\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 220, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/220\hfil Front tracking]
{Front tracking for a $2\times2$ system of conservation laws}
\author[P. Baiti, E. Dal Santo\hfil EJDE-2012/220\hfilneg]
{Paolo Baiti, Edda Dal Santo} % in alphabetical order
\address{Paolo Baiti \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Udine, Via delle Scienze 206,
Udine 33100, Italy}
\email{paolo.baiti@uniud.it}
\address{Edda Dal Santo \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Udine, Via delle Scienze 206,
Udine 33100, Italy}
\email{edda.dalsanto@uniud.it}
\thanks{Submitted August 3, 2012. Published November 29, 2012.}
\subjclass[2000]{35L65, 35L67, 35A01}
\keywords{Systems of conservation laws; hyperbolic; large data}
\begin{abstract}
This article studies a front-tracking algorithm for $2\times2$
systems of conservation laws. After revisiting the classical results
of DiPerna \cite{diperna} and Bressan \cite{bressan},
we address the case of a $2\times2$ system arising in the study of
granular flows \cite{shen}. For the latter we prove the well-definiteness
of a simplified front-tracking algorithm and its convergence to a
weak entropic solution of the system, in the case of large BV initial data.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks
\section{Introduction}
Let us consider a Cauchy problem associated with a $2\times2$ system of
conservation laws
\begin{equation}\label{0}
u_t + f(u)_x=0, \quad (x,t)\in\mathbb{R}\times[0,\infty[,
\end{equation}
and initial conditions
\begin{equation}\label{1}
u(x,0)=\overline{u}(x).
\end{equation}
Assume that the system is strictly hyperbolic, with smooth coefficients and
defined on an open set $\Omega\subseteq\mathbb{R}^2$. Moreover, suppose that each
characteristic field is either genuinely nonlinear or linearly degenerate.
Given a function $\overline{u}$ with sufficiently small total variation, one can
prove the global in time existence of a weak, entropy-admissible solution.
The first proof of this result was due
to Glimm and his celebrated random choice algorithm \cite{glimm}.
Nowadays, one of the most largely used methods to achieve the same result is
front-tracking, described in~\cite{bressan}, that consists in constructing a
sequence of piecewise constant approximate solutions,
a subsequence of which converges to
a weak solution of the Cauchy problem~\eqref{0}-\eqref{1}.
The basic ideas involved were introduced by Dafermos in~\cite{dafermos} for
scalar equations and DiPerna in~\cite{diperna} for $2\times 2$ systems, then
extended by Bressan in \cite{bressan2,bressan} to general $n\times n$ systems with genuinely
nonlinear or linearly degenerate characteristic fields. The construction of the
approximate solutions starts at time $t=0$ by taking a piecewise constant
approximation $\tilde{u}(x)$ of the initial data $\overline{u}(x)$. At each point of
discontinuity a piecewise constant approximate solution of the
corresponding Riemann problem is chosen so that it coincides with the exact one
if it contains only shocks or contact discontinuities. Otherwise, if centred
rarefaction waves are present, they are replaced with rarefaction fans
containing several small jumps traveling with speed close to the characteristic
one. The approximate solution $\tilde{u}$ is then prolonged until the first time two
wave-fronts interact. Since at this time $\tilde{u}$ is still a piecewise constant
function, the corresponding Riemann problems can be approximately solved again
within the class of piecewise constant functions and so on.
For $n\times n$ systems the main source of technical difficulty derives from the
fact that the number of lines of discontinuity may approach infinity in finite
time, in which case the construction would break down. This is due mostly to
the fact that at each interaction point there are two incoming fronts, while
the number of outgoing ones is $n$ or even larger if rarefaction waves are
involved. To overcome this difficulty, the algorithm in~\cite{bressan,baiti} adopts
two different procedures to approximately solve an emerging Riemann problem:
an \emph{Accurate Riemann Solver} which introduces several new fronts, and a
\emph{Simplified Riemann Solver}, which involves at most two physical outgoing
fronts and collect the remaining new waves into a single
``non-physical'' front traveling with a speed strictly larger than all
characteristic speeds.
In \cite{bressan,baiti} the
algorithm is proved to converge to a weak solution of \eqref{0}-\eqref{1}
at least in the case of small BV data. Afterwards these results were extended to
systems of conservation laws whose characteristic fields are
neither genuinely nonlinear nor linearly degenerate \cite{ancona,ancona2,ancona3}.
Nevertheless, when dealing with a $2\times2$ system satisfying the previous
assumptions it is possible to avoid non-physical fronts and always use an
accurate solver to construct approximate solutions. This was initially proved for
the front-tracking introduced by DiPerna in \cite{diperna}, in the case of small
BV data. However his construction is quite tricky and less used than the one
proposed by Bressan in \cite{bressan2} and refined in \cite{baiti},
a slight modification of which can avoid the introduction of non-physical
fronts in the $2\times 2$ case, too.
In a few words, in the $n=2$ case the only problem comes from the fact that
rarefaction waves can be partitioned generating several fronts and this is
the unique way the total number of waves could increase.
In the first part of the paper we will revisit Bressan's construction
and propose a further slight modification of the algorithm, still avoiding
the non-physical fronts, which will be used in the second part.
The second part of the paper will be devoted to the case study of a
$2\times2$ system of balance laws modeling granular flows,
discussed by Amadori and Shen \cite{shen}. In their paper
they prove the existence of a weak solution for the initial value problem
associated to that system, in the case of possibly large BV data.
Unfortunately the front-tracking algorithm they refer to
leads to the introduction of another kind of non-physical fronts; since
the subsequent part of their paper strongly depends on the analysis along
characteristics, that algorithm seems not well suited for their purposes.
