# Longitudinal Shock Waves in a Class of Semi-infinite Stretch-Limited Elastic
Strings

Casey Rodriguez<EMAIL_ADDRESS>
###### Abstract.

In this paper, we initiate the study of wave propagation in a recently
proposed mathematical model for stretch-limited elastic strings. We consider
the longitudinal motion of a simple class of uniform, semi-infinite, stretch-
limited strings under no external force with finite end held fixed and
prescribed tension at the infinite end. We study a class of motions such that
the string has one inextensible segment, where the local stretch is maximized,
and one extensible segment. The equations of motion reduce to a simple and
novel shock front problem in one spatial variable for which we prove existence
and uniqueness of local-in-time solutions for appropriate initial data. We
then prove the orbital asymptotic stability of an explicit two-parameter
family of piece-wise constant stretched motions. If the prescribed tension at
the infinite end is increasing in time, our results provide an open set of
initial data launching motions resulting in the string becoming fully
inextensible and tension blowing up in finite time.

##  1\. Introduction

The exact equations for planar motion of mechanical strings were derived by
Euler (1751) and for general motion in three-dimensional space by Lagrange
(1762), and they have been a rich source of research for natural philosophers
and mathematicians (see [2]). In standard treatments, a string is _elastic_ if
the tension is given as an explicit function of the stretch of the string (see
[1, 2]). This then implies the existence of a stored energy functional which
by the first law of thermodynamics implies mechanical work is not converted to
heat during the string’s motion (i.e. there is no dissipation).

In a series of papers [10, 12, 11], Rajagopal recognized within the context of
full three-dimensional continuum mechanics that the absence of dissipation is
not predicated upon assuming the stress variable is an explicit function of
the strain variable. More precisely, if we define _elasticity_ as the
incapability of dissipation, then elastic bodies can be modeled by a wider
class of relations between the stress variable and strain variable than those
where the stress is a function of strain. This generalized notion of
elasticity has had numerous applications in the modeling of electro and
magneto-elastic bodies [4, 5], fracture in brittle materials [8, 3], gum metal
[13] and many other materials (see the recent review [6] for further
references).

In [14], we initiated the study of these generalized relations for strings by
considering a class of _stretch-limited_ elastic strings. For these strings,
the stretch is an explicit function of the tension of the string, but not vice
versa, and the stretch is constrained between two limiting values. Therefore,
segments of stretch-limited strings can neither be stretched nor compressed
beyond certain lengths. Put another way, if one fixes one end and pulls hard
enough on the other end of a stretch-limited string, then the string becomes
completely taut and incapable of further stretching (a common experience with
real strings). These properties are incapable of being modeled within the
standard framework for elastic strings. In [14] we studied stationary
configurations of finite stretch-limited strings suspended between two points
under the force of gravity (catenaries) containing both inextensible segments
(where the stretch is maximized) and extensible segments.

In this work we study the dynamic longitudinal motion of a simple class of
uniform, semi-infinite, tensile, stretch-limited elastic strings with finite
end held fixed, prescribed tension at the infinite end (the end at infinity
being pulled taut), and containing one inextensible segment and one extensible
segment. A brief overview of the formulation of the problem is as follows (see
Section 2 for more details). Let $I=[0,\infty)$ parameterize the particles of
the string and denote the tension by $N$ and stretch by $\nu$. We denote the
position of the particle $s$ along the $\boldsymbol{i}$ axis at time $t$ by
$\chi(s,t)$ so that the stretch $\nu(s,t)=\chi_{s}(s,t)$. The variables
$\chi_{s}$ and $N$ are related via

$\displaystyle\chi_{s}(s,t)=\begin{cases}\frac{1}{E}N(s,t)&\mbox{ if
}0<N(s,t)\leq N_{1},\\\ \nu_{1}&\mbox{ if }N(s,t)\geq N_{1},\end{cases}$ (1.1)

where $\nu_{1}>1$ is the maximal stretch of the string, $N_{1}>0$ is the
threshold tensile force resulting in inextensibility and $E=N_{1}/\nu_{1}$.
The equations of motion reduce to the simple balance law:

$\displaystyle\begin{split}&\gamma\chi_{tt}(s,t)=N_{s}(s,t),\quad(s,t)\in[0,\infty)\times[0,T],\\\
&\chi(0,t)=0,\quad N(\infty,t)=\zeta t+\tau,\end{split}$ (1.2)

where $\gamma>0$ is the mass density and $\zeta\geq 0$, $\tau\in(0,N_{1})$
determine the prescribed tension at the infinite end. When the tension is a
function of stretch, $N(s,t)=\hat{N}(\chi_{s}(s,t))$, as in the classical
treatment of elastic strings, (1.2) falls within the well-studied realm of
hyperbolic balance laws of one spatial variable (see e.g. [7]). However, the
balance law (1.2) with relation between the state variables given by (1.1) is,
to the author’s knowledge, novel. We assume that the string for each time $t$
contains an inextensible segment parameterized by $s\in[0,\sigma(t)]$ where
the tension $N(\cdot,t)\in C^{1}([0,\sigma(t)])$ exceeds the threshold for
inextensibility

$N(s,t)>N_{1}>0,\quad s\in[0,\sigma(t)],$

and an extensible segment parameterized by $s\in[\sigma(t),\infty)$ where
$N(\cdot,t)\in C^{1}([\sigma(t),\infty))$ and

$0<N(s,t)=E\chi_{s}(s,t)<N_{1},\quad s\in[\sigma(t),\infty).$

We further assume the (generic) condition

$\displaystyle N(\sigma^{-}(t),t)>N_{1}>N(\sigma^{+}(t),t)$ (1.3)

holds. Then the point of transition $s=\sigma(t)$ between segments is a shock
front, and therefore, these motions do dissipate energy due to entropy
production (see Proposition 3.5) in contrast to motions which have continuous
state variables (see Section 2).

The organization and main results of this work are as follows. In Section 2 we
briefly review the general theory of mechanical strings, give the precise
formulation of the problem we consider and derive the reduced equations of
longitudinal motion. In Section 3 we define the class of classical shock
solutions we study and prove the local-in-time existence and uniqueness of
shock solutions to (1.2) for classical shock front initial data
$(\chi,\partial_{t}\chi,N)|_{t=0}$ (see Theorem 3.3). The boundary condition
at $s=0$ and the assumption of longitudinal motion effectively reduces the
proof to solving a linear wave equation for $\chi$ in the extensible region
$\\{(s,t):t\in[0,T],s\in[\sigma(t),\infty)\\}$ with
$\chi(\sigma(t),t)=\nu_{1}\sigma(t)$. The condition (1.3) implies the shock
speed $\sigma^{\prime}$ is greater than the speed of propagation for this wave
equation (a version of _Lax’s entropy inequality_), and the result is then
proved via the implicit function theorem and finite speed of propagation
arguments. Such a simplification is not present for the full equations of
motion, and the local-in-time existence and uniqueness of solutions outside of
longitudinal motion is an open question. Finally, in Section 4 we prove the
orbital asymptotic stability of an explicit family of piece-wise constant
stretched motions,

$\displaystyle\chi_{s}(s,t)=\begin{cases}\nu_{1}&\mbox{ if
}s\in[0,\sigma(t))\\\ \frac{\zeta t+\tau}{E}&\mbox{ if
}s\in(\sigma(t),\infty),\end{cases}$ (1.4)

parameterized by the initial shock front $\sigma(0)$ and shock speed
$\sigma^{\prime}(0)$ (see Theorem 4.2 and Theorem 4.3). When $\zeta>0$, the
final time of existence for this family and its perturbations is
$T(\zeta)=(N_{1}-\tau)/\zeta<\infty$, the time when the tension at infinity
exceeds the threshold for inextensibility, $N(\infty,T(\zeta))=N_{1}$. Since
the tension at the end $s=\infty$ is growing to the threshold value $N_{1}$
when $\zeta>0$, stability in this case is more delicate than $\zeta=0$. One
must show the stretch in the extensible segment grows in a controlled way like
$(\zeta t+\tau)/E$ to leading order so that a second inextensible segment does
not form. This requires a careful analysis of the relationship between the
shock front $\sigma$ and the size of the perturbation in the extensible region
$[\sigma(t),\infty)$. For $\zeta>0$ our results prove the existence of an open
set of initial data launching motions having one inextensible segment and one
extensible segment for $t<T(\zeta)$ and becoming fully inextensible
$(\sigma(t)\rightarrow\infty)$ as $t\rightarrow T(\zeta)$. Moreover, these
motions result in the blow-up of the tension: for all $s\in[0,\infty)$,

$\displaystyle\lim_{t\rightarrow T(\zeta)^{-}}N(s,t)=\infty.$ (1.5)

Whether or not the family of piece-wise constant stretched motions are
orbitally asymptotically stable outside of purely longitudinal motion is an
open question.

