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\title {The Method of Quasidecoupling for Discontinuous Solutions \\
to Conservation Laws}
\author {Gui-Qiang Chen\thanks{Department of Mathematics, The University 
 of Chicago}
\thanks{Email: CHENG@ZAPHOD.UCHICAGO.EDU }}
\maketitle
                  
\centerline{\bf Table of Contents}
%\thispagestyle{myheadings}


\bigskip

\noindent
1.  Introduction

\medskip

\noindent
2. Entropy, Solutions, and the Young Measure

\medskip

\noindent
3.  Quasidecoupling Method

a.  Quasidecoupling Transformation

b.  Framework for the Uniqueness and Stability of Generalized Solutions

c.  Framework for the Limit Behavior of Approximate Solutions

\medskip

\noindent
4.  Systems with Fully Contact Fields

a.  Propagation of Oscillations and Existence of Generalized Solutions

b.  Uniqueness and Stability of Generalized Solutions


\medskip
\noindent
5.  Systems with One Contact Field and One Line Field

a.  Uniqueness and Stability of Generalized Solutions

b.  Systems Arising in Magnetohydrodynamics and Elasticity

c.  A System Arising in Oil Recovery

\medskip

\noindent
6.  Systems of Electromagnetic Plane Waves



\medskip
\noindent
7.  Remarks

\medskip
\noindent
References

\newpage

\section{Introduction}

We are concerned with the existence, uniqueness, and qualitative behavior 
of discontinuous
solutions to nonlinear conservation laws.
The main objectives of this paper are to describe an analytical
method --- the quasidecoupling method --- and to study its applications to
these problems for discontinuous solutions to  
hyperbolic systems of conservation laws with
linearly degenerate characteristic fields.
This method is developed from the ideas in \cite{ch2,ch3}
about propagation and cancellation of initial oscillations for
conservation laws and is motivated by the transformation between
Euler coordinates and Lagrangian coordinates in fluid
dynamics (cf. \cite{cf,w,s3}).
The novel feature of the method described here is the quasidecoupling
of the entropy conservative inequalities and, to a lesser extent,
the particular forms of entropy conservative inequalities for
conservation laws.

We consider a hyperbolic system of conservation laws
\begin{equation}
u_t + f(u)_x = 0 , \qquad - \infty < x < \infty ,
\end{equation}
where $u = u(x,t) \in \R ^ n$ and $f$ is a smooth nonlinear
mapping from $\R^n$ to $\R^n$.  The condition of hyperbolicity
requires that the Jacobian $\nabla f$ of $f$ has $n$ real
eigenvalues $\lambda _ i (u)$, $1 \leq i \leq n$, and $n$
distinct right eigenvectors $r_i (u)$, $1 \leq i \leq n$,
\begin{equation}
\nabla f(u) r_i (u) = \lambda _i (u) r_i (u);
\end{equation}
that is, the Jacobian $\nabla f$ is diagonalizable for any value of
$u$.  An eigenvalue $\lambda _i$ is either genuinely nonlinear or
linearly degenerate in the sense of {\sc Lax} \cite{la2} if its derivative in
the corresponding eigendirection is either
\begin{equation}
r_i \cdot \nabla \lambda_i \neq 0 \qquad \mbox{   or   } \qquad
r_i \cdot \nabla \lambda _i \equiv 0 .
\end{equation}

In the classical theory of hyperbolic equations, an efficient
method is the diagonalization method for studying the qualitative
behavior of smooth solutions to quasilinear hyperbolic systems.
The systems are reduced to corresponding systems of diagonal forms
that consist of the transport equations
\begin{equation}
\partial_t w_i + \lambda_i (w) \partial _x w_i  = 0 , \qquad 1 \leq i \leq n ,
\end{equation}
where 
the functions $w_i (u)$, $1 \leq i \leq n$, are Riemann invariants
satisfying
\begin{equation}
\nabla w_i \cdot \nabla f = \lambda _i \nabla w_i .
\end{equation}
A necessary and sufficient condition for the existence of the
Riemann invariants $w_i$ for strictly hyperbolic systems is the
well-known Frobenius condition:
\begin{equation}
l_i \{ r_j , r_k \} = 0, \qquad \mbox{   for any   }
j,k \neq i ,
\end{equation}
where $l_i$ denotes the left eigenvector corresponding to
$\lambda _i$ and $\{ \cdot , \cdot \}$ is the Poisson bracket
of vector fields in $u$-space.
For $n=2$, the Frobenius condition always holds.
It is well known that the Cauchy problem for (1.1) does not, 
in general, have  globally defined smooth solutions because of the
nonlinear structure of the eigenvalues; hence only discontinuous
solutions may exist in the large.
For discontinuous solutions, the system (1.1) is not equivalent to the
system (1.4) in general, and this method no longer works.
In Section 3, we shall introduce a related method --- the quasidecoupling
method --- and shall  establish several frameworks for studying the
uniqueness and qualitative behavior of discontinuous solutions to
hyperbolic systems of conservation laws.

The uniqueness problem for conservation laws has been the topic of
many papers (see the references and those cited therein).  A complete
theory of the uniqueness and $L^1$ stability of
solutions to scalar conservation laws has been established in the space 
$L^\infty$ or $L^\infty \cap BV$.
We refer the reader to the work of {\sc Oleinik} \cite{o1}, 
{\sc Vol'pert} \cite{v},
{\sc Keyfitz} \cite{ke1}, and {\sc Kruzkov} \cite{kr} on this problem for 
a scalar
conservation law.  Under certain smoothness (cf. piecewise
Lipschitz) and ordering assumptions on generalized solutions, some
uniqueness theorems for strictly hyperbolic systems of
conservation laws were established by {\sc Oleinik} \cite{o2},
{\sc Rozhdestvenskii} \cite{ro}, {\sc Godunov} \cite{go}, 
{\sc Hurd} \cite{h1,h2}, {\sc Liu} \cite{li1}, and
{\sc DiPerna} \cite{di1} in the space $L^\infty \cap BV$.  
These results impose assumptions on generalized
solutions that have not been translated into conditions on
the initial data.  For an isotachophoresis model, {\sc Dafermos and
Geng} \cite{dg} established a uniqueness theorem of a $BV$ solution with
small oscillation $BV$ initial data in which generation
points of centered rarefaction waves of the two
families are strictly separated.  We also refer the reader to
{\sc Douglis} \cite{go}, {\sc Lyapidevskii} \cite{ly} 
and {\sc Tveito and Winther} \cite{tw} for stability results
within certain special classes of solutions.
In Subsection 3b, we shall first establish a framework (Framework
(A)) for the uniqueness and stability of $L^\infty$ solutions of
the model Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
(a_0(w-k)P(x,t))_t + (a_0(w-k)\lambda (x,t)P(x,t))_x \leq 0 , \qquad
\mbox{for any  } k \in (- \infty , \infty ), \\
w|_{t=0} = w_0(x) ,
\end{array} \right.
\end{equation}
where $a_0(w)$ satisfies
\begin{equation}
\left\{ \begin{array}{ll}
a_0(0) = 0 , & a_0(-w) = a_0(w), \\
a_0(w) > 0 , & \mbox{for} \qquad w \neq 0,
\end{array} \right.
\end{equation}
and the $L^\infty$ vector field $(P(x,t), \lambda(x,t)P(x,t))$, 
$P(x,t) \geq 0$, $P(x,0) \geq P_0 > 0$,
is divergence free:
\begin{equation}
\mbox{div} (P(x,t), \lambda (x,t)P(x,t)) = 0 .
\end{equation}
The major finding is that the solution of the Cauchy problem
(1.7) is independent of the choice of the $L^\infty$ function
$P(x,t) > 0$ with the same initial value $P(x,0)$.
This is essential for  using the framework to solve the uniqueness
problem for conservation laws (see Sections 4--6).  We remark
that Framework (A) does not require any regularity of $L^\infty$
solutions in contrast to, for example, $BV$ regularity that is the basic 
assumption
of all uniqueness theorems established for systems of conservation
laws, as in \cite{dg,di1,h2,li1,o2,v}.


The study of the limit behavior of dissipative approximate
solutions to conservation laws is another objective in this
paper.  Essentially, this problem is equivalent to the oscillation
problem, that is, to studying the dynamic behavior of initial
oscillations of $O(1)$ amplitude with high frequency, as
time evolves.
More specifically, we study the limit behavior of the solutions
corresponding to the oscillatory sequence of large initial data for
nonlinear hyperbolic systems of conservation laws.
This study can provide insight into how the behavior of
the system at the microscopic level affects the behavior of the
system at the macroscopic level.
Such results are important for understanding both the structure
of solutions for the Cauchy problem and the limit behavior of
numerical and analytical approximations for the Cauchy problem
with oscillatory initial data.