Moreover, the previous results \cite{baiti,bressan,ancona3} are not applicable
since the data in \cite{shen} may have large total variation.
The second goal of this paper is to prove that the simplified version, without
non-physical fronts, of the front-tracking algorithm for $2\times 2$ conservation
laws works also for the system proposed in \cite{shen} and in the presence of
large BV data; this will be accomplished in Theorem~\ref{welldef2}.
\section{Front-tracking for $2\times2$ systems}
In this section we briefly recall the front-tracking algorithm, for full
details the reader is referred to \cite{bressan}.
In general we can consider a strictly hyperbolic $n \times n$ system of
conservation laws \eqref{0} with $u\in\Omega\subseteq\mathbb{R}^{n}$ in which
each characteristic family is either genuinely nonlinear or linearly degenerate,
and where the flux $f$ is $C^2(\Omega)$.
Given two states $u^-,u^+$ sufficiently close, the corresponding Riemann
problem, that is the Cauchy problem with initial data given by
\begin{equation} \label{riemann}
\overline{u}(x)=
\begin{cases}
u^{-} &\text{if }x<0 \\
u^{+} &\text{if }x>0,
\end{cases}
\end{equation}
admits a self-similar solution given by at most $n+1$ constant states
separated by shocks, contact discontinuities
or rarefaction waves \cite{bressan}. More precisely, there exist $C^2$
curves $\sigma\mapsto\psi_i(\sigma)(u^-)$, $i=1,\dots,n$,
parametrized by arclength, such that
\begin{equation}
\label{psi}
u^+=\psi_n(\sigma_n)\circ\cdots\circ\psi_1(\sigma_1)(u^-),
\end{equation}
for some $\sigma_1,\dots,\sigma_n$. We define $u_0\doteq u^-$ and
$u_i\doteq \psi_i(\sigma_i)\circ \cdots\circ\psi_1(\sigma_1)(u_0)$.
When $\sigma_i$ is positive (negative) and the $i$-th characteristic family
is genuinely
nonlinear, the states $u_{i-1}$ and $u_i$ are separated by an $i$-rarefaction
($i$-shock) wave. If the $i$-th characteristic family is linearly degenerate
the states
$u_{i-1}$ and $u_i$ are separated by a contact discontinuity. The
\textsl{strength} of the $i$-wave is defined as $|\sigma_i|$. Therefore, the
solution of \eqref{0},\eqref{riemann} is given by the concatenation of at
most $n$ waves, which are discontinuous functions in the case of shocks or
contact discontinuities, or Lipschitz
continuous functions in the case of rarefactions.
The construction of approximate solutions by front-tracking proceeds as
follows: at time $t=0$ we approximate the initial data $\overline{u}$ with
a piecewise constant function. At each point of discontinuity the resulting
Riemann problems are solved as above in terms of elementary waves.
If a rarefaction is present in the solution, we replace it with a rarefaction fan
containing several small jumps of strength less than a fixed size $\eta$
and traveling with speed close to the characteristic one.
Piecing together the solution of all the Riemann problems, we obtain
an approximate solution of \eqref{0}-\eqref{1} defined on a small time interval
$[0,\bar{t}]$. Every front is then
prolonged until the first time two wave-fronts interact: at this time we
approximately solve the emerging Riemann problem within the class of piecewise
constant functions and so on.
As recalled in the introduction, to show that this procedure can
be carried on for all times, thus generating an approximate solution
globally defined on $\mathbb{R}\times[0,+\infty[$, the main problem is to provide
a uniform (in time) bound on the number of fronts and interactions that can be
generated at each time. This bound would prevent the number to approach infinity in
finite time.
For $n\times n$ systems, in~\cite{bressan,baiti} to achieve this goal two
different procedures are used to approximately solve a Riemann problem emerging
at time $t>0$ : an \emph{Accurate Riemann Solver} which introduces several new
fronts, and a \emph{Simplified Riemann Solver} which, roughly speaking, lets
the incoming waves pass through each other, if of different characteristic family,
or stick together, if of the same family, and collects all the remaining newly
born waves belonging to the other characteristic families in one single front
traveling with speed strictly greater than all characteristic speeds; for
this reason this front is in some sense non-physical, since it is not directly
related to elementary waves/speeds of the system. The first procedure is used
when the product of the strengths of the incoming waves is greater than a fixed
threshold, the other one when it is smaller. The main difficulties in proving
that the approximate solutions converge (up to subsequences) to a weak solution
of \eqref{0}-\eqref{1} come from controlling the overall error generated by the
introduction of non-physical fronts.
However, if $n=2$ the second procedure can be avoided; this was first
observed by DiPerna for his algorithm in \cite{diperna}. Here we want to verify
that this is still true for a slight modification of the algorithm proposed in
\cite{bressan}, in the case of small BV data. The simplified algorithm
$\mathcal{ALG}$\label{algorithm} can be simply described as follows:
\begin{itemize}
\item at time $t=0$ the initial datum is approximated by a piecewise
constant function having a finite number of jumps. Then we solve the
Riemann problems arising at each discontinuity point.
If a newly generated rarefaction front has strength $|\sigma|$ greater
than a small parameter $\eta>0$, fixed at the beginning, then it is
partitioned in a number of $\lfloor |\sigma|/\eta\rfloor+1$
(entropy violating) discontinuities;
\item the fronts are prolonged until two of them interact, the emerging
Riemann problem is approximately solved and so on. By slightly perturbing
the speed of just one front we can assume that at each time a single interaction
occurs, involving two incoming waves only.
For interaction times $t>0$ we always partition an
outgoing rarefaction, except when a rarefaction of
the same characteristic family is present also before the interaction.
In that case the outgoing rarefaction will not be split but
substituted by a single jump with the same strength.