We remark that our main results still hold with minor modifications if instead
of assuming a linear relation between stretch and tension (1.1) we assume a
linear relation between _strain_ $e:=\nu-1$ and tension, i.e. Hooke’s law. In
this case, (1.1) is replaced with

$\displaystyle\chi_{s}(s,t)-1=\begin{cases}\frac{1}{E}N(s,t)&\mbox{ if
}0<N(s,t)\leq N_{1},\\\ \nu_{1}-1&\mbox{ if }N(s,t)\geq N_{1},\end{cases}$
(1.6)

where $E:=N_{1}/(\nu_{1}-1)$ is _Young’s modulus_ for the string. We expect
our main results can be generalized for longitudinal motion involving more
general stretch-limiting constitutive relations than (1.1) or (1.6) with the
possible exception of Theorem 4.2 due to a potential second shock forming in
the extensible segment for small perturbations.

##  2\. Formulation of the problem

In this section we review of the general theory of mechanical strings, state
the precise assumptions for the problem we consider and derive the reduced
equations of motion.

###  2.1. Mechanical strings

A brief overview of mechanical strings is as follows (see [1, 2] for a more
extensive exposition). Let
$\\{\boldsymbol{i},\boldsymbol{j},\boldsymbol{k}\\}$ be a fixed right-handed
orthonormal basis for Euclidean space $\mathbb{E}^{3}$. Let
$I\subset\mathbb{R}$ be a fixed reference interval which parameterizes the
_particles_ or _material points_ of the string. The _configuration_ of the
string at time $t$ is the curve $I\ni
s\mapsto\boldsymbol{r}(s,t)\in\mathbb{E}^{3}$, and the tangent to the curve
$\boldsymbol{r}(\cdot,t)$ at $s$ is
$\boldsymbol{r}_{s}(s,t)=\partial_{s}\boldsymbol{r}(s,t)$. The _stretch_
$\nu(s,t)$ of the string at $(s,t)$ is

$\displaystyle\nu(s,t):=|\boldsymbol{r}_{s}(s,t)|.$

We require that the configuration is regular so that stretch is always
positive throughout the motion of the string. As a result of balance of linear
momentum, the general _classical equations of motion_ for a string are given
by:

$\displaystyle(\rho
A)(s)\boldsymbol{r}_{tt}(s,t)=\boldsymbol{n}_{s}(s,t)+\boldsymbol{f}(s,t),\quad(s,t)\in
I\times[0,T].$ (2.1)

Here $(\rho A)(s)$ is the mass per unit reference length at $s$,
$\boldsymbol{f}(s,t)$ is the body force per unit reference length at $(s,t)$,
and $\boldsymbol{n}(s,t)$ is the contact force at $(s,t)$. A defining
characteristic of a string is that the contact force is tangential to the
configuration of the string: there exists a scalar-valued function $N(s,t)$,
the _tension_ , such that

$\displaystyle\boldsymbol{n}(s,t)=N(s,t)\frac{\boldsymbol{r}_{s}(s,t)}{|\boldsymbol{r}_{s}(s,t)|}.$

The mechanical properties of a string are modeled by specifying a relation
between the stretch $\nu$ and tension $N$. Classically, a string is said to be
_elastic_ if there exists a function $\hat{N}(\nu,s)$ such that

$\displaystyle N(s,t)=\hat{N}(\nu(s,t),s),$ (2.2)

with $\hat{N}(1,s)=0$, $\nu\mapsto\hat{N}(\nu,s)$ increasing,
$\lim_{\nu\rightarrow 0}\hat{N}(\nu,s)=-\infty$ and
$\lim_{\nu\rightarrow\infty}\hat{N}(\nu,s)=\infty$ for all $s$. These
mathematical restrictions on $\hat{N}$ reflect the physically reasonable
assumptions that an un-stretched configuration is not in a state of tension,
an increase in tension leads to an increase in stretch, a state of zero
stretch requires infinite compression force and a state of infinite stretch
requires infinite tensile force. Thus, $\hat{N}(\cdot,s)$ has an inverse
function $\hat{\nu}(\cdot,s)$, and the constitutive relation between stretch
and tension takes the form

$\displaystyle\nu(s,t)=\hat{\nu}(N(s,t),s).$ (2.3)

###  2.2. Longitudinal motion for stretch-limited elastic strings

In this paper we study the dynamics of a simple class of uniform, semi-
infinite stretch-limited elastic strings introduced in [14]. The constitutive
relation can be expressed via (2.3) but not via (2.2), and we assume that the
string becomes inextensible once a threshold value of tensile force is
reached. Mathematically, the string satisfies:

  * •

for all $(s,t)\in[0,\infty)\times[0,T]$,

$\displaystyle(\rho A)(s)=\gamma>0,\quad\nu(s,t)=\hat{\nu}(N(s,t)),$ (2.4)

where $\hat{\nu}:\mathbb{R}\rightarrow(0,\infty)$ is absolutely continuous,
nondecreasing and $\hat{\nu}(0)=1$,

  * •

there exist $N_{1}>0$ and $\nu_{1}>1$ such that $\hat{\nu}\in
C^{\infty}([0,N_{1}];[1,\nu_{1}])$ is strictly increasing and for all $N\geq
N_{1}$,

$\displaystyle\hat{\nu}(N)=\nu_{1}.$ (2.5)

For simplicity we assume the constitutive relation is linear for all values of
tension near the threshold: there exists $\eta>0$ such that for all
$N\in[\eta,N_{1}]$

$\displaystyle\hat{\nu}(N)=\frac{1}{E}N,$ (2.6)

where $E:=N_{1}/\nu_{1}$. As mentioned in the introduction, our main results
hold with minor modifications if instead of (2.6) we assume a linear relation
between the _strain_ $e:=\nu-1$ and the tension (Hooke’s law): there exists
$\eta\geq 0$ such that for all $N\in[\eta,N_{1}]$

$\displaystyle e=\hat{e}(N)=\frac{1}{E}N,$ (2.7)

where $E:=N_{1}/(\nu-1)$ is _Young’s modulus_ for the string. For
definiteness, our statements and proofs will be in the setting of the first
relation (2.6).

Strings satisfying the previous assumptions admit a stored energy functional
for motions satisfying $N>\eta$ and may be described as _elastic_ : there is
no dissipation for motions that are sufficiently smooth. Indeed, define the
stored energy functional via

$\displaystyle W(N,\nu)=1_{\\{N\leq N_{1}\\}}\frac{E}{2}\nu^{2}+1_{\\{N\geq
N_{1}\\}}\frac{E}{2}\nu_{1}^{2},$ (2.8)

the total stored energy of the material segment $[a,b]$,

$\displaystyle E(t):=\int_{a}^{b}W(N(s,t),\nu(s,t))ds.$ (2.9)

and the kinetic energy of the material segment $[a,b]$,

$\displaystyle
K(t):=\int_{a}^{b}\frac{\gamma}{2}|\boldsymbol{r}_{t}(s,t)|^{2}ds.$ (2.10)

Then as long as the state variables
$(\boldsymbol{r}_{t},\boldsymbol{r}_{s},\boldsymbol{n})\in
C^{1}([a,b]\times[0,T];\mathbb{R}^{9})$ and $N>\eta$ during the motion, an
easy calculation using (2.1) and (2.6) shows that for all $t\in[0,T]$

$\displaystyle
K^{\prime}(t)+E^{\prime}(t)=P(t):=\boldsymbol{n}(b,t)\cdot\boldsymbol{r}_{t}(b,t)-\boldsymbol{n}(a,t)\cdot\boldsymbol{r}_{t}(a,t)$
(2.11)

where $P(t)$ is the power exerted on the segment $[a,b]$ by the tensile forces
in the string. By the first law of thermodynamics (2.11) implies that no
mechanical work is converted to heat.