For genuinely nonlinear systems of conservation laws, many
compactness theorems have been established in \cite{di2,ch1,dcl,rs}
(see also the references cited therein).
These theorems indicate that, for $2 \times 2$ genuinely nonlinear systems,
the initial oscillations will instantaneously cancel as time evolves.
The basic strategy is to transfer the compactness problem of the solution
operator into proving that the corresponding Young measures are the
Dirac masses by solving the 
Tartar-Murat functional equation for the Young measures.
However, for the systems with at least a linearly degenerate field, the
Tartar-Murat equation cannot describe a complete structure for
the corresponding Young measures. 
This causes an analytical difficulty 
(see {\sc DiPerna} \cite{di3}, 
{\sc Rascle} \cite{ra} and {\sc Serre} \cite{s2}).
A suitable framework must 
be introduced to pick up
the lost information for the Young measures.
For a $2 \times 2$ system with one linearly degenerate
field, a rigorous framework is successfully established in \cite{ch1,ch2}, 
which
indicates that the initial oscillations propagate along the linearly
degenerate field and cancel along the genuinely nonlinear field.
We also refer the reader to  \cite{s2} and \cite{b1}
for some evidence of large oscillations for strictly hyperbolic
and linearly degenerate systems.
In Subsection 3c, we shall develop a more general framework
(Framework (B)), which will be shown to be useful for conservation
laws in Sections 4--6.

Section 4 is concerned with $n \times n$ systems with fully contact
fields, i.e., linearly degenerate characteristic fields, which are
described in \cite{s4}.  In  \cite{s4}, {\sc  Serre} established 
a global existence and
uniqueness theorem of  smooth solutions for  the Cauchy problem for
such  systems by finding a hierarchy of transport equations
involving higher derivatives of Riemann invariants $w_i$'s and by using the 
local
existence and uniqueness theorem of {\sc Kato} \cite{ka}.  
For $n = 2$, existence
theorems of global generalized solutions were established in
\cite{s1} for $BV$ solutions and in  \cite{ch3} for 
$L^\infty$ solutions.
In Section 4 we shall study well-posedness problems in the space
$L^\infty$ and the space $L^\infty \cap BV$:  the
existence, uniqueness, and stability of generalized solutions,
and the propagation problem of initial oscillations for such
$n \times n$ systems by using the quasidecoupling method established in
Section 3.

We remark that the basic assumption of all existing uniqueness
theorems for systems of conservation laws as in \cite{di1,dg,li1,o2}
is that the
solutions lie in the space $BV \cap L^\infty$.  In Section 4,
our uniqueness theorems are established only in the space $L^\infty$,
which is much broader than $BV \cap L^\infty$.

In Section 5 we shall establish uniqueness and stability
theorems for generalized solutions to systems with one contact field and
one line field, identified by {\sc Temple} \cite{te1}, with the aid of the
quasidecoupling method.
The physical background of such systems is several important models
arising in the fields of elasticity theory (cf. \cite{ke2,bh}), enhanced oil
recovery (cf. \cite{is,te2}), and magnetohydrodynamics (cf. \cite{fr,bh}). 
In this connection we note that 
some existence
theorems of generalized solutions  
have been established as in \cite{te2,ch2}, the propagation and
cancellation of  initial oscillations have been identified in
\cite{ch2,ch3} and a wellposedness result is obtained for
a polymer flooding model within a special class of solutions in \cite{tw}
for such systems.

Section 6 describes an example of how to reduce the uniqueness
and qualitative behavior problems of solutions for large systems
with linearly degenerate fields into the corresponding problems
of solutions for much smaller systems by using the quasidecoupling
method.  This example is the system of electromagnetic plane
waves, derived from the Maxwell equations by {\sc Serre} \cite{s3},
where he succeeded in solving global existence of $L^\infty$ solutions for
this system by using a related technique.
We believe from this example that the quasidecoupling method can
be useful for dealing with large systems of conservation laws with
linearly degenerate fields arising in many areas such as
magnetohydrodynamics, electromagnetic theory, and elasticity theory.

In Section 7 we shall give several remarks for further applications of the 
quasidecoupling method to some important problems and methods for 
conservation laws.




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\section{Entropy, Solutions, and the Young Measure}

We shall begin with several remarks concerning entropy, 
solutions for conservation laws, and the Young measure, which
motivate part of the subsequent development.
We recall that $u$ is called a weak solution of (1.1) with initial
data $u_0(x)$ if the integral relation
\begin{equation}
{\int \int} _ {\Pi_T} (u \phi _t + f(u) \phi _x )dxdt + \int_{-\infty} ^ \infty
\phi (x,0)u_0 (x)dx = 0
\end{equation}
holds for any function $\phi \in C_0^1 (\Pi_T)$, where
$\Pi_T = (-\infty , \infty ) \times [0,T]$.

One of the main features of conservation laws is that
uniqueness is lost within the broader class of weak solutions; many
weak solutions may share the same initial data.  Thus the problem
arises of identifying appropriate class of weak solutions (i.e., admissibility 
criteria) to single out physically relevant solutions.

In this connection we recall that a function $\eta : \R^n \to \R$ is
an entropy for (1.1) with entropy flux $q: \R^n \to \R$ if all
smooth solutions satisfy an additional conservation law of the form
$$
\eta(u)_t + q(u)_x = 0 ;
$$
that is,
\begin{equation}
\nabla q = \nabla \eta \nabla f .
\end{equation}

Taking the inner product of (2.2) with the right eigenvectors
$r_i$ of $\nabla f$ produces the characteristic form
\begin{equation}
(\lambda _i \nabla \eta - \nabla q ) \cdot r_i = 0,
\end{equation}
and
\begin{equation}
\lambda_i \eta _ {w_i} = q_{w_i},
\end{equation}
provided that the system (1.1) is endowed with globally defined
Riemann invariants $w_j$.
For $n=2$, the linear hyperbolic system (2.2) is equivalent
to (2.4), while the system (2.4) can be reduced to a linear 
second-order hyperbolic equation for which the Goursat problem is
well posed in the coordinates of Riemann invariants.
The entropy space is of infinite dimension and is
represented by two family functions of one variable $\alpha(w_1)$ and
$\beta(w_2)$, provided $\lambda _1 < \lambda _2$.
For $n \geq 3$, the system (2.2) is overdetermined and generally
prevents the existence of nontrivial entropy.
{\sc Friedrichs and Lax} \cite{fl} observed that most of the
conservative systems that result from continuum mechanics (in
one and several dimensions) are endowed with a globally defined,
strictly convex entropy.
A description of the systems endowed with rich entropy has been
addressed by {\sc Serre} \cite{s4}.
The algebraic characterization for such systems is
\begin{equation}
\partial_{w_k} ((\lambda _j - \lambda_i )^{-1} \partial_{w_j} \lambda _i ) =
\partial_{w_j} ((\lambda _k - \lambda _i )^{-1} \partial_{w_k}\lambda_i ),
\end{equation}
for distinct $i$, $j$, and $k$.

For the class of systems (1.1) endowed with a strictly convex
entropy $\eta$, {\sc Lax} \cite{la1} postulated the following entropy criterion:
a weak solution $u$ is admissible if
\begin{equation}
\eta (u)_t + q(u)_x \leq 0,
\end{equation}
in the sense of distribution.
The entropy inequality (2.6) is a proper statement of the
second law of thermodynamics for the equations of fluid dynamics:
the entropy increases along thermodynamic processes.  It is easy to
check that, on the discontinuity of $u$ with speed $\sigma$,
left state $u_-$ and right state $u_+$, (2.6) reduces to the local
condition
\begin{equation}
\sigma(\eta (u_+) - \eta(u_- )) - (q(u_+) - q(u_- )) \geq 0 .
\end{equation}
Note that the inequality (2.7) imposes a restriction on each point
of discontinuity, a fact that suggests that admissible solutions of (1.1)
may be determined by means of shock criteria. {\sc Lax} \cite{la1} proposed
the Lax
shock criterion for systems where each of the eigenvalues is either genuinely
nonlinear or linearly degenerate:
\begin{equation}
\lambda _i(u_+) \leq s \leq \lambda_i (u_-).
\end{equation}
The criterion governs the number and type of characteristics impinging
on a shock wave.
For more general systems, a comprehensive shock criterion has been
proposed by {\sc Liu} \cite{li2}:
\begin{equation}
s(\tau^+;u_-) \leq s(\tau; u_- ) , \qquad 0 \leq \tau
\leq \tau^+ ,
\end{equation}
where $s(\tau;u_-)$ denotes the speed of the shock that joins
$u(\tau;u_-)$ with $u_-$ determined by the Rankine-Hugoniot
condition.
On the other hand, it is known that the second law of thermodynamics is not
powerful enough to rule out all undesirable solutions in the
general thermodynamics processes. {\sc Dafermos} \cite{da1} proposed the
entropy rate criterion, which recognizes as admissible those solutions that 
maximize the rate of entropy production.

{\sc Dafermos} \cite{da2,da3} analyzed these criteria and showed that all these
criteria are equivalent when applied to weak solutions with
nondegenerate shock speed; for shocks of moderate strength, the 
shock criterion (2.9) and the  entropy rate criterion are
equivalent and generally more discriminating than either the Lax shock
criterion (2.8) and the entropy criterion (2.6) or (2.7).  For
shocks of great strength, the relationship among these criteria
is generally complicated; some peculiar phenomena for them have been
discussed in \cite{ch2,li3}.