\end{itemize}
In contrast to \cite{bressan}, here we prefer to distinguish cases according to
the size of the outgoing rarefactions instead of the size of the interaction
potential between interacting waves. Indeed, we will shortly verify that
every time a newly generated rarefaction has strength greater than $\eta$ the
potential decreases by a fixed amount. Therefore this algorithm can differ from
that in \cite{bressan} only for interacting waves with sufficiently large
potential and producing small rarefactions.
In order to ensure that the total number of wave-fronts and interactions
remains finite even in this case, it suffices to verify
that the number of interactions creating possibly large rarefactions is finite.
Indeed, this is what is required in the following lemma (for a proof see
\cite[Lemma 2.3]{amadori}).
\begin{lemma}\label{23}
Let a wave front-tracking pattern be given in $[0,T[\times\mathbb{R}$, made of segments
of the two families. Assume that the velocities of the fronts of the first
family lay between two constants $a_1x_{\alpha}$ and belonging to
the family $i_{\beta}$ if either $i_{\alpha}>i_{\beta}$ or $i_{\alpha}=i_{\beta}$ and
at least one of them is a shock; moreover, for every couple of
interacting waves $\sigma_{i}'$, $\sigma_{i}''$ belonging to the same $i$-th
family, the strength $|\sigma_{k}|$ of the outgoing wave of family $k\neq i$
satisfies \cite[Lemma 7.2]{bressan}
\begin{equation}
\label{esti.three}
|\sigma_k| \leq \mathcal{O}(1) |\sigma_i' \sigma_i''|\big(|\sigma_i'|+|\sigma_i''|\big).
\end{equation}
Suppose that at a time $\tau>0$ an interaction takes place between
two fronts of the same family and sizes $\sigma_\alpha$ and $\sigma_\beta$.
It is well-known that as long as a piecewise constant
approximate front-tracking solution is defined, if the initial data have
sufficiently small total variation the interaction potential $Q$ is decreasing in time, more
precisely $\Delta Q(\tau)\leq-{|\sigma_\alpha\sigma_\beta|}/{2}$
(see \cite[(7.56) and (7.57)]{bressan}).
In the case where $|\sigma|>\eta$ is the strength of the outgoing
rarefaction wave of the other family,
the interaction estimate \eqref{esti.three} yields
$$
\eta <|\sigma|\leq \mathcal{O}(1)|\sigma_\alpha\sigma_\beta|(|\sigma_\alpha|
+|\sigma_\beta|)\leq C_{1}|\sigma_\alpha\sigma_\beta|
\leq -2C_{1}\Delta Q(\tau),
$$
for a suitable constant $C_{1}$, as long as the total variation remains bounded
(and small).
Then $\Delta Q(\tau) <- \eta/(2C_{1})$ and
this means that whenever such interactions occur, the potential $Q$ decreases by
a fixed positive amount, and this can happen only finitely many times (since $Q(0)$
is bounded and $Q(t)$ is decreasing). Therefore, applying Lemma~\ref{23} we
obtain that the total number of wave-fronts and interactions remains finite in
time and the algorithm is well-defined. In conclusion we have verified the
following result.
\begin{proposition}
\label{welldef}
There exists $\delta>0$ such that for all initial data $\overline{u}$ with
$\operatorname{T.V.}(\overline{u})\leq\delta$ and
for every $\eta>0$ the algorithm $\mathcal{ALG}$
is well-defined and provides a piecewise constant approximate solution
$u_{\eta}$ defined for all $t>0$.
Choosing $\eta=\eta_{\nu}\to0$, the corresponding sequence $u_{\eta_{\nu}}$
converges in $L^{1}_{loc}$, up to subsequences, to a weak, entropy-admissible solution
of \eqref{0}-\eqref{1}.
\end{proposition}
%
\begin{proof}
The well-definiteness of the algorithm was proved above. The convergence
of the sequence is standard, since the approximate solutions here defined are
$\varepsilon$-approximate front-tracking
solutions in the sense of \cite[Definition 7.1]{bressan}.
Indeed, the only thing still to prove is that the maximum size of the
rarefactions can be bounded by a small quantity converging to zero
as $\eta\to0$. This can be
done as in \cite[pp.\ 138--139]{bressan}. In addition, each limit point coincides
with the corresponding trajectory of the Standard Riemann semigroup solution in
the sense of \cite[Definition~9.1]{bressan}, hence the whole sequence converges
to that trajectory.
\end{proof}
Notice that the main ingredient in the previous analysis was the existence
of a decreasing functional $Q$ which controls the strength of the newly generated
rarefactions. However, in general $Q$ is proved to be decreasing only for small
BV data. With the aim of extending this result to possibly large BV data,
we will not always be able to rely on $Q$, and this also justifies our choice
to distinguish cases according only to the size of the outgoing rarefactions.
\section{Front-tracking for a $2\times2$ system of balance laws modeling
granular flows}
In this section we show an example of a $2\times 2$ system, arising in modeling
granular flows and proposed by Amadori and Shen in \cite{shen}, for which the
simplified algorithm $\mathcal{ALG}$ works even in the presence of large BV data.
In \cite{shen}, one is concerned with the construction of global BV solutions for a Cauchy
problem associated with the
system of balance laws
\begin{equation}\label{20}
\begin{gathered}
h_t-(h p)_x= (p-1)h,\\
p_t+((p-1)h)_x=0,
\end{gathered}
\end{equation}
(where $h=h(x,t)\geq0$ and $p=p(x,t)>0$) and initial data
\begin{equation}\label{25}
h(x,0)=\overline{h}(x), \quad p(x,0)=\overline{p}(x),
\end{equation}
having arbitrarily large total variation and a small $L^\infty$ bound only on the
$h$ coordinate. Unlike what has been seen in the previous section, the
system~\eqref{20} contains source terms and the first characteristic field is
neither genuinely nonlinear nor linearly degenerate in the domain. In general,
this last property may lead to the appearance of composite waves of the first
family, but this is not the case. Indeed, the line $p=1$ separates the domain
into two invariant regions from the point of view of the Riemann problem, where
both characteristic fields are genuinely nonlinear,
and it is verified that an interaction of two waves
generates at most two simple waves, one from each family.
