assume that

$\lambda_{1}<\lambda_{2},$

and genuinely nonlinear, that means we assume that

$\frac{\partial\lambda_{1}}{\partial v_{1}}\neq
0,\quad\frac{\partial\lambda_{2}}{\partial v_{2}}\neq 0.$

Then for data

$v_{1}(0,x)\doteq v_{1;0}(x),\quad v_{2}(0,x)\doteq v_{2;0}(x),\quad
x\in\mathbb{R},$

for which there exists some $x_{0}\in\mathbb{R}$ where

$\mbox{either }v_{1;0}^{\prime}(x_{0})<0\mbox{ or }v_{2;0}^{\prime}(x_{0})<0,$

the system (8.2) has a unique $C^{1}$ solution $(v_{1},v_{2})$ in
$\\{t\in[0,t_{0}],x\in\mathbb{R}\\}$ and

$\limsup_{t\rightarrow
t_{0}}\left(\|\partial_{x}v_{1}\|_{C^{0}}+\|\partial_{x}v_{2}\|_{C^{0}}\right)=\infty.$

Under some additional assumptions on $\lambda_{1}$ and $\lambda_{2}$, it can
be shown that the solution $(v_{1},v_{2})$ blows up at $(t_{0},x_{0})$ for
$x_{0}<\infty$, and moreover that in a neighbourhood $\mathcal{N}_{0}$ of
$(t_{0},x_{0})$ we have that

$\begin{split}&|v_{1}(t,x)-v_{1}(t_{0},x_{0})|\leq
c_{1}(|t-t_{0}|+|x-x_{0}|)\mbox{ for all $(t,x)\in\mathcal{N}_{0}$,}\\\
&|\partial_{x}v_{1}(t,x)-\partial_{x}v_{1}(t_{0},x_{0})|\leq
c_{1}(|t-t_{0}|+|x-x_{0}|)\mbox{ for all $(t,x)\in\mathcal{N}_{0}$,}\\\
&|v_{2}(t,x)-v_{2}(t_{0},x_{0})|\leq
c_{2}(|t-t_{0}|^{3}+|x-x_{0}(t)|^{2})^{1/6}\mbox{ for all
$(t,x)\in\mathcal{N}_{0}$,}\\\ &|\partial_{x}v_{2}(t,x)|\leq
c_{2}(|t-t_{0}|^{3}+|x-x_{0}(t)|^{2})^{-1/3}\mbox{ for all
$(t,x)\in\mathcal{N}_{0}\setminus(t_{0},x_{0})$,}\\\
&|\partial_{xx}^{2}v_{2}(t,x)|\leq
c_{3}(|t-t_{0}|^{3}+|x-x_{0}(t)|^{2})^{-5/6}\mbox{ for all
$(t,x)\in\mathcal{N}_{0}\setminus(t_{0},x_{0})$,}\end{split}$

for $c_{1},c_{2},c_{3}>0$ and for $x_{0}(t)$ the characteristic corresponding
to $\lambda_{1}$.

The analysis in [19], [9] and [18] relies on a careful examination of the
characteristics of the system (8.2), that results in the treatment of a system
of ODEs (something similar in spirit takes place in [36] where instead of a
system provided by the Riemann invariants, an approximate system is examined
following [3], [2]).

One can try treating the isentropic and irrotational system of compressible
Euler equations in the same way. We can rewrite the equation (2.2) for the
potential flow as a quasilinear $2\times 2$ system

$\partial_{y}u+A_{e}(u)\partial_{x}u=0,$ (8.3)

where

$A_{e}(u)=\begin{bmatrix}-\frac{2u_{1}u_{2}}{c^{2}-u_{1}^{2}}&\frac{c^{2}-u_{2}^{2}}{c^{2}-u_{1}^{2}}\\\
-1&0\end{bmatrix},$

for

$u=(u_{1}.u_{2})=(\partial_{y}\phi,\partial_{x}\phi),$

in the domain $\mathcal{D}$ as defined before

$\mathcal{D}\doteq\\{(x,y)\quad|\quad y\geq 0\mbox{ for $|x|\geq 1$, and
}y\geq f(x)\mbox{ for $|x|<1$}\\},$

where $f$ is a smooth concave function such that

$f(x)>0\mbox{ for $|x|<1$, }f(1)=f(-1)=0,$

and which defines the airfoil $\mathcal{P}$ by

$\mathcal{P}\doteq\\{(x,y)\quad|\quad|x|<1,y=f(x)\\}.$

In this formulation one can set data as in Theorem 5.1, compute the Riemann
invariants (which do not seem to have a very clean form, and in the most
abstract setting depend on $c$) and try to prove a result similar to 8.1. It
should be noted though as the eigenvalues of $A_{e}(u)$ are given by

$\lambda_{\pm}(u)=\dfrac{-u_{1}u_{2}\pm\sqrt{c^{2}(-c^{2}+u_{1}^{2}+u_{2}^{2})}}{c^{2}-u_{1}^{2}},$

there is also the scenario of degeneration of hyperbolicity (and formation of
a sonic line as in Theorem 5.1). To this end, we are not aware of any results
for quasilinear $2\times 2$ systems (only for wave equations as in [34] and
the references therein).

Moreover, a more interesting problem, is to try and construct (using either
the first order $2\times 2$ system formulation, or the quasilinear wave
equation for the potential flow) the full shock surface for appropriate data,
and attempt to continue the solution past this surface for long enough time
till a sonic line forms. In this direction see the notes [1] and the
references therein.

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