In view of the existence theory of conservation laws 
(cf. {\sc Glimm} \cite{gl}),
it is natural to pose the class of solutions in the space $L^\infty$
or $L^\infty \cap BV$.
Here $BV(\Omega)$ denotes the space of functions that have locally
bounded total variation in the sense of Cesari; that is, a real-valued 
function $u = u(y) \in \R^n$ defined on a region
$\Omega \subset \R^m$ is an element of $BV(\Omega )$ if it is
 locally integrable and if its gradient is a locally finite Borel
measure, namely,
$$
\int _ \Omega u \nabla \phi dy = - \int_\Omega \phi d \mu,
$$
for all $\phi \in C_0^\infty (\Omega )$, where $\mu$ is locally
finite Borel measure.  On the other hand, $u \in BV(\Omega)$ if
and only if the local total variation in each $x_i$ is locally
integrable in $x_i ' = (x_1 , \ldots , x_{i-1} , x_{i+1} , \ldots ,
x_n )$:
$$
\int _ {|x_i '| \leq M} TVu(x_i', \cdot )dx_i ' < \infty,
$$
for all $M < \infty$.  It is known (cf. see {\sc Vol'pert} \cite{v}) that $u$
induces a partition of $\R^m$ into the union of three 
pairwise-disjoint subsets:
the set of points of Lebesgue approximate continuity of $u$, the
set $\Gamma(u)$ of points of Lebesgue
approximate jump discontinuity of $u$ contained
in the countable union of Lipschitz arcs, and the set of irregular
points whose  $(m-1)$-dimensional Hausdorff measure $H_{(m-1)}$ is zero.

In the context of generalized solutions $u \in BV$, the set
$\Gamma (u)$ is referred to as the set of shock waves and contact
discontinuities of solution $u$.
For simplicity, we assume that, for every
fixed $\bar{t} \in [0, \infty )$, $u( \cdot, \bar{t} )$ is a function
of bounded variation on $(-\infty, \infty )$ with one-sided limits
$u(\bar{x} \pm , \bar{t} )$ for every $\bar{x} \in (-\infty, \infty)$, 
and $u$ is equal to the
symmetric mean of the solution:  $u = \bar{u}$.  Here
$$
\bar{u} \equiv \lim _ {m \to \infty} \psi _m * u 
$$
is defined $H_1$ almost everywhere and coincides with $u$ $H_2$ almost
everywhere, where $\{ \psi _m \}$ is a sequence of radially symmetric
functions approximating the $\delta$-function.  Consider the
$B$-measure defined by
\begin{equation}
B_u = \eta (u) _t + q(u) _x ,
\end{equation}
where $(\eta ,q)$
is a  pair of $C^2$ entropy-entropy flux of (1.1).
Using the generalized Green's theorem for measure \cite{f}, the $B$-measure
can be expressed by
\begin{equation}
B_u(E) = B_u(E \cap \Gamma(u)) = \int _ {E \cap \Gamma (u)}
(n _t [\eta] + n _x [q])dH_1 .
\end{equation}
Here $\vec n = (n _x ,n _t )$ denotes the unit normal to
$E \cap \Gamma (u)$ in the sense of {\sc Federer} \cite{f}, and
the bracket denotes the jump in the enclosed quantity across
$E \cap \Gamma (u)$ in the direction of $\vec n$:
$$
[\eta] = [\eta] (P) = \eta \{ l_{\vec n} u(P)\} - \eta \{ l_{-\vec n} u(P) \} ,
$$
where $l_{\pm \vec n} u(P)$ denote the approximate limits of $u$ at
$P \in E \cap \Gamma (u)$ with respect to the half-planes
$\{ Q: < Q-P, \pm \vec n > \geq 0 \}$.

Normalize the direction of $\vec n$ by requiring that $n _x < 0$, and
refer to
$$
u_- \equiv l_{\vec n} u \qquad \mbox{  and  }  \qquad u_+ \equiv l_{-\vec n} u
$$
as the left and right approximate limits of $u$ at $P$, respectively.
Then the bracket denotes the jump from left to right across a shock
or a contact discontinuity.  We note that the set of irregular points
and points on which $n _x = 0$ for $u \in BV(\R_+^2)$ are a one-dimensional
Hausdorff zero measure set.  Therefore, we can write (2.11) as
\begin{equation}
B_u (E) = \int _ {E \cap \Gamma (u)} (\sigma [\eta] - [q] )dt ,
\end{equation}
where $\sigma \equiv - n _t (P) / n _x (P)$ denotes the speed of
propagation of $\Gamma (u) \cap E$ at $P$ and $dt \equiv - n _t dH_1$.

The Young measure provides an appropriate tool for studying
oscillations in solutions to (1.1), that is, the behavior of
oscillations of perturbed solutions $u^\epsilon$ as the
perturbation parameter $\epsilon$ goes to zero.
We recall from {\sc Tartar} \cite{ta1} and {\sc Murat} \cite{m1,m2} that, 
for an
arbitrary sequence of measurable maps
$$
u^\epsilon : \R^m \to \R^n, \qquad  \|u^\epsilon\|_{L^\infty} \le M < \infty,
$$
that converges in the weak-star topology of $L^\infty$ to a function
$u$,
$$
w^* - \lim _ {\epsilon \to 0} u^\epsilon = u,
$$
there exists a subsequence (still labeled) $u^\epsilon$ and a family
of Young measures
$$
\nu _y \in \mbox{Prob} (\R^n), \quad \mbox{supp } \nu_y \subset
\{ \lambda : |\lambda| \leq M \} , \qquad y \in
\R^m,
$$
such that for all continuous functions $g$,
$$
w^* - \lim _ {\epsilon \to 0} g(u^\epsilon (y)) = \int _ {\R^n}
g(\lambda )d\nu _y \equiv <\nu _y ,g(\lambda )>,
$$
for almost all points $y \in \R^m$.
In particular, $u^\epsilon $ converges strongly to $u$ if and
only if the Young measure $\nu _y$ at almost all points $y$ is equal to
a Dirac mass concentrated at $u(y)$.  Then, we have the following theorem.

\begin{theorem}
Let $\Omega \subset \R_+^2$ be a bounded open set and $u^\epsilon :
\Omega \to \R^n$ be measurable functions satisfying

$\mbox{(i)  }  \|u^\epsilon \|_{L^\infty} \leq M.$ 

$\mbox{(ii)   For continuous function pairs } (\eta_i(u), q_i(u)) (i=1,2),$ 

\begin{equation}
 \frac{\partial \eta _i (u^\epsilon )}{\partial t} +
\frac{\partial q_i(u^\epsilon )}{\partial x} , \qquad \mbox{  compact in  }
H_{\rm {loc}} ^{-1} (\Omega) .
\end{equation}
Then there exists a subsequence (still labeled) $u^\epsilon$ and a
family of Young measures \\
$\{ \nu _{x,t} : \R_+^2 \to \mbox{Prob}
(\R^n)\}$, $\mbox{supp} \nu_{x,t} \subset \{ \lambda : |\lambda|
 \leq M \}$, such that for all continuous functions $g$,
\begin{equation}
w^* - \lim_{\epsilon \to 0} g(u^\epsilon (x,t)) = \int_{\R_+^2} g(
\lambda )d \nu_{x,t} \equiv < \nu_{x,t} ,g(\lambda )>,
\end{equation}
and
\begin{equation}
\left< \nu_{x,t} , \left| \begin{array}{ll}
\eta_1 (\lambda ) & q_1(\lambda ) \\
\eta_2 (\lambda ) & 
q_2(\lambda ) \end{array}
\right| \right> = \left| \begin{array}{ll}
<\nu_{x,t} ,\eta_1 (\lambda )> & <\nu_{x,t} , q_1(\lambda )> \\
<\nu_{x,t} , \eta_2 (\lambda )> & <\nu_{x,t} ,q_2(\lambda )>
\end{array} \right| , \qquad \mbox{a.e. } \R_+^2 .
\end{equation}
\end{theorem}

Next, we consider the system (1.1) endowed with globally defined
Riemann invariant coordinates and satisfying the formula (2.5).
Suppose that one of its eigenvalues $\lambda _i$ is linearly
degenerate in the sense of {\sc Lax} \cite{la2}:
\begin{equation}
\partial_{w_i} \lambda _i \equiv 0 , \qquad \mbox{  i.e.,  } \qquad
\lambda _i = \lambda _i (w_i ' ) ,
\end{equation}
where $w_i ' = (w_1 , \ldots , w_{i-1} , w_{i+1} , \ldots ,
w_n ) \in \R^{n-1}$.
Then we deduce from (2.4) that there exists a family of entropy-entropy
pairs $(\eta _i , q_i )$ corresponding to the linearly degenerate
field which are of the form
\begin{equation}
\left\{ \begin{array}{l}
\eta_i = N_i a(w_i ) , \\
q_i = N_i \lambda_i (w_i ' ) a(w_i ) ,
\end{array} \right.
\end{equation}
where $a$ is any continuous function of the one-dimensional variable
$w_i$ and
\begin{equation}
N_i = \exp \left\{ \int^{w_j} \frac{\partial_{w_j} \lambda_i}
{\lambda_j - \lambda_i} dw_j \right\} , \qquad j \neq i,
\end{equation}
is independent of $j(\neq i)$ because of (2.5).
If one can choose $a(w_i)$ such that the associated
$\eta_i$ is convex, then we have
\begin{equation}
\partial_t (N_i a(w_i)) + \partial_x (\lambda_i
N_i a(w_i)) \leq 0,
\end{equation}
in the sense of distribution.
Generally, there always exists an $a_0(w_i) \geq 0$ such that
\begin{equation}
\partial _t (N_i a_0 (w_i)) + \partial_x (\lambda_i N_i
a_0 (w_i)) = 0,
\end{equation}
derived from one equation of the system (1.1).