In \cite{shen} approximate solutions to the Cauchy problem~\eqref{20}-\eqref{25}
are defined by an operator splitting method.
Consider the sequence of times $t_k=k\Delta t$, where $\Delta t\geq0$
is a fixed time step. On each time interval $[t_{k-1},t_k[$ an approximation
$(h,p)$ of the system of conservation laws
\begin{equation}\label{30}
\begin{gathered}
h_t-(hp)_x=0,\\
p_t+((p-1)h)_x=0,
\end{gathered}
\end{equation}
is constructed by means of a front-tracking algorithm; while at each time $t_k$
the functions $(h,p)$ are redefined in order to account for the source term.
In \cite{shen} the authors suggest the use of an algorithm similar to
\cite{ancona}; the latter extends previous results of Bressan and Colombo
\cite{colombo} to non genuinely nonlinear systems with small BV data.
However these results could not be applied directly to system \eqref{30}, since
\eqref{30} does not satisfy the assumptions in \cite{ancona} and allows
for large BV and $L^{\infty}$ data.
In addition, the study of the full balance system \eqref{20} in
\cite{shen} depends on the analysis along the characteristics of the related
homogeneous system \eqref{30}. Hence it is important that all the waves in
the approximate solutions be physical, i.e.\ shocks or rarefactions.
In fact the algorithm proposed in \cite{colombo,ancona} does not introduce
non-physical fronts but relies on a suitable interpolation between
shock and rarefaction curves. This leads to the generation of a sort of new
interpolated waves (which are neither shocks nor rarefactions), and these
should have been taken into account in the analysis of interactions
patterns/estimates, but actually are not.
In the following, we prove that the front-tracking algorithm $\mathcal{ALG}$
applied to system \eqref{30} between instants $t_{k-1}$ and $t_k$ (indeed on
all $[0,+\infty[$) is well-defined, even allowing for large data, as in the
assumptions in \cite{shen}.
%
Towards this goal, we need to control the number of waves and interactions,
and to understand how the
strength of a rarefaction front varies over the time interval $[t_{k-1},t_k]$.
\subsection{The number of interactions and wave-fronts is finite}\label{number}
To claim that the front-tracking algorithm used is well-defined, one
must demonstrate that the total number of waves and interactions remains
finite. Recalling Lemma~\ref{23}, it suffices to show that within the interval
considered the number of interactions that generate more than one outgoing
wave of the same family is finite, since the other requests of the lemma are
easily satisfiable. As already pointed out, these are the interactions that give
rise to a new rarefaction front of strength $|\sigma|$ larger than a small
parameter $\eta>0$ and that should be split into a number
$\left\lfloor|\sigma|/\eta\right\rfloor+1$ of fronts, each of strength at most
$\eta$. Since incoming rarefactions are never partitioned, the cases when the
above situation may occur are the following:
\begin{itemize}
\item when two waves of the first family interact with each other, they may
generate a rarefaction of the second family of strength larger than $\eta$;
\item similarly, when two waves of the second family interact with each other,
they may give rise to a rarefaction of the first family of strength larger
than $\eta $;
\item when a wave of the second family that crosses the line $p=1$ interacts
with a shock of the first family, it gives rise to a rarefaction of the
first family that possibly has strength larger than $\eta $.
\end{itemize}
Observe that for an interaction involving a wave of the first family and one of
the second family that does not cross $p =1$ (i.e.\ for an interaction that takes
place entirely in $\{p <1\}$ or $\{p> 1\}$), the outgoing waves have the same
sign of the incoming ones (w.r.t.\ the characteristic family), hence they
are of the same type (see \cite{shen}). Notice that in general this may not be
true when the initial data are large; in this case it holds thanks to the special
structure of system \eqref{30} and to the
fact that the first characteristic field is genuinely nonlinear in each of the
two invariant regions of the domain separated by the line $p=1$. Thus, in these
interactions there can not be any partitioning of rarefactions. The same holds
for any outgoing rarefaction of the same family of the two incoming ones.
As before, to control the number of the new waves created we will try
to find a non-increasing functional that decreases
by a fixed amount at each interaction where more than two waves appear.
In contrast to the previous section, the interaction potential
$\mathcal{Q}$ used in \cite{shen} will not be useful, since
when two waves of the first family and of different sign interact one has
$\Delta \mathcal{Q}=\mathcal{O}(1)\|h\|_{L^{\infty}}|\sigma_{\alpha}\sigma_{\beta}|$
and $\mathcal{Q}$ may increase \cite[p.\ 1024]{shen}. In fact our functional will be based
on the functional $\mathcal{S}$ also introduced in \cite{shen} and which will be
briefly recalled below. As it will be clear in Lemma~\ref{results},
$\mathcal{S}$ is a decreasing functional which, by Lemma~\ref{intesti}, is used to
control the change in wave-strengths after interactions.
Unfortunately, $\mathcal{S}$ does not always control the strength of the newly
generated rarefactions since there exist interactions between (arbitrarily) large
shocks of the first family and (arbitrarily) small waves of the second family
for which the interaction potential is very small but still a big $1$-rarefaction is
created. Hence $\mathcal{S}$ will not decrease by a \emph{fixed} amount after those
interactions, and we will have to modify it.