In this work we specialize our study to the longitudinal motion for a uniform,
tensile, stretch-limited string under no external body force, fixed at the end
$s=0$, pulled taut at the end at $s=\infty$ and containing an inextensible
segment and an extensible segment:

  * •

for all $(s,t)$,

$\displaystyle\boldsymbol{r}(s,t)=\chi(s,t)\boldsymbol{i},\quad\chi_{s}(s,t)>0,$
(2.12)

so

$\displaystyle\nu(s,t)=\chi_{s}(s,t),\quad\boldsymbol{n}(s,t)=N(s,t)\boldsymbol{i},$
(2.13) $\displaystyle\chi_{s}(s,t)=\begin{cases}\frac{1}{E}N(s,t)&\mbox{ if
}N(s,t)\in[\eta,N_{1}],\\\ \nu_{1}&\mbox{ if
}N(s,t)\in[N_{1},\infty),\end{cases}$ (2.14)

  * •

there is no body force, $\boldsymbol{f}(s,t)=\boldsymbol{0}$,

  * •

the boundary conditions

$\displaystyle\boldsymbol{r}(0,t)=\boldsymbol{0},\quad\boldsymbol{n}(\infty,t)=(\zeta
t+\tau)\boldsymbol{i},$ (2.15)

where $\zeta\geq 0$ and $\tau\in(\eta,N_{1})$,

  * •

there exists $\sigma\in C^{2}([0,T];(0,\infty))$ such that
$\sigma^{\prime}\geq 0$ and

$\displaystyle\forall s\in[0,\sigma(t)),\quad N(s,t)\in(N_{1},\infty),$ (2.16)
$\displaystyle\forall s\in(\sigma(t),\infty),\quad N(s,t)\in(\eta,N_{1}),$
(2.17)

  * •

and the generic condition that for all $t\in[0,T]$,

$\displaystyle N(\sigma^{-}(t),t)>N_{1}>N(\sigma^{+}(t),t).$ (2.18)

We refer to $[0,\sigma(t)]$ as the inextensible (material) segment,
$[\sigma(t),\infty)$ as the extensible (material) segment and the point
$s=\sigma(t)$ as the shock front. We note that the assumption
$\sigma^{\prime}\geq 0$ reflects the physical assumption that the inextensible
segment is _growing_ in time.

We consider weak solutions to the equations of motion (2.1) on
$[0,\infty)\times[0,T]$ which are also satisfied classically on

$\displaystyle\mathcal{I}_{\sigma}:=\\{(s,t):t\in[0,T],s\in[0,\sigma(t)]\\}$
(2.19)

and

$\displaystyle\mathcal{E}_{\sigma}:=\\{(s,t):t\in[0,T],s\in[\sigma(t),\infty)\\}$
(2.20)

separately. The property of $(\boldsymbol{r},\boldsymbol{n})$ being a weak
solution is equivalent to the Rankine-Hugoniot jump conditions:

$\displaystyle\llbracket\boldsymbol{n}\rrbracket+\sigma^{\prime}\llbracket\boldsymbol{r}_{t}\rrbracket=\boldsymbol{0},$
(2.21)

where
$\llbracket\boldsymbol{y}\rrbracket(\sigma(t),t)=\boldsymbol{y}(\sigma(t)^{+},t)-\boldsymbol{y}(\sigma(t)^{-},t)$
is the jump across the point $s=\sigma(t)$ at time $t$.

###  2.3. Reduced equations of motion and jump conditions

We now derive the reduced equations of motion from (2.1), the Rankine-Hugoniot
conditions (2.21) and our assumptions from the previous subsection. For
$(s,t)\in\mathcal{E}_{\sigma}$, where the string is extensible, (2.1) and the
boundary condition (2.15) reduce via (2.6) to the wave equation

$\displaystyle\begin{split}&\chi_{tt}(s,t)=\frac{E}{\gamma}\chi_{ss}(s,t),\\\
&\chi_{s}(\infty,t)=\frac{\zeta t+\tau}{E},\end{split}$ (2.22)

For $(s,t)\in\mathcal{I}_{\sigma}$, where the string is inextensible and
$\chi_{s}(s,t)=\nu_{1}$, we have

$\displaystyle\chi(s,t)=\int_{0}^{s}\nu_{1}d\xi=\nu_{1}s,$ (2.23)

and (2.1) reduces to $N_{s}=0$ so,

$\displaystyle\begin{split}N(s,t)&=N(\sigma^{-}(t),t)\\\
\chi(s,t)&=\nu_{1}s.\end{split}$ (2.24)

The Rankine-Hugoniot conditions (2.21) reduce to

$\displaystyle\llbracket
N\rrbracket+\gamma\sigma^{\prime}(t)\llbracket\chi_{t}\rrbracket=0.$ (2.25)

Imposing the condition that $\chi(s,t)$ is continuous across $s=\sigma$ (for
brevity we drop the dependence of $\sigma$ on $t$) implies

$\displaystyle\chi(\sigma^{+},t)=\nu_{1}\sigma=\frac{N_{1}}{E}\sigma$ (2.26)

and thus

$\displaystyle\chi_{t}(\sigma^{+},t)$
$\displaystyle=\frac{\sigma^{\prime}}{E}(N_{1}-N(\sigma^{+},t)).$ (2.27)

Since $\chi_{t}(s,t)=0$ for $s\in[0,\sigma]$, we conclude

$\displaystyle\gamma\sigma^{\prime}\llbracket\chi_{t}\rrbracket=\frac{\gamma(\sigma^{\prime})^{2}}{E}(N_{1}-N(\sigma^{+},t))$
(2.28)

which by (2.25) yields

$\displaystyle
N(\sigma^{-},t)=N(\sigma^{+},t)+\frac{\gamma(\sigma^{\prime})^{2}}{E}(N_{1}-N(\sigma^{+},t)).$
(2.29)

Thus, in the extensible region $\mathcal{I}_{\sigma}$,

$\displaystyle\begin{split}N(s,t)&=N(\sigma^{+},t)+\frac{\gamma(\sigma^{\prime})^{2}}{E}(N_{1}-N(\sigma^{+},t)),\\\
\chi(s,t)&=\nu_{1}s.\end{split}$ (2.30)

In summary, the equations of motion (2.1) and Rankine-Hugoniot conditions
(2.21) are equivalent to the equations (2.22) in the region
$\mathcal{E}_{\sigma}$, the equations (2.30) in the region
$\mathcal{I}_{\sigma}$, the condition (2.26) and our assumption (2.18).

From (2.18) and (2.30), it follows that the shock speed is greater than the
speed of propagation in the extensible region to the right of $\sigma(t)$:

$\displaystyle\infty>\sigma^{\prime}(t)>\sqrt{E/\gamma}.$ (2.31)

Since the speed of propagation of the inextensible region to the left of
$\sigma(t)$ is infinite, (2.31) reflects that the solutions we study satisfy a
generalized version of _Lax’s entropy inequality_ [9]. By choosing appropriate
units, we may assume that

$\displaystyle\gamma=E=1,$ (2.32)

so that $N_{1}=\nu_{1}$ and for all $(s,t)\in\mathcal{E}_{\sigma}$

$\displaystyle N(s,t)=\chi_{s}(s,t).$ (2.33)

##  3\. Existence and Uniqueness of Shock Solutions

We now define precisely the class of initial data and shock solutions to the
equations of motion that we study in this work and prove our existence and
uniqueness result.

###  3.1. Preliminaries

######  Definition 3.1.

We say a triple of real-valued functions

$\displaystyle(\chi_{0},\chi_{1},N_{0})\in C([0,\infty))\times
L^{\infty}([0,\infty))\times L^{\infty}([0,\infty)).$ (3.1)

is a set of classical _shock front initial data_ if there exists a unique
$\sigma_{0}\in(0,\infty)$ (the initial shock front) such that

  1. (1)

$(\chi_{0},\chi_{1})\in C^{2}\times C^{1}([\sigma_{0},\infty))$ and $N_{0}\in
C^{1}([\sigma_{0},\infty))$,

  2. (2)

for all $s\in[\sigma_{0},\infty)$, $N_{0}(s)=\chi_{0}^{\prime}(s)$ and

$\displaystyle\eta<N_{0}(s)<N_{1},$ (3.2)

  3. (3)

$\chi_{0}^{\prime}(\infty)=\tau,\quad\chi_{1}^{\prime}(\infty)=\zeta,\quad\chi_{0}(\sigma_{0})=N_{1}\sigma_{0}$,

  4. (4)

$\sigma_{1}:=\frac{\chi_{1}(\sigma_{0}^{+})}{N_{1}-\chi_{0}^{\prime}(\sigma_{0}^{+})}>1,$

  5. (5)

for all $s\in[0,\sigma_{0}]$,

$\displaystyle\begin{split}\chi_{0}(s)&=N_{1}s,\\\ \chi_{1}(s)&=0,\\\
N_{0}(s)&=N_{0}(\sigma_{0}^{+})+(\sigma_{1})^{2}(N_{1}-N_{0}(\sigma_{0}^{+}))>N_{1}.\end{split}$
(3.3)

We remark that a simple class of shock front initial data is given by

$\displaystyle\chi_{0}(s)=N_{1}\sigma_{0}+\tau(s-\sigma_{0}),\quad
s\in[\sigma_{0},\infty).$ (3.4)

and _any_ $\chi_{1}\in C^{1}([\sigma_{0},\infty))$ satisfying

$\displaystyle\chi_{1}^{\prime}(\infty)=\zeta,\quad\chi_{1}(\sigma_{0})>(N_{1}-\tau).$
(3.5)

(This data $(\chi_{0},\chi_{1},N_{0})|_{[\sigma_{0},\infty)}$ is extended to
$[0,\sigma_{0}]$ via (3.3).)