Using the notation of the Young measures $\nu_{x,t} \in
\mbox{Prob}(\R^n)$ for the dissipative approximate solutions, 
we can obtain the associated average
inequality of (2.19) and equality of (2.20).
In particular, if the Young measures $\nu_{x,t}$ are of the following property:
for any continuous function $a(w_i)$, there exists an $L^\infty$
function $\bar{\lambda}_i(x,t)$ such that
\begin{equation}
<\nu_{x,t} , \lambda_i (w_i')a(w_i)N_i> = \bar{\lambda} _i (x,t)
<\nu_{x,t} ,a(w_i)N_i> \qquad \mbox{ a.e. } ,
\end{equation}
then we have
\begin{equation}
\partial_t<\nu_{x,t} ,N_ia(w_i)> + \partial_x(\bar{\lambda}_i
(x,t) < \nu_{x,t} ,N_ia(w_i)>) \leq 0 ,
\end{equation}
\begin{equation}
\partial_t<\nu_{x,t} ,N_ia_0(w_i) > + \partial_x (\bar{\lambda}
_i(x,t) < \nu_{x,t} ,N_ia_0(w_i) > ) = 0,
\end{equation}
in the sense of distribution.

Finally, we remark that, if $S_0 = \{ (x,t): N_i(x,t) = 0 \} \neq 
\phi$ for nonstrictly hyperbolic cases, we can generalize the sense of 
(2.19) to a stronger sense that
the inequality (2.19) holds in the generalized sense of distribution;
that is, for any
$\phi \in C_0^1 (\Pi_T - S_0 ) \cap BV(\Pi _T),  \phi \geq 0,
$
\begin{equation}
\int \int _ {\Pi _T} N_i a(w_i)(\phi_t + \lambda _i \phi _x)dxdt +
\int _ {- \infty}^\infty (N_ia(w_i))(x,0)\phi (x,0)dx \geq 0.
\end{equation}
Similarly we can also assume that the inequality (2.22), and the equalities
(2.20) and (2.23) hold in the same sense.




\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{lemma}{0}
\setcounter{corollary}{0}
\section{Quasidecoupling Method}

In this section we  describe the quasidecoupling method
for studying the existence, uniqueness, and qualitative behavior of solutions
to systems of conservation laws with linearly degenerate fields.

\vspace{.3in}
\noindent
{\bf  a.  Quasidecoupling Transformation}
\vspace{.2in}
\noindent

Let $w: \R_+^2 \to \R$ be a bounded and measurable function
satisfying the following inequality:
\begin{equation}
\partial_t(a(w;x,t)P(x,t)) + \partial_x(a(w;x,t)\lambda(x,t)
P(x,t)) \leq 0,
\end{equation}
with the initial condition
\begin{equation}
w(x,0) = w_0(x),
\end{equation}
in the generalized sense of distribution; that is, 
the inequality
\begin{equation}
\int \int_{\Pi_T} a(w;x,t)P(x,t)(\phi_t+\lambda(x,t)\phi_x)dxdt
+ \int_{-\infty}^\infty a(w_0(x);x,0)P(x,0)\phi(x,0)dx \geq 0 
\end{equation}
holds  for any
nonnegative function $\phi \in C_0^1 (\Pi_T - \{ P(x,t)=0\}) \cap
BV(\Pi_T)$. 
Here the function $a(w;x,t)$ is continuous with respect to $w$ and uniformly
bounded with respect to $(x,t)$, and
$(P(x,t),\lambda(x,t)P(x,t))$, $ P(x,t) \geq 0$, is a bounded and
measurable vector field whose divergence vanishes in the sense
of distribution
\begin{equation}
\mbox{div}(P(x,t), \lambda(x,t)P(x,t)) = 0,
\end{equation}
and
\begin{equation}
\int_0^\infty P(x,0)dx = \int_{-\infty}^0 P(x,0)dx = + \infty .
\end{equation}
Then we have the following theorem.
\begin{theorem}
There exists a globally defined coordinate transformation
$$
T: (x,t) \to (p,t) \equiv (T_t(x),t)
$$
such that, in the new coordinate system $(p,t) = T(x,t)$,
\begin{equation}
\left\{ \begin{array}{l}
\partial_t a(w(T^{-1}(p,t));T^{-1}(p,t)) \leq 0 , \\
w|_{t=0} = w_0(T_0^{-1}(p)),
\end{array} \right.
\end{equation}
in the sense of distribution; that is,
\begin{equation}
\int \int_{\Pi_T} a(w(T^{-1}(p,t));T^{-1}(p,t)) \psi(p,t)_t dpdt 
+ \int_{- \infty}^ \infty a(w_0(T_0^{-1}(p));T^{-1}(p,0))\psi(p,0)dp \geq 0,
\end{equation}
for all nonnegative functions $\psi \in C_0^1(\Pi_T)$, where $T_0^{-1}(p)$
is the  inverse map of $T_t(x)|_{t=0}= p(x,0) = \int_0^x P(\xi,0)d\xi$,
uniquely determined by $P(x,0)$.
\end{theorem}
{\bf Remark}.
Under the transformation $T$, the divergence form
of the vector field
$$
(a(w;x,t)P(x,t), a(w;x,t)\lambda(x,t)P(x,t))
$$ 
is transformed
to a complete single differential form 
$\partial_ta(w(T^{-1}(p,t));T^{-1}(p,t))$, and
the inequality in (3.6) is formally independent of the vector fields
$(P,\lambda P)$ in the new coordinate system $(p,t)$.
For these reasons, we call the transformation $T$ a ``quasidecoupling
transformation'' corresponding to the vector field $(P, \lambda P)$.

\begin{lemma}
Let the $L^\infty$ vector field $(P(x,t),Q(x,t))$, $P(x,t) \geq 0$,
be  divergence free in the sense of distribution in $\Pi_T$ and
$$
\int_\alpha ^\beta P(x,0)dx = + \infty , \qquad - \infty \le \alpha
< \beta \le + \infty .
$$
Then, for almost all $t \in (0,T)$,
$$
\int_\alpha^\beta P(x,t)dx = +\infty .
$$
\end{lemma}
{\bf Proof}.
The proof is simple.  
Similar to \cite{w}, we choose
two sequences of nonnegative smooth functions $\phi_\epsilon (\tau)$
and $\psi_\delta (x)$ with compact support for any $t \in (0,T)$, 
$$
\begin{array}{l}
\phi_\epsilon (\tau ) = \left\{ \begin{array}{ll}
1, & \tau \in [0,t] , \\
0, & \tau \in [t+\epsilon , T] ,
\end{array} \right. \\
\psi_\delta (x) = \left\{ \begin{array}{ll}
1, & x \in [\alpha , \beta ] , \\
0, & x \in (-\infty ,\alpha - \delta ] \cup [\beta + \delta , + \infty ),
\end{array} \right.
\end{array}
$$
such that
$$
\int_0^T |\phi _ \epsilon ' (\tau)|d \tau = \int_{- \infty}^\infty
|\psi _ \delta ' (x)|dx = 2 .
$$

Since $\mbox{div}(P(x,t),Q(x,t)) = 0$, we have
$$
\int \int _ {\prod _T} (\phi _ \epsilon ' (\tau)\psi_\delta (x)
P(x,t) + \phi _ \epsilon (\tau)\psi _\delta ' (x)Q(x,t))dxdt +
\int_{-\infty}^\infty \psi_\delta (x)P(x,0)dx = 0 .
$$
Set $\epsilon \to 0$.  We obtain
$$
\left| \int_{-\infty}^\infty \psi _ \delta (x)(P(x,t) -
P(x,0))dx \right| \leq 2 \| Q \| _ {L^\infty} t,
$$
for any Lebesgue point $t$ of all locally integrable functions
$t \to \int \psi _ \delta (x)P(x,t)dx$.
Set $\delta \to 0$.
We have that, for almost all $t \in (0,T)$,
$$
\left| \int _ \alpha ^ \beta (P(x,t)-P(x,0))dx \right| \leq 2 \| Q \| _
{L^\infty} t.
$$
Therefore
$$
\int_\alpha^\beta P(x,t)dx = \int_\alpha^\beta P(x,0)dx = + \infty .
$$
{\bf Proof of Theorem 3.1.}
Notice that the vector field $(P(x,t),\lambda(x,t)P(x,t))$ is
divergence free.
We introduce a coordinate transformation
$$
T: (x,t) \to (p,t),
$$
determined by
\begin{equation}
\left\{ \begin{array}{l}
\frac{\partial p}{\partial x} =P(x,t), \\
\frac{\partial p}{\partial t} = -\lambda (x,t)P(x,t),\\
p(x,0) =\int_0^x P(\xi,0)d\xi.
\end{array} \right.
\end{equation}
Then the transformation $T$ is well defined, and there exists a
globally defined Lipschitz solution $(p,t) = T(x,t)$ in the space
of distribution.  From (3.5) and Lemma 3.1, we have
$$
\int_0^\infty P(x,t)dx = \int_{-\infty}^0 P(x,t)dx = +\infty .
$$
Then $T$ is proper and onto in $\R_+^2$.