%
First we need some definitions: given a piecewise constant
approximate solution $u=(h,p):\mathbb{R}\times[t_{k-1},t_{k}[\to\mathbb{R}^{2}$, let
$\sigma_{\alpha}=\sigma_{\alpha}(t)$ be the size of the waves of $u(\cdot,t)$
at time $t$, of family $i_{\alpha}$ ($i_{\alpha}\in\{1,2\}$)
and positioned at $x_{\alpha}$. Let
$$
V(t)=\sum_{\alpha}|\sigma_{\alpha}|
$$
be the total strength of waves in $u(\cdot,t)$ and,
with the notations of \cite{shen},
\begin{equation}
\label{Qfunc}
\mathcal{Q}(t)=\mathcal{Q}_{hh}(t)+\mathcal{Q}_{pp}(t)+\mathcal{Q}_{ph}(t)
\end{equation}
be the interaction potential between couples of approaching waves.
The strength of the $1$-waves and $2$-waves (which will also
be called $h$-waves and $p$-waves, respectively) are
measured in suitable Riemann coordinates $(H,P)$.
The functionals $\mathcal{Q}_{hh}$, $\mathcal{Q}_{pp}$ and $\mathcal{Q}_{ph}$ are the interaction
potentials between pairs of $h$-$h$ waves, $p$-$p$ waves and $p$-$h$ waves,
respectively. More precisely
$$
\mathcal{Q}_{hh}(t)=\sum_{\substack{i_{\alpha}=i_{\beta}=1 \\ x_{\alpha}0$, there exist $\delta>0$ and $c,\delta_{0}>0$ such
that for every initial data $(\overline{h},\overline{p})$ with
\begin{gather*}
\operatorname{T.V.}(\overline{h})\leq M,\quad\operatorname{T.V.}(\overline{p})\leq M,\\
\|\overline{h}\|_{L^{1}}\leq M,\quad\|\overline{p}-1\|_{L^{1}}\leq M,
\quad
\overline{p}\geq p_{0}>0,
\end{gather*}
if $\|\overline{h}\|_{L^{\infty}}\leq \delta$ and as long as the
front-tracking
approximation of \eqref{30}-\eqref{25} is defined, the function
$$
\mathcal{S}(t):=V(t)+c\mathcal{Q}(t)
$$
is non-increasing in time. More precisely, given two waves
$\sigma_{\alpha}$ and $\sigma_{\beta}$ interacting at time $t$ the
following hold:
\begin{itemize}
\item[(i)] if they are both $h$-shocks then
$\Delta \mathcal{S}(t)\leq -w_{\alpha,\beta}c|\sigma_{\alpha}\sigma_{\beta}|/4$;
\item[(ii)] if they are a $h$-shock and a $h$-rarefaction then
$\Delta \mathcal{S}(t)\leq -\min\{|\sigma_{\alpha}|,|\sigma_{\beta}|\}$;
\item[(iii)] if they are a $p$- and a $h$-wave then
$\Delta \mathcal{S}(t)\leq -c|\sigma_{\alpha}\sigma_{\beta}|/4$;
\item[(iv)] if they are both $p$-waves then
$\Delta \mathcal{S}(t)\leq -c|\sigma_{\alpha}\sigma_{\beta}|/4$.
\end{itemize}
\end{lemma}
For a proof of the above lemma, see \cite[Section~4.4]{shen}.
A \emph{wave crossing $p=1$} is a wave connecting states $(h_{l},p_{l})$
and $(h_{r},p_{r})$ such that $(p_{l}-1)(p_{r}-1)\leq0$ and at least one
of $p_{l}$ and $p_{r}$ is different from $1$; it is easy to see that
these waves must be $p$-waves.
Now, let $u$ be an approximate solution of \eqref{30}-\eqref{25} obtained by
$\mathcal{ALG}$ and denote by $y_i=y_i(t)$ the wave-fronts of the second family crossing
$p=1$ at time $t\in[t_{k-1},t_{k}[$, which are finite in number.
Introduce the functional $\mathcal{M}$
\begin{equation}\label{35}
\mathcal{M}(t) :=\mathcal{S}(t) + b \,\mathcal{N} (t),
\end{equation}
where $b$ is a constant to be determined and $\mathcal{N}$ is defined as follows:
\begin{equation}\label{40}
\mathcal{N}(t):=\sum_{\alpha}N^{2k_\alpha},
\end{equation}
where the summation ranges over the wave-fronts $x_\alpha$ at time $t$, $N$ is
the constant
\begin{equation}\label{45}
N:=\left\lfloor\frac{\sup_\tau\|(H,P)(\cdot,\tau)\|_{L^\infty}}{\eta}
\right\rfloor+ 1
\end{equation}
and $k_\alpha$ is the cardinality of the finite set of fronts crossing $p=1$
and to the left of $x_{\alpha}$, i.e.\ the set
$$
\left\{y_i: \text{$y_i$ is a front crossing $p=1$ at time $t$
and $y_i0$ such that $\Delta\mathcal{M}(t)\leq0$
for every $t\in[t_{k-1},t_k[$,
that is, $\mathcal{M}$ is non-increasing;
\item there exists a constant $\mu>0$ such that whenever a rarefaction of
strength $|\sigma|>\eta$ is created at $t>t_{k-1}$, then
$$
\Delta\mathcal{M}(t)\leq-\mu<0.
$$
\end{enumerate}
\end{lemma}
\begin {proof}
The first statement is easy to check. Indeed, the number of fronts crossing
$p=1$ can not increase, as it remains constant when there is an interaction
between one of those and a front of the first family or one of the second family
which does not cross the line, while it decreases by one when two of those fronts
interact with each other leading to a cancellation.