######  Definition 3.2.

Let $(\chi_{0},\chi_{1},N_{0})$ be a set of classical shock front initial data
with initial shock front $\sigma_{0}\in(0,\infty)$. We say a pair

$\displaystyle(\chi,N)\in C([0,\infty)\times[0,T])\times
L^{\infty}([0,T]\times[0,\infty))$ (3.6)

is a classical _shock solution_ on the time interval $[0,T]$ with initial data
$(\chi_{0},\chi_{1},N_{0})$ if $\partial_{t}\chi\in
L^{\infty}([0,\infty)\times[0,T])$ and there exists a unique curve

$\sigma\in C^{2}([0,T];[\sigma_{0},\infty)),$

the shock front, with $\sigma(0)=\sigma_{0}$ such that

  1. (1)

$\chi\in C^{2}(\mathcal{E}_{\sigma})$ and $N\in C^{1}(\mathcal{E}_{\sigma})$
where, as before,

$\displaystyle\mathcal{E}_{\sigma}:=\\{(s,t):t\in[0,T],s\in[\sigma(t),\infty)\\},$
(3.7)

  2. (2)

on $\mathcal{E}_{\sigma}$, $N(s,t)=\chi_{s}(s,t)$ and

$\displaystyle\eta<N(s,t)<N_{1},$ (3.8)

  3. (3)

$\chi(\sigma^{+},t)=N_{1}\sigma$,

  4. (4)

$\sigma^{\prime}=\frac{\chi_{t}(\sigma^{+},t)}{N_{1}-N(\sigma^{+},t)}>1$,

  5. (5)

on $\mathcal{I}_{\sigma}:=\\{(s,t):t\in[0,T],s\in[0,\sigma]\\}$,

$\displaystyle\begin{split}\chi(s,t)&=N_{1}s,\\\
N(s,t)&=N(\sigma^{+},t)+(\sigma^{\prime})^{2}(N_{1}-N(\sigma^{+},t))>N_{1}.\end{split}$
(3.9)

  6. (6)

on $\mathcal{E}_{\sigma}$, $\chi$ satisfies the wave equation

$\displaystyle\begin{split}&\chi_{tt}(s,t)=\chi_{ss}(s,t),\\\
&\chi_{s}(\infty,t)=\zeta t+\tau,\\\
&\chi(s,0)=\chi_{0}(s),\chi_{t}(s,0)=\chi_{1}(s),\end{split}$ (3.10)

###  3.2. Existence and uniqueness of classical shock solutions

We now prove the existence and uniqueness of classical shock solutions for
given classical shock front initial data.

######  Theorem 3.3.

Suppose $(\chi_{0},\chi_{1},N_{0})$ is a set of classical shock front initial
data with initial shock front $\sigma_{0}\in(0,\infty)$. Then there exist
$T=\hat{T}(\chi_{0},\chi_{1})>0$ and a unique classical shock solution
$(\chi,N)$ to the equations of motion on the time interval $[0,T]$.

###### Proof.

(Existence) We first extend the initial data

$(\chi_{0},\chi_{1})\in C^{2}\times C^{1}([\sigma_{0},\infty))$

to a pair

$(\hat{\chi}_{0},\hat{\chi}_{1})\in C^{2}\times C^{1}(\mathbb{R})$

such that for all $s\in[\sigma_{0}-\delta,\infty)$

$\displaystyle N_{0}<\hat{\chi}_{0}^{\prime}(s)<N_{1}.$ (3.11)

Let $\hat{\chi}\in C^{2}(\mathbb{R}^{2}_{s,t})$ be the unique solution to the
wave equation

$\displaystyle\hat{\chi}_{tt}(s,t)=\hat{\chi}_{ss}(s,t),\quad(s,t)\in\mathbb{R}^{2}_{s,t},$
(3.12)
$\displaystyle\hat{\chi}(s,0)=\hat{\chi}_{0}(s),\hat{\chi}_{t}(s,0)=\hat{\chi}_{1}(s),\quad
s\in\mathbb{R}.$ (3.13)

By D’Alembert’s formula

$\displaystyle\hat{\chi}(s,t)=\frac{1}{2}\Bigl{(}\hat{\chi}_{0}(s+t)+\hat{\chi}_{0}(s-t)\Bigr{)}+\frac{1}{2}\int_{s-t}^{s+t}\hat{\chi}_{1}(\xi)d\xi$
(3.14)

and the assumptions on the data, we have for all $t\in\mathbb{R}$,

$\displaystyle\hat{\chi}_{s}(\infty,t)=\zeta t+\tau.$ (3.15)

By continuity and choosing $T_{0}>0$ sufficiently small, we have for all
$(s,t)\in[\sigma_{0}-\delta,\infty)\times[-T_{0},T_{0}]$,

$\displaystyle\eta<\hat{\chi}_{s}(s,t)<N_{1}.$ (3.16)

Consider the function

$\displaystyle
f(\sigma,t)=N_{1}\sigma-\hat{\chi}(\sigma,t),\quad(\sigma,t)\in\mathbb{R}\times[-T_{0},T_{0}].$
(3.17)

Then $f(\sigma_{0},0)=N_{1}\sigma_{0}-\chi_{0}(\sigma_{0})=0$, and

$\displaystyle
f_{\sigma}(\sigma_{0},0)={N_{1}}-\chi_{0}^{\prime}(\sigma_{0}^{+})=N_{1}-N_{0}(\sigma_{0}^{+})>0.$
(3.18)

By the implicit function theorem, there exists $T_{1}\in(0,T_{2})$,
$\epsilon>0$ such that for all $t\in[-T_{1},T_{1}]$ there exists a unique
$\hat{\sigma}(t)\in[\sigma_{0}-\epsilon,\sigma_{0}+\epsilon]$ such that
$f(\hat{\sigma}(t),t)=0$. Moreover, by differentiating the equation
$f(\hat{\sigma}(t),t)=0$ in time, we have

$\displaystyle\hat{\sigma}^{\prime}(t)=\frac{\hat{\chi}_{t}(\hat{\sigma}(t),t)}{N_{1}-\hat{\chi}_{s}(\hat{\sigma}(t),t)}$
(3.19)

which implies $\hat{\sigma}^{\prime}(t)$ is $C^{1}$ and

$\displaystyle\hat{\sigma}^{\prime}(0)=\sigma_{1}=\frac{\chi_{1}(\sigma_{0}^{+})}{N_{1}-N_{0}(\sigma_{0}^{+})}>1.$
(3.20)

By choosing $\hat{T}<T_{1}$ sufficiently small, we can ensure that for all
$t\in[0,\hat{T}]$

$\displaystyle\hat{\chi}(\hat{\sigma}(t),t)={N_{1}}\hat{\sigma}(t),\quad\hat{\sigma}^{\prime}(t)>1.$
(3.21)

The latter inequality along with $N_{1}>\hat{\chi}_{s}(\hat{\sigma}(t),t)$
implies

$\displaystyle\hat{\chi}_{s}(\hat{\sigma}(t),t)+{(\hat{\sigma}^{\prime}(t))^{2}}(N_{1}-\hat{\chi}_{s}(\hat{\sigma}(t),t))>N_{1}.$
(3.22)

Defining $T:=\hat{T}$, $\sigma:=\hat{\sigma}$,
$(\chi,N)|_{\mathcal{E}_{\sigma}}:=(\hat{\chi},\hat{\chi}_{s})|_{\mathcal{E}_{\sigma}}$
and $(\chi,N)|_{\mathcal{I}_{\sigma}}$ via (3.9) we obtain a shock solution on
$[0,T]$ with initial data $(\chi_{0},\chi_{1},N_{0})$.

(Uniqueness) Suppose $(\check{\chi},\check{N})$ is another shock solution on
$[0,T]$ with initial data $(\chi_{0},\chi_{1},N_{0})$ and initial shock front
$\sigma_{0}$. Let $\bar{\chi}$ be the unique $C^{2}$ solution to the wave
equation (3.10) on $\\{(s,t):t\in[0,T],s\in[\sigma_{0}+t,\infty)\\}$. By
finite speed of propagation, $\bar{\chi}$ is a $C^{2}$ extension of
$\chi|_{\mathcal{E}_{\sigma}}$ and
$\check{\chi}|_{\mathcal{E}_{\check{\sigma}}}$, and thus, $\sigma$ and
$\check{\sigma}$ solve the ordinary differential equation

$\displaystyle\bar{\sigma}^{\prime}=\frac{\bar{\chi}_{t}(\bar{\sigma},t)}{N_{1}-\bar{\chi}_{s}(\bar{\sigma},t)},\quad\bar{\sigma}(0)=\sigma_{0},$
(3.23)

which implies $\sigma=\check{\sigma}$. We conclude
$\chi|_{\mathcal{E}_{\sigma}}=\bar{\chi}|_{\mathcal{E}_{\sigma}}=\check{\chi}|_{\mathcal{E}_{\sigma}}$,
proving uniqueness. ∎

We note that our uniqueness proof does not rely on $T$ being the same as that
from the existence proof. Therefore, we define the _final time of existence_
$T_{+}$ for a shock solution $(\chi,N)$ to be the largest time such that
$(\chi,N)$ is a shock solution on $[0,T_{+})$. We refer to $[0,T_{+})$ as the
_maximal interval of existence_ for $(\chi,N)$.