If $P(x,t) > 0$, then $T$ is a bi-Lipschitz homeomorphism with
the Jacobian $P(x,t) > 0$.
Then we have from (3.3) that, for any nonnegative function
$\psi \in C_0^\infty (\Pi_T)$,
\begin{equation}
\int \int_{\Pi_T} a(w;x,t)P(x,t)(\psi_t + \lambda(x,t)\psi_x)dxdt 
+ \int_{-\infty}^\infty a(w_0(x); x,0)P(x,0)\psi(x,0)dx \geq 0.
\end{equation}
Make the coordinate transformation (3.8) for the integrals in (3.9).
We immediately obtain
$$
\int \int _ {\Pi_ T} a(w(T^{-1}(p,t));T^{-1}(p,t))\psi(p,t)_tdpdt 
+ \int_{-\infty}^
\infty a(w_0(T_0^{-1}(p)); T^{-1}(p,0)) \psi (p,0)dp \geq 0.
$$
This arrives at (3.7).

If the set $\{(x,t): P(x,t) = 0 \} \neq \phi$, we can verify the above
process of proof under the assumption that (3.1) holds in the generalized
sense of distribution and prove that the inequalities (3.1) and (3.6)
are equivalent by arguments similar to {\sc Wagner} \cite{w}.  We omit
the details.

A useful fact is the following corollary. 

\begin{corollary}
Let $T$ be the quasidecoupling transformation (3.8).  Then for
any $g \in L^1(\Pi _T)$,
$$
\int_{-\infty}^\infty g(p,t)dp = \int_{-\infty}^{\infty} g(T(x,t))
P(x,t)dx
$$
holds for almost all $t \in (0,T)$.
\end{corollary}

\vspace{.3in}
\noindent
{\bf b.  Framework for the Uniqueness and Stability of Solutions}

\vspace{.2in}
\noindent

Suppose that $L^\infty$ functions $w(x,t)$ and $z(x,t)$ satisfy the
following Framework (A).

\noindent
A1:  $w(x,t)$ is the solution of the Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
\partial _t (a_0(w-k)P_1(x,t)) + \partial _x (a_0(w-k)\lambda_1(x,t)
P_1(x,t)) \leq 0, \mbox{  for any  } k \in (-\infty , \infty ), \\
w(x,0) = w_0(x),
\end{array} \right.
\end{equation}
in the generalized sense of distribution in $\Pi_T$.

\noindent
A2:  $z(x,t)$ is the solution of the Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
\partial_t (a_0(z-l)P_2(x,t)) + \partial_x (a_0
(z-l)\lambda_2 (x,t)P_2 (x,t)) \leq 0, \mbox{  for any  } l \in
(- \infty , \infty ) , \\
z(x,0) = z_0 (x),
\end{array} \right.
\end{equation}
in the generalized sense of distribution in $\Pi_T$.\\
Here $a_0$ is a nonnegative and even Lipschitz function
satisfying $a_0(0)=0, a_0(w)>0$ for $w \ne 0$, and
the vector fields $(P_i(x,t),\lambda_i(x,t)P_i(x,t))$, 
$ P_i(x,t) \geq 0$, $i=1,2$, are two divergence-free fields in 
$(x,t)$-space
and satisfy
$$
P_1(x,0) \geq P_0 > 0, \qquad P_2(x,0) \geq P_0 > 0.
$$
Then we have the following theorem.

\begin{theorem}
Suppose that $L^\infty$ functions $w(x,t)$ and $z(x,t)$ satisfy
Framework (A) in $\Pi_T$.  Then 
\begin{equation}
w(x,t) = w_0(T_0^{-1}(p(x,t))), \qquad z(x,t) = z_0(\bar{T}_0^{-1}(q(x,t))), 
\qquad \mbox{ a.e.  } t \in [0,T),
\end{equation}
and, for any positive constant $N$,
\begin{equation} \left \{\begin{array}{l}
\int_{|x| \leq N} 
a_0 (w(x,t)-z(\bar{T}_t^{-1}T_t(x), t))
P_1(x,t)P_2(\bar{T}_t^{-1}T_t(x), t)dx \\
\mbox{        }\leq \|P_1\|_{L^\infty}
\|P_2\|_{L^\infty} \int _
{|x| \leq K_0(N + Lt)} a_0 (w_0 (x)-z_0 (\bar{T}_0^{-1}T_0(x)))dx , \\
\int_{|x| \leq N} a_0 (w(T_t^{-1}\bar{T}_t(x),t)-z(x,t))
P_1(T_t^{-1}\bar{T}_t(x),t)P_2(x,t)dx \\
\mbox{       } 
\leq \|P_1\|_{L^\infty}\|P_2\|_{L^\infty}
\int_{|x| \leq K_0(N + Lt)} a_0 (w_0 (T_0^{-1}\bar{T}_0(x))-z_0(x))dx,
\end{array}  \right.
\end{equation}
or,
\begin{equation} \left \{\begin{array}{l}
\int_{|x| \leq N}
a_0(w(x,t)-z(\bar{T}_t^{-1}\bar{T}_0T_0^{-1}T_t(x),t))
P_1(x,t)P_2(\bar{T}_t^{-1}\bar{T}_0T_0^{-1}T_t(x),t)dx,\\
\mbox{        }\le \|P_1\|_{L^\infty}\|P_2\|_{L^\infty}
\int_{|x| \le K_0(N + Lt)} a_0(w_0(x) - z_0(x))dx \\
\int_{|x| \le N} a_0(w(T_t^{-1}T_0\bar{T}_0^{-1}\bar{T}_t(x),t)-z(x,t))
P_1(T_t^{-1}T_0\bar{T}_0^{-1}\bar{T}_t(x),t)P_2(x,t)dx \\
\mbox{        } \le \|P_1\|_{L^\infty}\|P_2\|_{L^\infty}
\int_{|x| \le K_0(N+Lt)}a_0(w_0(x)-z_0(x))dx,
\end{array}  \right.
\end{equation}
for almost all $t \in [0,T)$, where $(p, t) = T(x,t) \equiv (T_t(x),t)$ 
and $(q,t) = \bar{T}(x,t) \equiv (\bar{T}_t(x),t)$ 
are the quasidecoupling transformations corresponding to the vector fields
$(P_i, \lambda_i P_i), i=1,2,$ respectively, 
$L = \max _ {i=1,2} (\sup _
{\Pi_T} |\lambda _i(x,t)|)$, and $K_0 = max_{i=1,2}\|P_i\|_{L^\infty}/P_0$.
\end{theorem}

We first introduce a lemma from \cite{ch3}.

\begin{lemma}
Suppose that functions $u(x,t) \in L^\infty ([0,T];L^1(-\infty ,\infty ))$ and $u_0(x) \in L^\infty$ satisfy
$$
\int \int _ {\Pi_T} u(x,t) \psi ' (t)dxdt + \psi (0) \int _ {- \infty}
^ \infty u_0(x)dx \geq 0,
$$
for any nonnegative function $\psi (t) \in C_0^1 [0,T)$.  Then
$\lim _ {t \to 0} \int _ {-\infty}^\infty u(x,t)dx$ exists and
satisfies
$$
\int _ {- \infty}^\infty u(x,t)dx \leq \lim _ {t \to 0} \int _
{-\infty}^\infty u(x,t)dx \leq \int_{-\infty}^\infty u_0(x)dx ,
\qquad \mbox{a.e.  } t \in [0,T).
$$
\end{lemma}

\noindent
{\bf Proof of Theorem 3.2.}
Using the quasidecoupling transformation (3.8) for (3.10) and (3.11), 
respectively, we obtain from Theorem 3.1 that the function
$w(T^{-1}(p,t))$ satisfies
\begin{equation} \left \{\begin{array}{l}
\partial_t a(w(T^{-1}(p,t))-k) \le 0, \\
w|_{t=0} = w_0(T_0^{-1}(p)),
\end{array} \right.
\end{equation}
in the sense of distribution under the  $(p,t)=T(x,t)$ coordinates
for any $k \in \R^1$,
and the function $z(\bar{T}^{-1}(q,t))$ satisfies
\begin{equation} \left\{\begin{array}{l}
\partial_t a(z(\bar{T}^{-1}(q,t))-l) \le 0,\\
z|_{t=0} = z_0(\bar{T}_0^{-1}(q)),
\end{array} \right.
\end{equation}
in the sense of distribution under the $(q,t)= \bar{T}(x,t)$ coordinates
for any $l \in \R^1$.