To prove the second statement, we must consider several cases. In particular,
suppose that at a time $t$ an interaction between two fronts not crossing $p=1$
takes place. In the following, we denote by $\sigma_\alpha$ and $\sigma_\beta$
the sizes of the incoming waves and by $\sigma_h$ and $\sigma_p$ those of
the outgoing ones, respectively of the first and second characteristic family.
First, note that, since no front $y_i$ is involved in the interaction, one has
\begin{equation}\label{50}
k_h = k_p = k_\alpha = k_\beta.
\end{equation}
Suppose that the interaction is between two shocks of the first family and that
the outgoing wave of the second family is either a shock or a rarefaction of
strength smaller than $\eta$ (i.e.\ it does not need to be partitioned). In this
case, by Lemma~\ref{results} one has $\Delta\mathcal{S}<0$, while for
$\mathcal{N}$ it holds $\Delta \mathcal{N} =0$,
so that $\Delta\mathcal{M}<0$. On the other hand, if the outgoing wave of the
second family is a rarefaction of strength $|\sigma_p|>\eta$, then it is
partitioned into at most $N$ fronts. Thus,
\begin{equation}
\label{Nesti}
\Delta \mathcal{N}\leq N^{2k_h}+N N^{2k_p}-N^{2k_\alpha}-N^{2k_\beta}
=N^{2k_\alpha} (N-1)\leq N^{2\widetilde{k}}(N-1).
\end{equation}
Moreover, from (i) of Lemma~\ref{results}, one has
$$
\Delta\mathcal{S}\leq -\frac{c}{4}w_{\alpha,\beta}|\sigma_\alpha
\sigma_\beta|,
$$
and, thanks to the interaction estimates \eqref{hh-int},
$$
\eta <|\sigma_p|\leq O(1)\|h\|_{L^\infty}w_{\alpha,\beta}
|\sigma_\alpha\sigma_\beta|\leq w_{\alpha,\beta}|
\sigma_\alpha\sigma_\beta|\leq-\frac{4}{c}\Delta\mathcal{S}
$$
if $\|h\|_{L^{\infty}}$ is small enough.
Consequently, $\Delta\mathcal{S} <-c\eta/4$ and
$$
\Delta\mathcal{M}=\Delta\mathcal{S} + b\,\Delta \mathcal{N} <-\frac{c}{4}\eta
+b\,N^{2\widetilde{k}}(N -1).
$$
If the constant $b$ is chosen so that
\begin{equation}\label{60}
b \leq\frac{c\eta}{8N^{2\widetilde{k}} (N-1)},
\end{equation}
one finally gets
\begin{equation}
\label{esti1}
\Delta\mathcal{M}<-\frac{c}{8}\eta<0.
\end{equation}
When the interaction involves two waves of the first family but of different
sign (with $|\sigma_\alpha|<|\sigma_\beta|$), the functional $\mathcal{M}$
decreases trivially if the outgoing wave of the second family is a shock or a
rarefaction of strength $|\sigma_p|\leq\eta$. On the other hand, if it is a
rarefaction of strength larger than $\eta$, it is again partitioned into at
most $N$ fronts. Then, $\mathcal{N}$ satisfies \eqref{Nesti}, while from ii) of
Lemma~\ref{results} $\mathcal{S}$ satisfies
$$
\Delta\mathcal{S}\leq -|\sigma_\alpha|<-\eta,
$$
since, thanks to the interaction estimates \eqref{hh-int}, one has
$$
\eta<|\sigma_p|\leq O(1)\|h\|_{L^\infty}|\sigma_\alpha\sigma_\beta|
\leq|\sigma_\alpha|,
$$
again by the smallness assumption on $\|h\|_{L^{\infty}}$ and the upper bound
on $|\sigma_{\beta}|$, that is on $\|p\|_{L^{\infty}}$ and $\|h\|_{L^{\infty}}$,
proved in \cite{shen}. This time, if $b$ is chosen so that
\begin{equation}
\label{70}
b\leq\frac{\eta}{2N^{2\widetilde{k}}(N-1)},
\end{equation}
finally we get
\begin{equation}
\label{esti2}
\Delta\mathcal{M}<-\eta+b\,N^{2\widetilde{k}} (N-1)\leq-\frac{\eta}{2}.
\end{equation}
In a similar way we prove also that $\mathcal{M}$ decreases when the interaction
involves two waves of the second family both not crossing $p=1$.
If the interaction, instead, involves two waves of different families and the
incoming one of the second family does not cross $p=1$, we have already noticed
that there can not be any partitioning and $\mathcal{M}$ decreases trivially.
Now suppose that the interaction involves (only) one of the fronts crossing $p=1$,
of size $\sigma_\alpha$. Let $\sigma_\beta$ be the size of the other
incoming wave and $\sigma_h$ and $\sigma_p$ those of the outgoing waves, as
before. If $\sigma_\beta$ is a rarefaction of the first family, then the outgoing
wave of the second family will maintain the sign of $\sigma_\alpha$, while the one of
the first family will be forced to be a shock. In this case,
\begin{equation}\label{65}
k_\alpha=k_h=k_p\quad \text{and}\quad k_\beta=k_\alpha+ 1.
\end{equation}
Then, being that
$$
\Delta \mathcal{N} = N^{2k_h}+N^{2k_p}-N^{2k_\alpha}-N^{2k_\beta}
=N^{2k_\alpha}(1-N^2)\leq -(N^2-1)<0
$$
and $\Delta\mathcal{S}<0$, one has $\Delta\mathcal{M}\leq-b(N^2-1)<0$. On the other hand,
if $\sigma_\beta$ is a shock of the first family, then the outgoing wave of the
first family is a rarefaction that could be of strength larger than $\eta$.