By finite speed of propagation we have the following simple observation that
we can identify a classical shock solution $(\chi,N)$ on
$\mathcal{E}_{\sigma}$ with the restriction of a solution to the wave equation
defined on the strictly larger domain
$\\{(s,t):t\in[0,T],s\in[\sigma(0)+t,\infty)\\}$.

######  Proposition 3.4.

Suppose that $(\chi,N)$ is a classical shock solution on a time interval
$[0,T)$ with shock front $\sigma$ and initial data
$(\chi_{0},\chi_{1},N_{0})$. Let $\bar{\chi}$ be the unique $C^{2}$ solution
to the wave equation (3.10) on
$\\{(s,t):t\in[0,\infty),s\in[\sigma(0)+t,\infty)\\}$. Then

$\displaystyle\mathcal{E}_{\sigma}\subset\\{(s,t):t\in[0,\infty),s\in[\sigma(0)+t,\infty)\\},$
(3.24)

and $\chi_{\mathcal{E}_{\sigma}}=\bar{\chi}_{\mathcal{E}_{\sigma}}$.
Consequentially, if

$\displaystyle\sigma(T):=\lim_{t\rightarrow T}\sigma(t)<\infty,$ (3.25)

and,

$\displaystyle\eta<\bar{\chi}_{s}(s,T)<N_{1},\quad s\in[\sigma(T),\infty),$
(3.26)
$\displaystyle\frac{\bar{\chi}_{t}(\sigma(T),T)}{N_{1}-\bar{\chi}_{s}(\sigma(T),T)}>1,$
(3.27)

then $(\chi,N)$ extends uniquely to a shock solution on $[0,\bar{T}]$ with
$\bar{T}>T$.

###  3.3. Dissipation

We conclude this section by showing that these motions dissipate the
mechanical energy of material segments.

######  Proposition 3.5.

Suppose $(\chi,N)$ is a shock solution to the equations of motion on the time
interval $[0,T]$ with shock front $\sigma$, and suppose that
$a,b\in(0,\infty)$ with $a<\sigma(t)<b$. Define the total energy (the sum of
the total kinetic energy and total stored energy of the string) of the
material segment $[a,b]$ via

$\displaystyle K(t)+E(t)$
$\displaystyle:=\int_{\sigma}^{b}\frac{1}{2}(\chi_{t}(s,t))^{2}ds+\int_{a}^{\sigma}\frac{1}{2}N_{1}^{2}ds+\int_{\sigma}^{b}\frac{1}{2}(N(s,t))^{2}ds.$
(3.28)

Then for all $t\in[0,T]$,

$K^{\prime}(t)+E^{\prime}(t)=P(t)+Q(t)$

where $P(t)$ is the total power exerted by the tensile forces on the segment
$[a,b]$,

$\displaystyle
P(t):=N(b,t)\chi_{t}(b,t)-N(a,t)\chi_{t}(a,t)=N(b,t)\chi_{t}(b,t),$ (3.29)

and $Q(t)$ is the total heat power,

$\displaystyle
Q(t):=-\frac{\sigma^{\prime}}{2}(N_{1}-N(\sigma^{+},t))^{2}((\sigma^{\prime})^{2}-1)<0.$
(3.30)

###### Proof.

Using the relation $N=\chi_{s}$ on $\mathcal{E}_{\sigma}$, the equations of
motion (3.10) and integration by parts we deduce

$\displaystyle K^{\prime}+E^{\prime}=P+Q$ (3.31)

where

$\displaystyle-Q(t)=\chi_{t}(\sigma^{+},t)N(\sigma^{+},t)+\frac{\sigma^{\prime}}{2}(\chi_{t}(\sigma^{+},t))^{2}+\frac{\sigma^{\prime}}{2}(N(\sigma^{+},t)^{2}-N_{1}^{2}).$
(3.32)

By the Rankine-Hugoniot conditions and the relation

$\llbracket\chi_{t}\rrbracket(\sigma^{+},t)=\chi_{t}(\sigma^{+},t)=\sigma^{\prime}(N_{1}-N(\sigma^{+},t))$

we deduce

$\displaystyle\chi_{t}(\sigma^{+},t)N(\sigma^{+},t)$
$\displaystyle+\frac{1}{2}\sigma^{\prime}(\chi_{t}(\sigma^{+},t))^{2}$ (3.33)
$\displaystyle=\llbracket\chi_{t}\rrbracket(\sigma^{+},t)\left(N(\sigma^{+},t)+\frac{1}{2}\sigma^{\prime}\llbracket\chi_{t}\rrbracket(\sigma^{+},t)\right)$
(3.34)
$\displaystyle=\llbracket\chi_{t}\rrbracket(\sigma^{+},t)\frac{N(\sigma^{+},t)+N(\sigma^{-},t)}{2}$
(3.35)
$\displaystyle=\frac{\sigma^{\prime}}{2}(N_{1}-N(\sigma^{+},t))(N(\sigma^{+},t)+N(\sigma^{-},t)).$
(3.36)

By Definition 3.2 we conclude

$\displaystyle-Q(t)$
$\displaystyle=\frac{\sigma^{\prime}}{2}\Bigl{(}(N_{1}-N(\sigma^{+},t))(N(\sigma^{+},t)+N(\sigma^{-},t))$
(3.37) $\displaystyle\qquad-(N_{1}^{2}-N(\sigma^{+},t)^{2})\Bigr{)}$ (3.38)
$\displaystyle=\frac{\sigma^{\prime}}{2}(N_{1}-N(\sigma+,t))(N(\sigma^{-},t)-N_{1})$
(3.39)
$\displaystyle=\frac{\sigma^{\prime}}{2}(N_{1}-N(\sigma^{+},t))^{2}((\sigma^{\prime})^{2}-1)>0$
(3.40)

as desired. ∎

##  4\. Piece-wise constant stretched motions

In this section we prove the orbital asymptotic stability of an explicit two-
parameter family of piece-wise constant stretched motions.

###  4.1. The two-parameter family of piece-wise constant stretched motions

We define

$\displaystyle T(\zeta):=\begin{cases}\infty&\mbox{ if }\zeta=0,\\\
\frac{N_{1}-\tau}{\zeta}&\mbox{ if }\zeta>0.\end{cases}$ (4.1)

When $\zeta>0$, this is precisely the time when the tension at $s=\infty$
surpasses the threshold tensile value, $N_{1}$, for inextensibility. In
particular, the solutions to the equations of motion having one inextensible
segment and one extensible segment satisfy $T_{+}\leq T(\zeta)$.

We first exhibit the solutions which have piece-wise constant stretch,

$\displaystyle\nu(s,t)=\chi_{s}(s,t)=\begin{cases}N_{1}&\mbox{ if
}s\in[0,\sigma(t)],\\\ \zeta t+\tau&\mbox{ if
}s\in[\sigma(t),\infty),\end{cases}$ (4.2)

where $\sigma$ is the shock front separating the inextensible segment
$[0,\sigma]$ from the extensible segment $[\sigma,\infty)$. Via simple
calculations, we have the following.

######  Proposition 4.1.

Let $(\sigma_{0},\sigma_{1})\in(0,\infty)\times(1,\infty)$. Define a shock
front $\sigma$ and pair of functions for $(s,t)\in\mathcal{E}_{\sigma}$ by

$\displaystyle\begin{split}\sigma(t;\zeta,\sigma_{0},\sigma_{1})&=\sigma_{0}+\frac{\sigma_{1}(N_{1}-\tau)t}{N_{1}-(\zeta
t+\tau)},\\\ \chi(s,t;\zeta,\sigma_{0},\sigma_{1})&=N_{1}\sigma_{0}+(\zeta
t+\tau)(s-\sigma_{0})+\sigma_{1}(N_{1}-\tau)t,\\\
N(s,t;\zeta,\sigma_{0},\sigma_{1})&=\zeta t+\tau,\end{split}$ (4.3)

extended to $\mathcal{I}_{\sigma}$ via (3.9). Then (4.3) defines the unique
solution to the equations of motion on the maximal interval of existence
$[0,T(\zeta))$ satisfying (4.2), $\sigma(0)=\sigma_{0}$ and
$\sigma^{\prime}(0)=\sigma_{1}$.