First we show (3.12). 
Suppose that $\psi (p,t; p') \geq 0$ is a $C^\infty (\Pi_T \times
\R^1)$ function and, for any fixed $p' \in \R^1 (p \in \R^1)$,
 $\psi (p,t;p') \in C_0^\infty (\Pi_T)$.
Now we choose $k = w_0(T_0^{-1}(p'))$ and the test function $\psi (\cdot ,
\cdot ; p')$ for (3.15).
We obtain
\begin{equation} \begin{array}{l}
\int \int _ {\Pi_T} a_0 (w(T^{-1}(p,t))-w_0(T_0^{-1}(p'))) \psi _t (p,t; p')
dpdt \\ 
+ \int _{- \infty}^\infty a_0(w_0 (T_0^{-1}(p)) - w_0(T_0^{-1}(p')))
\psi (p,0;p')dp \geq 0 .
\end{array}
\end{equation}

Integrate (3.17) over $p' \in \R^1$.
We have
$$
\int_{-\infty}^{\infty}{\int \int} _ {\Pi_T} 
a_0 (w(T^{-1}(p,t)) - w_0(T_0^{-1}(p')))
\psi_t (p,t; p')dpdtdp'
$$
\begin{equation}
\mbox{          } + \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
a_0(w_0(T_0^{-1}(p))-w_0(T_0^{-1}(p')))\psi (p,0;p')dpdp' \geq 0.
\end{equation}

Motivated by {\sc Kruzkov} \cite{kr}, we choose
\begin{equation}
\psi(p,t;p') = \phi(p,t)\omega_h(p-p').
\end{equation}
Here
$$
\begin{array}{l}
\phi (p,t) \geq 0, \qquad \phi (p,t) \in C_0^\infty (\Pi_T ) , \\
\omega_h(s) = \frac{1}{h} \omega (\frac{s}{h} ) ,
\end{array}
$$
where $\omega (s)$ is the function satisfying
$$
\omega (s) \in C_0^\infty (-\infty , \infty ) , \quad \omega(s) \geq 0,
\quad \omega(s) = \omega (-s), \quad \mbox{supp } \omega (s) \subset
\{ s: |s| \leq 1 \},
$$
and
$$
\int _ {-\infty}^\infty \omega (s)ds = 1.
$$
Then we obtain from (3.18) that
$$
\int_0^T \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} 
a_0 (w(T^{-1}(p,t))-w_0(T_0^{-1}(p')))
\omega_h(p-p')\phi_t(p,t)dtdpdp'
$$
\begin{equation}
\mbox{   } + \int_{-\infty}^\infty \int_{-\infty}^\infty a_0
(w_0(T_0^{-1}(p))-w_0(T_0^{-1}(p')))\omega_h(p-p') \phi (p,0)dpdp' \geq 0.
\end{equation}

On the other hand, for any 
$\phi \in C_0^\infty (\Pi_T)$, $\mbox{supp }\phi(\cdot ,t) \subset
\{ p: |p| \leq N < \infty \}$, 
 we have
\begin{equation}
\begin{array}{l}
A_h  \equiv  \left| \int_0^T \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p')))\omega_h(p-p')\phi_t(p,t)dtdpdp'
 \right.
\\
\qquad \ \  
- \int_0^T \int_{-\infty}^{\infty} a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p)))
\phi_t(p,t)dpdt | \\
 \leq  \frac{C}{h} \int_0^T{\int\int}_{|p-p'| \leq h, |p| \leq N} 
|a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p')))
- a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p))) | dtdpdp'\\
 \leq  C   \int_{|\alpha| \le h}
H_h (\alpha )d\alpha,
\end{array}
\end{equation}
by using the Lipschitz condition of $a_0$, where
$$
H_h(\alpha) = \frac{1}{h} \int_{
|\beta| \leq h}|w_0(T_0^{-1}(\alpha))
- w_0(T_0^{-1}(\alpha - \beta))| d\beta.
$$
Since almost all points $\alpha$ of $[-N,N]$ are the Lebesgue points of 
$H_h(\alpha) $, we deduce 
$$
\lim _ {h \to 0} H_h (\alpha ) = 0 , \qquad \mbox{a.e. }
\alpha \in  [-N,N].
$$
Note that $|H_h (\alpha )| \leq  4 \ \mbox{sup}|w_0|$.
Using the Lebesgue dominated theorem, we conclude from (3.21) 
\begin{equation}
\lim _ {h \to 0} A_h = 0.
\end{equation}
Moreover, we have
\begin{equation}
\lim_{h \to 0} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
a_0(w_0(T_0^{-1}(p))-w_0(T_0^{-1}(p')))\omega_h(p-p')\psi(p,0)dpdp' = 0.
\end{equation}

Combining (3.20) with (3.21)-(3.23), we
finally obtain
\begin{equation}
{\int \int}_{\Pi_T} a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p)))\phi_t(p,t)dpdt 
\geq 0.
\end{equation}
Choose $\phi (p,t) = \phi (p) \psi (t)$, $\phi (p) \in C_0^\infty
(-\infty , \infty )$, $\psi(t) \in C_0^1 [0,T)$.
Then
\begin{equation}
\int_0^T \int_{-\infty}^\infty a_0 (w(T^{-1}(p,t))-w_0(T_0^{-1}(p))) \phi (p)
\psi ' (t)dpdt 
\geq 0,
\end{equation}
for  almost all $t \in [0, T)$.
Using Lemma 3.2, we have
\begin{equation}
\int_{-\infty}^\infty a_0(w(T^{-1}(p,t))-w_0(T_0^{-1}(p))) \phi (p)dp \leq 0,
\end{equation}
for almost all $t \in [0,T)$.
Therefore,
$$
w(T^{-1}(p,t)) = w_0(T_0^{-1}(p));
$$
that is,
$$
w(x,t) = w_0(T_0^{-1}(p(x,t))), \qquad  \mbox{ a.e.  } 
\{(x,t) \in \Pi_T: P_1(x,t) > 0\}.
$$
Similarly, we also have
$$
z(x,t) = z_0(\bar{T}_0^{-1}(q(x,t))), \qquad  \mbox{ a.e.  } 
\{(x,t) \in \Pi_T: P_2(x,t) > 0\}.
$$
This proves (3.12).

Notice that, for fixed $t$, both $p(x,t)$ and $q(x,t)$ are monotonically
increasing functions of $x$. If $x$ lies in $[-N,N]$, then both
intervals $[p(-N,t)/P_0, p(N,t)/P_0]$ and $[q(-N,t)/P_0, q(N,t)/P_0]$ lie in
$[-K_0(N+Lt), K_0(N+Lt)]$, where $L = \max_{i=1,2} \sup_{\Pi_T} |\lambda_i
(x,t)|$ and $K_0 = \max_{i=1,2}\|P_i\|_{L^\infty}/P_0$. Therefore, 
for any nonnegative function 
$\psi(t) \in C_0^{\infty}[0,T]$ and any positive constant $N$, 

$$
\begin{array}{l}
\int_0^T\int_{|x| \le N} 
a_0(w(x,t)-z(\bar{T}_t^{-1}T_t(x),t))
P_1(x,t)P_2(\bar{T}_t^{-1}T_t(x),t)\psi(t)dxdt\\
=\int_0^T \int_{|x| \le N} 
a_0(w_0(T_0^{-1}(T_t(x)))-z_0(\bar{T}_0^{-1}(T_t(x))))
P_1(x,t)P_2(\bar{T}_t^{-1}T_t(x),t)\psi(t)dxdt\\
\le \|P_2\|_{L^\infty}\int_0^T \psi(t)dt 
\int_{|p| \le N \|P_1\|_{L^\infty} + Lt}
a_0(w_0(T_0^{-1}(p))-z_0(\bar{T}_0^{-1}(p)))dp.
\end{array}
$$
Then, for almost all $t \in [0,T)$,
$$
\int_{|x| \leq N} 
a_0(w(x,t)-z(\bar{T}_t^{-1}T_t(x),t))
P_1(x,t)P_2(\bar{T}_t^{-1}T_t(x),t)dx 
$$
$$
\leq \|P_2\|_{L^\infty}\int_{|p| \leq N \|P_1\|_{L^\infty}+Lt}
a_0(w_0(T_0^{-1}(p))- z_0(\bar{T}_0^{-1}(p)))dp
$$
$$
\leq \|P_1\|_{L^\infty}\|P_2\|_{L^\infty} \int_{|x| \leq K_0(N + Lt)} 
a_0(w_0(x)-z_0(\bar{T}_0^{-1}T_0(x)))dx .
$$
Similarly, we can also prove other inequalities in (3.13) and (3.14).
This completes the proof of Theorem 3.2.

Now we consider the deviation between $(p(x,t),t)\equiv (p(x,t; w), t)
= T(x,t)$ and \\
$(q(x,t), t) \equiv (q(x,t; z), t) = \bar{T}(x,t)$.


\begin{theorem}
Suppose that  an $L^\infty$ function $\lambda(y, t; w_0(y))$
satisfies
\begin{equation}
\int_{|y| \le N_0}|\lambda(y,t; w_0(y))-\lambda(y,t; \bar{w}_0(y))|dy
\le \int_{|y| \le N_0}|w_0(y)-\bar{w}_0(y)|dy,
\end{equation}
and the Lipschitz functions $p(x,t)$ and $q(x,t)$ are solutions of 
the Cauchy problems
\begin{equation}\left\{\begin{array}{l}
p_t + \lambda(p, t; w_0(p))p_x = 0, \\
p|_{t=0} = p_0(x),
\end{array} \right.
\end{equation}
and
\begin{equation} \left \{\begin{array}{l}
q_t + \lambda(q,t; \bar{w}_0(q))q_x = 0, \\
q|_{t=0} = q_0(x),
\end{array} \right.
\end{equation}
respectively. Then, for almost all $t \in [0, T]$,
\begin{equation}
\int_{|x| \le N}|p(x,t)-q(x,t)|dx \le C\int_{|x|\le \bar{N} + Lt}
(|w_0(x)- \bar{w}_0(x)| + |p_0(x) - q_0(x)|)dx,
\end{equation}
for any $N > 0$, 
where $L = \max(\sup_{\Pi_T}|\lambda(y, t; w_0(y))|,  
\sup_{\Pi_T}|\lambda(y, t; \bar{w}_0(y))|)$, \\ 
$\bar{N} = \max(\sup_{[-N, N] \times [0, T]}|p_0(x)|, 
\sup_{[-N, N] \times [0, T]}|q_0(x)|, N)$, and
$C$ is the constant independent of $N$ and $L$.
In particular, if 
\begin{equation} \left \{\begin{array}{l}
w_0(y) = \bar{w}_0(y),  \qquad \mbox{a.e.}  \\
p_0(x) = q_0(x), \qquad \mbox{a.e.}
\end{array} \right.
\end{equation}
then
$$
p(x, t) = q(x, t), \qquad \mbox{  a.e.   } t \in [0, T].
$$
\end{theorem}