In that case,
\begin{align*}
\Delta \mathcal{N}
&\leq NN^{2k_h}+N^{2k_p}-N^{2k_\alpha}-N^{2k_\beta}
=N^{2k_\alpha+1}-N^{2(k_\alpha+1)}\\
&=N^{2k_\alpha}(N-N^2)
\leq-(N^2-N),
\end{align*}
thus $\Delta\mathcal{M}\leq -b(N^2-N)<0$.
If $\sigma_\beta$ is of the second family (not crossing $p=1$) and hits from the
right $\sigma_\alpha$, again we have~\eqref{65}. Moreover, if the outgoing wave
of the first family is a shock or a rarefaction of strength smaller than $\eta$,
it is easy to see that $\mathcal{M}$ decreases. If the outgoing wave of the
first family, instead, is a rarefaction of strength $|\sigma_h|>\eta$, as before,
one obtains $\Delta\mathcal{M}\leq -b(N^2-N)$. Otherwise, suppose $\sigma_\beta$
is of the second family and hits $\sigma_\alpha$ from the left: now we have
that~\eqref{50} holds and if the outgoing wave of the first family is a
rarefaction of strength $|\sigma_h|>\eta$, then
$$
\Delta \mathcal{N}\leq N^{2\widetilde{k}}(N-1)
$$
and
$$
\Delta\mathcal{M}\leq -\frac{c}{4}\eta+b\,N^{2\widetilde{k}}(N-1)
\leq-\frac{c}{8}\eta,
$$
choosing $b$ as in~\eqref{60}. The last case to consider (i.e.\ when the
interaction involves two fronts of the second family both crossing $p=1$) can be
treated in a similar way and leads to analogous results.
In conclusion, if $b$ satisfies \eqref{60} and \eqref{70} then $\mathcal{M}$ is non-increasing,
and choosing the constant $\mu$ as
$$
\mu=\min\big\{\frac{c}{8}\eta\,,\,\frac{\eta}{2}\,,\,b(N^2-N)\big\},
$$
the last statement is proven.
\end{proof}
The above lemma ensures that interactions generating more than two outgoing
fronts can occur only finitely many times. Therefore, we can apply Lemma~\ref{23} to
obtain that the total number of fronts and interactions is finite.
\subsection{The strength of a rarefaction}\label{raref}
To conclude the analysis of the algorithm we have to prove its
convergence (up to subsequences) to a weak entropic solution of
\eqref{30}-\eqref{25}. It is sufficient to show that the approximate
solutions are indeed $\varepsilon$-approximate front-tracking solutions in the sense of
\cite[Definition 7.1]{bressan}, and it only remains to prove that the
maximum size of the rarefactions can be bounded by a
small quantity.
To do this, we will need to adapt the analysis done in
\cite[pp.\ 138--139]{bressan}.
The strength of a rarefaction front can increase only when it interacts with
waves of a different characteristic family, while when interacting with a shock
of the same family its strength decreases. This is expected for $1$-waves
since $\|\overline{h}\|_{L^{\infty}}$ is chosen small, but it is not
straightforward for $2$-waves when dealing with data with large
$\|\overline{p}\|_{L^{\infty}}$, and it has to be proven. Indeed
this comes from the following interaction estimates derived in
\cite[Lemma 3]{shen}.
\begin{lemma} \label{intesti}
Consider two interacting wave-fronts, with left, middle, and right states
$(h_{l},p_{l})$, $(h_{m},p_{m})$, $(h_{r},p_{r})$ before interaction,
respectively, and $$
h_{\rm max}= \max\{h_{l},h_{m},h_{r}\}.
$$
\begin{itemize}
\item[(i)] If two $p$-waves of size $\sigma_{p}$ and $\tilde{\sigma}_{p}$
interact producing the outgoing waves of size $\sigma_{h}^{+}$ and
$\sigma_{p}^{+}$ then
\begin{equation}
\label{pp-int}
|\sigma_{h}^{+}|+|\sigma_{p}^{+}-(\sigma_{p}+\tilde{\sigma}_{p})|
=\mathcal{O}(1)\cdot h_{l}\cdot|\sigma_{p}\tilde{\sigma}_{p}|.
\end{equation}
\item[(ii)] If two $h$-waves of size $\sigma_{h}$ and $\tilde{\sigma}_{h}$
interact producing the outgoing waves of size $\sigma_{h}^{+}$ and
$\sigma_{p}^{+}$ then
\begin{equation}
\label{hh-int}
|\sigma_{h}^{+}-(\sigma_{h}+\tilde{\sigma}_{h})|+|\sigma_{p}^{+}|
=\mathcal{O}(1)\cdot\min\{|p_{l}-1|,|p_{m}-1|\}
\cdot(|\sigma_{h}|+|\tilde{\sigma}_{h}|)|\sigma_{h}\tilde{\sigma}_{h}|.
\end{equation}
\item[(iii)] If two waves of different family and size
$\sigma_{p}$ and $\sigma_{h}$
interact producing the outgoing waves of size $\sigma_{h}^{+}$ and
$\sigma_{p}^{+}$ then
\begin{equation}
\label{ph-int}
|\sigma_{h}^{+}-\sigma_{h}|+|\sigma_{p}^{+}-\sigma_{p}|
=\mathcal{O}(1)\cdot h_{\rm max}\cdot|\sigma_{h}\sigma_{p}|.