The values of the following state variables corresponding to a piece-wise
constant stretched motion with parameters
$(\sigma_{0},\sigma_{1})\in(0,\infty)\times(1,\infty)$ (and omitting the
dependence of the solution on $\zeta,\sigma_{0},\sigma_{1}$) are given by: for
$s\in[\sigma(t),\infty)$

$\displaystyle\chi_{s}(s,t)=\zeta
t+\tau,\quad\chi_{t}(s,t)=\zeta(s-\sigma_{0})+\sigma_{1}(N_{1}-\tau),$ (4.4)
$\displaystyle\sigma^{\prime}(t)=\sigma_{1}\frac{(N_{1}-\tau)^{2}}{(N_{1}-(\zeta
t+\tau))^{2}}=\frac{\zeta(\sigma(t)-\sigma_{0})+\sigma_{1}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)},$ (4.5)

and for $s\in[0,\sigma(t)]$

$\displaystyle N(s,t)=\zeta
t+\tau+\frac{[\zeta(\sigma(t)-\sigma_{0})+\sigma_{1}(N_{1}-\tau)]^{2}}{N_{1}-(\zeta
t+\tau)}.$ (4.6)

###  4.2. Perturbations of piece-wise constant stretched motions

We first note that the piece-wise constant stretched motions described in the
previous subsection are, in general, _unstable_ to small $L^{\infty}$
perturbations of the state variables. Indeed, suppose $\zeta>0$ and
$(\sigma_{0},\sigma_{1})\in(0,\infty)\times(1,\infty)$. Then for all
$\epsilon>0$ and $R>\sigma_{0}$ one can construct initial data
$(\chi_{0},\chi_{1},N)$ with initial shock front $\sigma_{0}$ such that

$\displaystyle\tau\leq\chi_{0}^{\prime}(s)$
$\displaystyle\leq\tau+\epsilon,\quad s\in[\sigma_{0},\infty),$ (4.7)
$\displaystyle\chi_{0}^{\prime}(s)$ $\displaystyle=\begin{cases}\tau&\mbox{ if
}s\in[\sigma_{0},R]\cup[4R,\infty)\\\ \tau+\epsilon&\mbox{ if
}s\in[2R,3R]\end{cases},$ (4.8) $\displaystyle\chi_{1}(s)$
$\displaystyle=\zeta(s-\sigma_{0})+\sigma_{1}(N_{1}-\tau),\quad
s\in[\sigma_{0},\infty).$ (4.9)

Then

$\displaystyle\|\chi_{0}^{\prime}-\tau\|_{L^{\infty}[\sigma_{0},\infty)}+\|\chi_{1}-[\zeta(s-\sigma_{0})+\sigma_{1}(N_{1}-\tau)]\|_{L^{\infty}}\leq\epsilon.$
(4.10)

By choosing $\epsilon$ sufficiently small, $R$ sufficiently large and using
finite speed of propagation, we can then conclude using Proposition 3.4 that

$\displaystyle T_{+}=\frac{N_{1}-(\tau+\epsilon)}{\zeta}<T(\zeta),$ (4.11)

and for $t\in[0,T_{+}]$, $s\in[2R+t,3R-t]$,

$\displaystyle\chi_{s}(s,t)=\zeta t+\tau+\epsilon,$ (4.12)

i.e. the string has formed a second inextensible material segment before the
time $T(\zeta)$. The previous argument also previews why proving stability of
piece-wise constant stretch states is more delicate when $\zeta>0$: the
increasing tension at $s=\infty$ and the nature of the perturbation may result
in a second inextensible segment.

We now prove piece-wise constant stretch motions are orbitally asymptotically
stable under small _weighted_ $L^{\infty}$ perturbations of the state
variables and their derivatives. We begin with the simpler case $\zeta=0$.

######  Theorem 4.2.

Assume $\zeta=0$. Let $(\sigma_{0},\sigma_{1})\in(0,\infty)\times(1,\infty)$.
There exist $\epsilon_{0}>0$ and $C>0$ such that for all
$\epsilon<\epsilon_{0}$, the following is true. Suppose
$(\chi_{0},\chi_{1},N_{0})$ is a set of initial data with initial shock front
$\sigma_{0}$ such that there exists $r>0$ such that

$\displaystyle B:=\sup_{s\in[\sigma_{0},\infty)}\Bigl{[}|$
$\displaystyle\chi_{0}^{\prime}(s)-\tau|+s^{r+1}|\chi_{0}^{\prime\prime}(s)|$
(4.13)
$\displaystyle+|\chi_{1}(s)-\sigma_{1}(N_{1}-\tau)|+s^{r+1}|\chi_{1}^{\prime}(s)|\Bigr{]}<\epsilon.$
(4.14)

Then the unique solution $(\chi,N)$ to the equations of motion with initial
data $(\chi_{0},\chi_{1},N_{0})$ satisfies $T_{+}=T(0)=\infty$ and the state
variables and shock speed remain close to and asymptotically approach those of
a piece-wise constant stretched motion as $t\rightarrow\infty$: there exist
$\sigma_{1}^{\infty}\in(1,\infty)$ such that
$\bigl{|}\frac{\sigma_{1}^{\infty}}{\sigma_{1}}-1\bigr{|}\leq C\epsilon$ and
for all $t\in[0,\infty)$

$\displaystyle\begin{split}\sup_{s\in[\sigma(t),\infty)}&s^{r+1}[|\chi_{ss}(s,t)|+|\chi_{st}(s,t)|]\leq
CB,\\\
\sup_{s\in[\sigma(t),\infty)}&\Bigl{[}|\chi_{s}(s,t)-\tau|++|\chi_{t}(s,t)-\sigma_{1}^{\infty}(N_{1}-\tau)|\Bigr{]}\leq
CB(1+t)^{-r},\\\
\Bigl{|}&\frac{\sigma^{\prime}(t)}{\sigma_{1}^{\infty}}-1\Bigr{|}\leq
CB(1+t)^{-r},\\\
\sup_{s\in[0,\sigma(t)]}&\Bigl{|}N(s,t)([\tau+(\sigma_{1}^{\infty})^{2}(N_{1}-\tau)])^{-1}-1\Bigr{|}\leq
CB(1+t)^{-r}.\end{split}$ (4.15)

###### Proof.

We first note that by (4.14) and the fundamental theorem of calculus,

$\displaystyle|\chi_{0}^{\prime}(s)-\tau|=\left|\int_{s}^{\infty}\chi_{0}^{\prime\prime}(\xi)d\xi\right|\leq\frac{B}{r}s^{-r},$
(4.16)

$\sigma_{1}^{\infty}:=\frac{1}{N_{1}-\tau}\lim_{s\rightarrow\infty}\chi_{1}(s)$
exists and

$\displaystyle|\chi_{1}(s)-\sigma_{1}^{\infty}(N_{1}-\tau)|\leq\frac{B}{r}s^{-r},$
(4.17)

for all $s\in[\sigma_{0},\infty)$. In particular, by (4.14),

$\displaystyle|\sigma_{1}(N_{1}-\tau)-\sigma_{1}^{\infty}(N_{1}-\tau)|=|\sigma_{1}(N_{1}-\tau)-\chi_{1}(\infty)|\leq\epsilon,$
(4.18)

proving $\sigma_{1}^{\infty}\in(1,\infty)$ and

$\displaystyle\Bigl{|}\frac{\sigma_{1}^{\infty}}{\sigma_{1}}-1\Bigr{|}\leq\frac{\epsilon}{N_{1}-\tau}$
(4.19)

for all $\epsilon$ sufficiently small.