\noindent
{\bf Proof.}
First we have from (3.28)-(3.29) that

$$
(p-q)_t + (\int_q^p \lambda(\xi, t; w_0(\xi))d\xi)_x
+( \lambda(q, t; w_0(q)) - \lambda(q, t; \bar{w}_0(q)))q_x = 0.
$$
Therefore we conclude, for almost all $(x,t) \in \Pi_T$,

$$
|p-q|_t + (\int_q^p \lambda(\xi, t; w_0(\xi))d\xi\mbox{sign}(p-q))_x 
+ \mbox{sign}(p-q)(\lambda(q, t; w_0(q)) - \lambda(q, t; \bar{w}_0(q)))q_x = 0.
$$

Finally, we can obtain from
the boundness of the dependent domain of $p-q$ 
that, 
for almost all $t \in [0, T]$,
$$
\begin{array}{l}
\int_{|x| \le N}|p(x,t) - q(x,t)| dx  \\ 
\le C \left \{\begin{array}{l}
	\int_{|x| \le N+Lt} |p_0(x) - q_0(x)|dx  \\
         + \int_0^t \int_{|x| \le N} 
|\lambda(q(x,s), s; w_0(q(x,s))) - \lambda(q(x,s), s; \bar{w}_0(q(x,s)))|dxds 
 \end{array}   \right\} \\
\le C
{\int_0^t \int_{|x| \le \bar{N} + Ls}|\lambda(x, s; w_0(x))
- \lambda(x, s; \bar{w}_0(x))|dxds
+ \int_{|x|\le N+Lt} |p_0(x) - q_0(x)|dx} \\
\le C \int_{|x| \le \bar{N} + Lt}(|w_0(x) - \bar{w}_0(x)| + |p_0(x)-q_0(x)|)dx.
\end{array}
$$
This completes the proof.


A direct corollary follows.

\begin{corollary}
Suppose that there exists an $L^\infty$ function $\lambda(y,t)$ such that
\begin{equation} \left \{\begin{array}{l} 
\lambda_1(x,t) = \lambda(p(x,t),t), \\
\lambda_2(x,t) = \lambda(q(x,t),t),
\end{array} \right.
\end{equation}
and suppose the function $a_0(w) = |w|^m$, $m > 0$, and $P_1(x,0)=P_2(x,0)$
in the Framework(A).
Then the weighted $L^m$-stability with weight $(P_1P_2)(x,t)$
 for functions $w(x,t)$ and $z(x,t)$ holds; that is, for
almost all $t \in [0,T)$,
\begin{equation}
\int_{|x| \leq N} |w(x,t)-z(x,t)|^m P_1(x,t)P_2(x,t)dx 
\leq \|P_1\|_{L^\infty}\|P_2\|_{L^\infty}
\int _ {|x| \leq K_0(N + Lt)} |w_0 (x)-z_0 (x)|^m dx,
\end{equation}
where $L = \max_{i=1,2} \sup_{\Pi_T} |\lambda _i (x,t)|$.
\end{corollary}

Furthermore, we also obtain a uniqueness theorem from Theorem 3.2 and 
Theorem 3.3.

\begin{theorem}(Uniqueness).  Suppose that the functions $w(x,t)$ and $z(x,t)$
satisfy Framework (A) with (3.27), (3.32), and 
suppose $a_0(0) = 0$ and $a_0(w) > 0$ 
for $w \neq 0$.  Then $w_0(x) = z_0(x)$ leads to
$$
w(x,t) = z(x,t) , \qquad \mbox{a.e. } \Pi_T - \{ (x,t):
 P_1(x,t) = P_2(x,t) = 0 \} .
$$
\end{theorem}

\noindent
{\bf Remark.}
Theorem 3.4  shows that the solution of the Cauchy problem
$$
\left\{ \begin{array}{l}
\partial_t (a_0(w-k)P(x,t)) + \partial_x (a_0(w-k)\lambda(x,t)P(x,t))
\leq 0 , \qquad \mbox{for any  } k \in (-\infty , \infty ) , \\
w(x,0) = w_0(x) ,
\end{array} \right.
$$
with $a_0(0) = 0$, $a_0(w) > 0$ for $w \neq 0$, is independent of the
choice of the function
$P(x,t) > 0$ with the same initial data $P(x,0)$.  
This finding is essential to solve the
uniqueness and stability problems of solutions to systems of
conservation laws in Sections 4-6.

\vspace{.3in}
\noindent
{\bf c.  Framework for the Limit Behavior of Approximate Solutions}
\vspace{.2in}
\noindent

We shall establish a framework for the limit dynamics behavior of
approximate solutions to conservation laws.  We recall  from
Section 2 that with each point $(x,t) \in \Pi_T$ is associated
a Young measure over the range space $\R^m$, which  corresponds
to  a sequence of functions $w^\epsilon (x,t)$ and describes the
oscillation behavior of $w^\epsilon$ at $(x,t)$.
For this reason, we shall establish the framework for a family of
Radon measures.

Consider a family of Radon measures $\{\mu_{x,t}: \Pi_T  \to \mbox
{Prob}(\R )\}$, $\mbox{supp } \mu_{x,t} \subset \subset (-\infty,
\infty )$, with a family of the initial Radon measures
$\{ \mu_{x,0} \}$, satisfying the following Framework (B):

\noindent
$B_1$:  There exists a continuous state variable set $\Lambda $ such
that, for any $a(\lambda ) \in \Lambda$,
\begin{equation}
\left\{ \begin{array}{l}
\partial _t (<\mu_{x,t} ,a(\lambda )> P(x,t)) + \partial_x(<\mu
_ {x,t} ,a(\lambda )> \lambda(x,t)P(x,t)) \leq 0 , \\
\mu_{x,t} | _ {t=0} = \mu _ {x,0} ,
\end{array} \right.
\end{equation}
in the generalized sense of distribution in $\Pi_T$.

\noindent
$B_2$:  The set $\Lambda$ is sufficiently large  such that, for any
function of bounded variation  $g(\lambda )$,
\begin{equation}
\int_\alpha^\beta a(\lambda )dg(\lambda ) \leq 0
\end{equation}
implies that $g(\lambda )$ is a monotonically decreasing function.

\noindent
Here the vector field $(P(x,t),\lambda(x,t)P(x,t))$, $P(x,t) \geq 0$,
is divergence free in the $(x,t)$-space
\begin{equation}
\mbox{div}(P(x,t),\lambda(x,t)P(x,t)) = 0 ,
\end{equation}
and
\begin{equation}
\int_{-\infty}^0 P(x,0)dx = \int_0^\infty P(x,0)dx = \infty.
\end{equation}
Then we have the following theorem.

\begin{theorem}
Suppose that a family of Radon measures $\{ \mu_{x,t}:  \Pi_T \to
\mbox{Prob}(\R )\}$, \\ 
$\mbox{supp }\mu_{x,t} \subset \subset
(-\infty ,\infty )$, with a family of the initial Radon measures
$\{\mu _ {x,0}\}$, satisfies Framework (B).  Then
\begin{equation}
\mu_{x,t} \leq \mu_{T_0^{-1}(p(x,t)),0} , \qquad \mbox{a.e. }
\{ (x,t) \in \Pi_T : P(x,t) > 0 \} ,
\end{equation}
where $p: \Pi_T \to \R^1$ is determined by the
quasidecoupling transformation, that is,
\begin{equation}
\left\{ \begin{array}{l}
\frac{\partial p(x,t)}{\partial x} =P(x,t),\\
\frac{\partial p(x,t)}{\partial t} 
= -\lambda (x,t)P(x,t), \\
p(x,0) = \int_0^x P(\xi,0)d\xi. 
\end{array} \right.
\end{equation}
\end{theorem}

\noindent
{\bf Remark 1.}
Suppose that the conclusion in $B_2$ is that $g(\lambda )$ is a monotonically 
increasing
function.
Then the inequality (3.38) of Theorem 3.5 is changed into
\begin{equation}
\mu_{x,t} \geq \mu_{T_0^{-1}(p(x,t)),0}, \qquad \mbox{a.e. }
\{ (x,t) \in \Pi_T: P(x,t) > 0 \} .
\end{equation}

\noindent
{\bf Remark 2.}
The formulae (3.38) and (3.40) express that, along the dynamic curve
\begin{equation}
C: \qquad   \frac{dx}{dt} = \lambda (x,t),
\end{equation}
the value of the measures $\mu_{x,t}$ is smaller or larger than
the value of the initial measures $\mu_{x,0}$.
The formula (3.38) indicates that the Radon measures $\mu_{x,t}$ are
Dirac measures, provided that $\mu_{x,0}$ are Dirac measures, which
shows convergence of the corresponding approximate solutions
in the strong topology in $\{ (x,t) \in \Pi_T: P(x,t) > 0 \}$.
The formula (3.40) indicates that the Radon measures $\mu_{x,t}$
are not Dirac measures, provided that $\mu_{x,0}$ are not Dirac
measures, which shows that the initial oscillations propagate along
the curve (3.41).
This framework is  useful for  studying the limit behavior of approximate
solutions to conservation laws, especially  the dynamic
behavior of the initial oscillations (see Sections 4--6).