\end{equation}
\end{itemize}
\end{lemma}
If $\sigma_{h}$ is a $h$-rarefaction interacting at time $\tau$ with a $h$-shock of
size $\tilde{\sigma}_{h}$, either the rarefaction is canceled or
by \eqref{hh-int} its size $\sigma_{h}^{+}$ satisfies
\begin{align*}
|\sigma_{h}^{+}|-|\sigma_{h}|
\leq|\sigma_{h}^{+}-\sigma_{h}-\tilde{\sigma}_{h}|-|\tilde{\sigma}_{h}|
&\leq\mathcal{O}(1)\|h\|_{L^{\infty}}|\sigma_{h}\tilde{\sigma}_{h}|
-|\tilde{\sigma}_{h}|\\
&\leq-|\tilde{\sigma}_{h}|\big(1-\mathcal{O}(1)\|h\|_{L^{\infty}}V(\tau-)\big),
\end{align*}
where it was also used that $|\sigma_h|>|\tilde \sigma_h|$, which is valid
thanks to the definition of the $\sigma$'s.
Using \eqref{pp-int} the same argument for $p$-rarefactions gives
\begin{align*}
|\sigma_{p}^{+}|-|\sigma_{p}|
&\leq-|\tilde{\sigma}_{p}|\big(1-\mathcal{O}(1)\|h\|_{L^{\infty}}V(\tau-)\big).
\end{align*}
As long as the approximate solution is defined the functional $V+c\mathcal{Q}$ is
decreasing, hence $V(t)$ is uniformly bounded. Since $\|h\|_{L^{\infty}}$
is of order $\mathcal{O}(1)\|\overline{h}\|_{L^{\infty}}$ (see \cite{shen}), by choosing
the constant $\delta$ in Lemma~\ref{results} sufficiently small we obtain that
the strength of both $h$- and $p$-rarefactions decreases after interacting with
waves of the same family. Hence the strength of a rarefaction can increase only
after interacting with waves of the other family. Now we proceed almost like
in \cite[pp.\ 138--139]{bressan}.
Let $\sigma_\alpha(t)$ be the size at time $t$
of a rarefaction front generated at $\tau_0\in\,]t_{k-1}, t_k[$. The aim
is to find a limitation to $|\sigma_\alpha(t)|$ by means
of $\mathcal{Q}$ and a functional $V_\alpha$ to be introduced.
In the case $\sigma_\alpha$ is of
the first characteristic family we define the quantity
\begin{equation}
\label{Valphah}
V_\alpha(t):=\sum_{\substack{i_\beta=2\\x_\betax_\alpha}}|\sigma_\beta|
+\sum_{\substack{i_\gamma=2\\ \sigma_\gamma\,shock}}
|\sigma_\gamma|.
$$
The only difference with \cite[p.\ 139]{bressan} comes from the fact that, because
of the structure of the system, a $h$-rarefaction may become a $h$-shock
after colliding with a $p$-wave; this means that, after an interaction,
a wave not approaching $\sigma_{\alpha}$ could turn into one which is
approaching.
Hence, two distinct $h$-waves are always potentially approaching, and this
justifies the choice of the indexes of summation in the second sum
in \eqref{Valphah}.
Now, suppose that at a time $\tau>\tau_0$ an interaction takes place.
There are two possibilities: either it involves two wave-fronts
different from the rarefaction under consideration, or it involves
$\sigma_\alpha$ itself. In any case, thanks to Lemma~\ref{results} and
Lemma~\ref{intesti} one can prove that either
$$
\Delta|\sigma_\alpha|\leq0, \quad
\Delta V_\alpha + c \,\Delta \mathcal{Q} \leq0,
$$
or
$$
\Delta|\sigma_\alpha|\leq C_{2} |\sigma_\alpha\sigma_\beta|,
\quad
\Delta V_\alpha=-|\sigma_\beta|,
\quad
\Delta \mathcal{Q}\leq 0,
$$
for a suitable constant $C_{2}$.
Hence we are in the same situation of \cite[p.\ 139]{bressan},
and it is easy to prove that the map defined by
$$
\Theta: t \mapsto|\sigma_\alpha(t)|
\exp\{C_{2}[V_\alpha(t)+c\,\mathcal{Q}(t)]\}
$$
is non-increasing for $t\in[\tau_0,\tau_1]\subseteq[t_{k-1},t_k]$,
where $\tau_0$
is the time of generation and $\tau_1$ the time of (possible)
cancellation of the rarefaction.
Consequently, for $t\geq\tau_0$
$$
|\sigma_\alpha(t)|\leq\Theta(t)\leq\Theta(\tau_0)
\leq|\sigma_\alpha(\tau_0)|\exp\{C_{2}[V_\alpha(t_{k-1})+
c\,\mathcal{Q}(t_{k-1})]\}
$$
and it is clear that the strength of each rarefaction between $t_{k-1}$
and $t_k$ remains small. Indeed, since $V_\alpha(t_{k-1})+c\,\mathcal{Q}(t_{k-1})$
is bounded and $|\sigma_\alpha(\tau_0)|\leq\eta$ (where $\eta$ is the small
parameter controlling the size of the newly generated rarefactions),
it follows that
$$
|\sigma_\alpha(t)|\leq \mathcal{O}(1)\eta.
$$
Finally, combining the results of subsection~\ref{number} and \ref{raref} we
obtain the following result.
\begin{theorem} \label{welldef2}
Under the same assumptions of Lemma~\ref{results},
for every $\eta>0$ the algorithm $\mathcal{ALG}$
applied to system \eqref{30} is well-defined and provides a piecewise
constant approximate solution of \eqref{30} defined for all
$t\in[t_{k-1},t_{k}[$ (indeed for all $t\geq t_{k-1}$ and all $t_{k-1}$).
Choosing a sequence $\eta_{\nu}\to0^{+}$, the corresponding sequence of
approximate solutions converges in $L^{1}_{loc}$, up to subsequences, to
a weak entropic solution of \eqref{30}-\eqref{25}.
\end{theorem}
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\end{document}