Let $(\chi,N)$ be the unique solution to the equations of motion with shock
front $\sigma$ and initial data $(\chi_{0},\chi_{1},N_{0})$ defined on a
maximal time interval of existence $[0,T_{+})$. Let $\bar{\chi}$ be the unique
$C^{2}$ function on
$\mathcal{D}_{\sigma_{0}}:=\\{(s,t):t\in[0,\infty),s\in[\sigma_{0}+t,\infty)\\}$
solving the wave equation

$\displaystyle\begin{split}&\bar{\chi}_{tt}(s,t)=\bar{\chi}_{ss}(s,t),\\\
&\bar{\chi}_{s}(\infty,t)=\tau,\end{split}$ (4.20)
$\displaystyle\bar{\chi}(s,0)=\chi_{0}(s),\bar{\chi}_{t}(s,0)=\chi_{1}(s).$
(4.21)

By D’Alembert’s formula:

$\displaystyle\bar{\chi}(s,t)=\frac{1}{2}\Bigl{(}\chi_{0}(s+t)+\chi_{0}(s-t)\Bigr{)}+\frac{1}{2}\int_{s-t}^{s+t}\chi_{1}(\xi)d\xi.$
(4.22)

By finite speed of propagation,

$\displaystyle\chi|_{\mathcal{E}_{\sigma}}=\bar{\chi}|_{\mathcal{E}_{\sigma}},$
(4.23)

and thus, the shock front $\sigma$ is the unique $C^{2}$ solution to

$\displaystyle\sigma^{\prime}(t)=\frac{\bar{\chi}_{t}(\sigma,t)}{N_{1}-\bar{\chi}_{s}(\sigma,t)},\quad\sigma(0)=\sigma_{0}.$
(4.24)

By D’Alembert’s formula we have for all
$t\in[0,\infty),s\in[\sigma(t),\infty)$

$\displaystyle|\bar{\chi}_{s}(s,t)$ $\displaystyle-\tau|$ (4.25)
$\displaystyle=\Bigl{|}\frac{1}{2}(\chi^{\prime}_{0}(s+t)-\tau)+\frac{1}{2}(\chi^{\prime}_{0}(s+t)-\tau)$
(4.26)
$\displaystyle\qquad+\frac{1}{2}(\chi_{1}(s+t)-\sigma_{1}^{\infty}(N_{1}-\tau))-\frac{1}{2}(\chi_{1}(s-t)-\sigma_{1}^{\infty}(N_{1}-\tau))\Bigr{|}$
(4.27) $\displaystyle\leq\frac{2B}{r}(\sigma-t)^{-r},$ (4.28)

and similarly

$\displaystyle|\bar{\chi}_{t}(s,t)-\sigma_{1}^{\infty}(N_{1}-\tau)|$
$\displaystyle\leq\frac{2B}{r}(\sigma-t)^{-r},$ (4.29) $\displaystyle
s^{r+1}[|\bar{\chi}_{ss}(s,t)|+|\bar{\chi}_{st}(s,t)|]$ $\displaystyle\leq
B(1+(s-t)^{-r-1}s^{r+1}).$ (4.30)

Since $\sigma-t\geq\sigma_{0}$, we conclude from (4.14), (4.28) and (4.29)
that there exists $C_{1}>0$ such that for all $\epsilon$ sufficiently small,
for all $(s,t)\in\mathcal{D}_{\sigma_{0}}$

$\displaystyle\eta<\tau-
C_{1}\epsilon<\bar{\chi}_{s}(s,t)<\tau+C_{1}\epsilon<N_{1},$ (4.31)
$\displaystyle\sigma_{1}^{\infty}(1-C_{1}\epsilon)<\frac{\bar{\chi}_{s}(\sigma,t)}{N_{1}-\bar{\chi}_{s}(\sigma,t)}<\sigma_{1}^{\infty}(1+C_{1}\epsilon).$
(4.32)

By Proposition 3.4, it follows that $T_{+}=T(0)=\infty$.

By (4.19), (4.24) and (4.32)

$\displaystyle\sigma-t$
$\displaystyle=\sigma_{0}+\int_{0}^{t}\sigma^{\prime}(\bar{t})d\bar{t}-t$
(4.33)
$\displaystyle\geq\sigma_{0}+\sigma_{1}\Bigl{(}1-\frac{\epsilon}{N_{1}-\tau}\Bigr{)}(1-C_{1}\epsilon)t-t$
(4.34) $\displaystyle\geq\sigma_{0}+\frac{1}{2}(\sigma_{1}-1)t$ (4.35)

for all $\epsilon$ sufficiently small. A similar argument also proves there
exists a constant $c>0$ depending on $\sigma_{0}$ and $\sigma_{1}$ such that
$(1-c)\sigma\geq t$ and thus for all $s\in[\sigma,\infty)$

$\displaystyle(s-t)^{-r-1}s^{r}\leq c^{-r},$ (4.36)

for all $\epsilon$ sufficiently small. Thus, by (4.23), (4.28), (4.29) and
(4.30) we conclude there exists $C>0$ such that for all $s\in[\sigma,\infty)$

$\displaystyle s^{r+1}[|\chi_{ss}(s,t)|+|\chi_{s,t}(s,t)|]$ $\displaystyle\leq
CB,$ (4.37)
$\displaystyle|\chi_{s}(s,t)-\tau|+|\chi_{t}(s,t)-\sigma_{1}^{\infty}(N_{1}-\tau)|$
$\displaystyle\leq CB(1+t)^{-r}.$ (4.38)

This proves the first two estimates in (4.15) which then immediately imply the
third and fourth estimates in (4.15) via (4.24) and (3.9). ∎

######  Theorem 4.3.

Assume $\zeta>0$. Let $(\sigma_{0},\sigma_{1})\in(0,\infty)\times(1,\infty)$.
There exist $\epsilon_{0}>0$ and $C>0$ such that for all
$\epsilon<\epsilon_{0}$, the following is true. Suppose
$(\chi_{0},\chi_{1},N_{0})$ is a set of initial data with initial shock front
$\sigma_{0}$ such that there exists $r>0$ such that

$\displaystyle B:=\sup_{s\in[\sigma_{0},\infty)}\Bigl{[}|$
$\displaystyle\chi_{0}^{\prime}(s)-\tau|+s^{r+2}|\chi_{0}^{\prime\prime}(s)|$
(4.39)
$\displaystyle+|\chi_{1}(s)-\sigma_{1}(N_{1}-\tau)|+s^{r+2}|\chi_{1}^{\prime}(s)|\Bigr{]}<\epsilon.$
(4.40)

Then the unique solution $(\chi,N)$ to the equations of motion with initial
data $(\chi_{0},\chi_{1},N_{0})$ satisfies $T_{+}=T(\zeta)$ and the state
variables and shock speed remain close to and asymptotically approach those of
a piece-wise constant stretched motion as $t\rightarrow T(\zeta)$: there exist
$\sigma_{1}^{\infty}\in(1,\infty)$ such that
$\bigl{|}\frac{\sigma_{1}^{\infty}}{\sigma_{1}}-1\bigr{|}\leq C\epsilon$ and
for all $t\in[0,T_{+}(\zeta))$

$\displaystyle\begin{split}\sup_{s\in[\sigma(t),\infty)}&s^{r+2}[|\chi_{ss}(s,t)|+|\chi_{st}(s,t)|]\leq
CB,\\\ \sup_{s\in[\sigma(t),\infty)}&\Bigl{[}|\chi_{s}(s,t)-(\zeta
t+\tau)|+|\chi_{t}(s,t)-[\zeta(s-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)]|\Bigr{]}\\\
&\qquad\leq CB(T(\zeta)-t)^{1+r},\\\
&\left|\sigma^{\prime}(t)\Bigl{(}\frac{\zeta(\sigma-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1\right|\leq CB(T(\zeta)-t)^{r},\\\
\sup_{s\in[0,\sigma(t)]}&\left|N(s,t)\Bigl{(}\frac{[\zeta(\sigma-\sigma_{0})+\sigma^{\infty}_{1}(N_{1}-\tau)]^{2}}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1\right|\leq CB(T(\zeta)-t)^{r}.\end{split}$ (4.41)

###### Proof.

As in the previous proof, it follows from (4.40) and the fundamental theorem
of calculus that there for all $\epsilon$ sufficiently small, there exists
$\sigma_{1}^{\infty}\in(1,\infty)$ with
$|\frac{\sigma_{1}^{\infty}}{\sigma_{1}}-1|\leq\frac{\epsilon}{N_{1}-\tau}$
such that for all $s\in[\sigma_{0},\infty)$

$\displaystyle\begin{split}|\chi_{0}^{\prime}(s)-\tau|&\leq\frac{B}{r+1}s^{-1-r},\\\
|\chi_{1}(s)-(\zeta(s-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau))|&\leq\frac{B}{r+1}s^{-1-r}.\end{split}$
(4.42)

In particular, there exists $C_{1}>0$ depending on
$\sigma_{0},\sigma_{1},\zeta$ and $N_{1}-\tau$ such that

$\displaystyle\Bigl{|}\frac{\sigma^{\prime}(0)}{\sigma_{1}^{\infty}}-1\Bigr{|}\leq
C_{1}BT(\zeta)^{r}$ (4.43)

Let $(\chi,N)$ be the unique solution to the equations of motion with shock
front $\sigma$ and initial data $(\chi_{0},\chi_{1},N_{0})$ defined on a
maximal time interval of existence $[0,T_{+})$. Let $\bar{\chi}$ be the unique
$C^{2}$ function on
$\mathcal{D}_{\sigma_{0}}:=\\{(s,t):t\in[0,T(\zeta)),s\in[\sigma_{0}+t,\infty)\\}$
solving the wave equation

$\displaystyle\begin{split}&\bar{\chi}_{tt}(s,t)=\bar{\chi}_{ss}(s,t),\\\
&\bar{\chi}_{s}(\infty,t)=\zeta t+\tau,\end{split}$ (4.44)
$\displaystyle\bar{\chi}(s,0)=\chi_{0}(s),\bar{\chi}_{t}(s,0)=\chi_{1}(s).$
(4.45)

As before, using (4.42) and D’Alembert’s formula, we conclude the bounds for
all $t\in[0,T(\zeta)),s\in[\sigma(t),\infty)$,

$\displaystyle\begin{split}|\bar{\chi}_{s}(s,t)-(\zeta
t+\tau)|&\leq\frac{2B}{r+1}(\sigma-t)^{-1-r},\\\
|\bar{\chi}_{t}(s,t)-[\zeta(s-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)]|&\leq\frac{2B}{r+1}(\sigma-t)^{-1-r},\\\
s^{r+1}[|\bar{\chi}_{ss}(s,t)|+|\bar{\chi}_{st}(s,t)|]&\leq
B(1+(s-t)^{-r-2}s^{r+2}),\end{split}$ (4.46)

To control the shock speed and state variables, we proceed via a bootstrap
argument. We claim that there exists a constant $C_{0}\geq C_{1}$ depending
only on $\sigma_{0},\sigma_{1},\zeta$ and $N_{1}-\tau$ such that for all
$\epsilon$ sufficiently small, the following is true: if $T<T_{+}$ and for all
$t\in[0,T]$ we have

$\displaystyle\left|\sigma^{\prime}(t)\Bigl{(}\frac{\zeta(\sigma(t)-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1\right|\leq 2C_{0}B(T(\zeta)-t)^{r},$ (4.47)

then for all $t\in[0,T]$ we have

$\displaystyle\left|\sigma^{\prime}(t)\Bigl{(}\frac{\zeta(\sigma(t)-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1\right|\leq C_{0}B(T(\zeta)-t)^{r}.$ (4.48)

Assume that (4.47) holds with $C_{0}$ to be determined. Let

$\displaystyle\delta(t)$
$\displaystyle:=\sigma^{\prime}(t)\Bigl{(}\frac{\zeta(\sigma-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1,$ (4.49) $\displaystyle f(t)$
$\displaystyle:=\int_{0}^{t}\frac{\zeta(1+\delta(\bar{t}))}{N_{1}-(\zeta\bar{t}+\tau)}d\bar{t},$
(4.50)

so

$\displaystyle f^{\prime}(t)=\frac{\zeta(1+\delta(t))}{N_{1}-(\zeta t+\tau)}.$
(4.51)

Then

$\displaystyle(\sigma-\sigma_{0})^{\prime}-\frac{\zeta(1+\delta(t))}{N_{1}-(\zeta
t+\tau)}(\sigma-\sigma_{0})=\frac{\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}(1+\delta(t))$ (4.52)

which implies

$\displaystyle\frac{d}{dt}[(\sigma-\sigma_{0})\exp(-f(t))]=\frac{\sigma_{1}^{\infty}(N_{1}-\tau)}{-\zeta}\frac{d}{dt}\exp(-f(t)).$
(4.53)

Integrating, we conclude that

$\displaystyle\sigma-\sigma_{0}$
$\displaystyle=\frac{\sigma_{1}^{\infty}(N_{1}-\tau)}{\zeta}(\exp f(t)-1)$
(4.54)
$\displaystyle=\frac{\sigma_{1}^{\infty}(N_{1}-\tau)}{\zeta(N_{1}-(\zeta
t+\tau))}\bigl{(}(N_{1}-\tau)\exp g(t)-(N_{1}-(\zeta t+\tau))\bigr{)}$ (4.55)
$\displaystyle=\frac{\sigma_{1}^{\infty}T(\zeta)}{T(\zeta)-t}\bigl{(}t+T(\zeta)(\exp
g(t)-1)\bigr{)}$ (4.56)

where

$\displaystyle
g(t):=\int_{0}^{t}\frac{\zeta\delta(\bar{t})}{N_{1}-(\zeta\bar{t}+\tau)}d\bar{t}$
(4.57)

satisfies, via (4.47),

$\displaystyle|g(t)|\leq\frac{2C_{0}(T(\zeta))^{r}}{r}\epsilon.$ (4.58)

Then for all
$\epsilon<\frac{r}{8C_{0}\sigma_{1}^{\infty}T(\zeta)^{r+1}}\min\\{\sigma_{0},T(\zeta)(\sigma_{1}^{\infty}-1)\\}$,

$\displaystyle(T(\zeta)-t)(\sigma-t)$
$\displaystyle=\sigma_{0}(T(\zeta)-t)+\sigma_{1}^{\infty}T(\zeta)(t+T(\zeta)(\exp
g(t)-1))$ (4.59) $\displaystyle\quad-T(\zeta)t+t^{2}$ (4.60)
$\displaystyle\geq\sigma_{0}(T(\zeta)-t)+T(\zeta)(\sigma_{1}^{\infty}-1)t+\sigma_{1}^{\infty}T(\zeta)^{2}(\exp
g(t)-1)$ (4.61) $\displaystyle\geq
T(\zeta)\min\\{\sigma_{0},T(\zeta)(\sigma_{1}^{\infty}-1)\\}-\frac{4\sigma_{1}^{\infty}T(\zeta)^{r+2}C_{0}}{r}\epsilon$
(4.62)
$\displaystyle\geq\frac{1}{2}T(\zeta)\min\\{\sigma_{0},T(\zeta)(\sigma_{1}^{\infty}-1)\\}$
(4.63)

Thus, for all $p>0$

$\displaystyle(\sigma-t)^{-p}\leq\Bigl{(}\frac{1}{2}T(\zeta)\min\\{\sigma_{0},T(\zeta)(\sigma_{1}^{\infty}-1)\\}\Bigr{)}^{-p}(T(\zeta)-t)^{-p}.$
(4.64)

Inserting (4.64) into (4.46) implies that there exists an explicit constant
$C_{2}>0$ depending on $\sigma_{0},\sigma_{1},\zeta$ and $N_{1}-\tau$ such
that

$\displaystyle\begin{split}\left|\frac{\bar{\chi}_{t}(\sigma,t)}{\zeta(\sigma-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}-1\right|&\leq
C_{2}B(T(\zeta)-t)^{1+r},\\\
\left|\frac{N_{1}-\bar{\chi}_{s}(\sigma,t)}{N_{1}-(\zeta
t+\tau)}-1\right|&\leq C_{2}B(T(\zeta)-t)^{r}.\end{split}$ (4.65)

Thus, via the relation
$\sigma^{\prime}=\frac{\bar{\chi}_{t}(\sigma,t)}{N_{1}-\bar{\chi}_{s}(\sigma,t)}$
we conclude that there exists an explicit constant $C_{3}>0$ depending on
$\sigma_{0},\sigma_{1},\zeta$ and $N_{1}-\tau$ such that for all $\epsilon$
sufficiently small

$\displaystyle\left|\sigma^{\prime}(t)\Bigl{(}\frac{\zeta(\sigma-\sigma_{0})+\sigma_{1}^{\infty}(N_{1}-\tau)}{N_{1}-(\zeta
t+\tau)}\Bigr{)}^{-1}-1\right|\leq C_{3}B(T(\zeta)-t)^{r}$ (4.66)

If we define $C_{0}:=\max\\{C_{1},C_{3}\\}$, then we have proved that (4.47)
implies (4.48) for all
$\epsilon<\frac{r}{8C_{0}\sigma_{1}^{\infty}T(\zeta)^{r+1}}\min\\{\sigma_{0},T(\zeta)(\sigma_{1}^{\infty}-1)\\}$
sufficiently small. This proves the claim.

By the claim and a continuity argument, if follows that the bound (4.47) and
its consequences hold on $[0,T_{+})$ for all $\epsilon$ sufficiently small. By
(4.47), (4.65) and Proposition 3.4, it follows that $T_{+}=T(\zeta)$ for all
$\epsilon$ sufficiently small. Then (4.47), (4.46), (4.64), (3.9) and the fact
$\chi|_{\mathcal{E}_{\sigma}}=\bar{\chi}|_{\mathcal{E}_{\sigma}}$ immediately
yield (4.41) for all $\epsilon$ sufficiently small. ∎

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Casey Rodriguez

Department of Mathematics, Massachusetts Institute of Technology

77 Massachusetts Ave, 2-246B, Cambridge, MA 02139, U.S.A.