\vspace{.1in}
\noindent
{\bf Proof of Theorem 3.5.}
First we have from (3.34) that
\begin{equation}
{\int \int}_{\prod_T} <\mu_{x,t} ,a(\lambda )>P(x,t)
\{ \phi_t + \lambda (x,t) \phi_x \} dxdt +
\int_{-\infty}^\infty <\mu_{x,0} ,a(\lambda )> P(x,0)\phi(x,0)dx \geq
0,
\end{equation}
for any $a \in \Lambda $ and any nonnegative function
$\phi \in BV(\Pi_T)\cap C_0^1 (\Pi_T - \{ (x,t): P(x,t) = 0 \})$.
Notice that the vector field $(P(x,t),\lambda (x,t)P(x,t))$ is
divergence free in the $(x,t)$-space and
$$
\int_{-\infty}^0 P(x,0)dx = \int_0^\infty P(x,0)dx = +\infty .
$$
Make the quasidecoupling transformation $T: (x,t) \to (p,t)$ in
the integral (3.42).
We conclude from Lemma 3.1 that the transformation is proper and
onto in $\Pi_T$.
We obtain from (3.42) that
$$
\int_0^T \int_{- \infty}^\infty <\mu_{T^{-1}(p,t)}, a(\lambda )>
\phi(p)\psi ' (t)dpdt + \psi(0) \int_{-\infty}^\infty
<\mu_{T_0^{-1}(p),0} ,a(\lambda )>\phi(p)dp \geq 0, \forall a \in \Lambda,
$$
for any $\phi (p) \in C_0^1 (-\infty ,\infty )$, $\psi (t) \in C_0^1
[0,T)$, $\phi (p) \psi (t) \geq 0$.

Using Lemma 3.2, we have
$$
\int_{-\infty}^\infty <\mu_{T^{-1}(p,t)},a(\lambda )> \phi (p)dp
\leq \int_{-\infty}^\infty <\mu_{T_0^{-1}(p),0} ,a(\lambda )> \phi (p)dp ;
$$
that is,
$$
\int _{- \infty}^\infty <\mu_{T^{-1}(p,t)} 
- \mu_{T_0^{-1}(p),0} ,a(\lambda )>\phi(p)dp \leq 0 ,
$$
for any nonnegative $\phi \in C_0^1(-\infty, \infty )$.

Using Corollary 3.1, we have
$$
\int_{-\infty}^\infty <\mu_{x,t} - \mu_{T_0^{-1}(p(x,t)),0},
a(\lambda ) > \phi (p(x,t))P(x,t)dx \leq 0,
$$
for almost all $t \in (0,T)$.
Therefore, we obtain
\begin{equation}
<\mu_{x,t} - \mu_{T_0^{-1}(p(x,t)),0},a(\lambda) > \leq 0 , \qquad
\mbox{a.e. } \{ (x,t) \in \Pi_T: P(x,t) > 0 \} .
\end{equation}

Notice that $\Lambda $ is sufficiently large such that the
assumption $(B_2)$ holds.  Thus we obtain from (3.43)
$$
\mu_{x,t} \leq \mu_{T_0^{-1}(p(x,t)),0}, \qquad \mbox{a.e. }
\{ (x,t): P(x,t) > 0 \} .
$$
Therefore, one  arrives at (3.38).


\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{lemma}{0}
\setcounter{corollary}{0}
\section{Systems with Fully Contact Fields}

In this section we  consider how the quasidecoupling method can
be applied to the Cauchy problem of $n \times n$ systems with
$n$ independent contact fields, that is, linearly degenerate fields,
which are described by {\sc Serre} \cite{s4}:
\begin{equation}
\left\{ \begin{array}{l}
u_t + f(u)_x = 0, \qquad u \in \R^n , \\
u|_{t=0} = u_0 (x) ,
\end{array} \right.
\end{equation}
with
\begin{equation}
\partial_{w_i} \lambda _i \equiv 0, \qquad \mbox{  for all  } 1 \leq i \leq n ,
\end{equation}
and
\begin{equation}
\partial_{w_k} ((\lambda _j - \lambda _i)^{-1} \partial_{w_j} \lambda_i)
= \partial_{w_j} ((\lambda_k - \lambda _i)^{-1} \partial _{w_k}\lambda _i),
\end{equation}
for distinct $i, j$, and $k$,
where $\lambda_i$ , $1 \leq i \leq n$, are eigenvalues of the Jacobian
$\nabla f(u)$, and $w_i$, $1 \leq i \leq n$, are corresponding Riemann
invariants.
For simplicity, we assume that
\begin{equation}
\lambda _1 < \lambda _2 < \ldots < \lambda_n , \qquad \lambda_i \neq
0, \qquad 1 \leq i \leq n,
\end{equation}
and the Riemann invariants $w_i$, $1 \leq i \leq n$, are smooth
functions of $u$.

The entropy-entropy flux fields for such systems are of the
following general representation from (2.17) (see also \cite{s4}):
\begin{equation}
\left\{ \begin{array}{l}
\eta_a = \sum _ {i=1}^m N_i a_i (w_i) , \\
q_a = \sum _ {i=1}^m \lambda _i N_i a_i (w_i ) ,
\end{array} \right.
\end{equation}
for any continuous vector function
\begin{equation}
a(w) = (a_1(w_1) , \ldots , a_m (w_m )) ,
\end{equation}
where
\begin{equation}
N_i = \mbox{exp} \left\{ \int^{w_j} \frac{\partial_{w_j} \lambda_i}
{\lambda _j - \lambda_i } dw_j \right\}
\end{equation}
is independent of $j$, $j \neq i$, because of the equality (2.5) for
all distinct $i$, $j$, and $k$, and
\begin{equation}
N_i (u) \geq c_0 > 0,
\end{equation}
on the parallelipiped $K = \prod_i [\inf w_i^0 , \sup w_i^0 ]$ 
provided (4.4) holds.

A simple $2 \times 2$ example of such systems is (see \cite{te1})
$$
\left( \begin{array}{c}
u_1 \\ u_2 \end{array} \right) _ t + \left( \begin{array}{c}
-u_1^2 - u_2 \\ u_1^3 + u_1 u_2 \end{array} \right) _ x = 0 .
$$

{\sc Serre} \cite{s4} proved a global existence and uniqueness theorem
of a smooth solution
for the Cauchy problem (4.1) by using a hierarchy of transport
equations involving higher derivatives of $w_i$, and the local
existence and uniqueness theorem of {\sc Kato} \cite{ka}.

\begin{theorem}
Assume that $\lambda_i$, $1 \leq i \leq n$, are smooth functions
of $w = (w_1 , \ldots , w_n)$.  Then there exists a unique smooth
solution $u(x,t)$ of the Cauchy problem (4.1), satisfying the
following regularity:  $w \in C^k (\R_+^2 ;K) \equiv \{ w \in C^k (\R_+^2):
w \in K \}$ if and only if $w^0 \in C^k (\R_+^2;K)$.
\end{theorem}

This result shows that the smooth regularity of the solution for the
Cauchy problem (4.1) is the same as the smooth regularity of the initial
data because of the degeneracy of the systems. 
With the aid of the quasidecoupling method,
we shall now study the
well-posedness of generalized solutions  for
the Cauchy problem with the initial data $w_0 \in K$
and $w_0 \in  BV (\R_+^2; K)$:
existence, uniqueness and stability as well as the dynamic behavior of the
initial oscillations for the generalized solutions. 

\vspace{.2in}
\noindent
{\bf Definition 4.1.}
A bounded measurable vector function $u(x,t) \in \R^n$ is called a
generalized solution of the Cauchy problem (4.1) in $\Pi_T$ if
the equality
\begin{equation}
{\int \int}_{\Pi_T} \{ \eta_a (u(x,t)) \phi_t + q_a (u(x,t))
\phi_x \} dxdt + \int_{- \infty}^\infty \eta_a (u_0(x)) \phi (x,0)dx = 0
\end{equation}
holds for any continuous vector function $a(w)$ and any smooth function
$\phi \in C_0^1 (\Pi_T)$.



\vspace{.3in}
\noindent
{\bf a.  Propagation of Oscillations and Existence of Generalized Solutions}

\vspace{.2in}
\noindent

We first study the dynamic behavior of the initial oscillations, which
are of $O(1)$ amplitude with high frequency, as time evolves.  That is,
we study the limit behavior of the smooth solutions $u^\epsilon (x,t)$
in Theorem 4.1 that corresponds to the oscillatory sequence of large
initial data:
\begin{equation}
\left\{ \begin{array}{l}
u_t + f(u)_x = 0 , \\
u|_{t=0} = u_0^\epsilon (x) \in C^k (\R_+^2;K), \qquad k \ge 0.
\end{array} \right.
\end{equation}
We have the following theorem.

\begin{theorem}
Consider a hyperbolic system of conservation laws with fully contact
fields satisfying (4.3) and (4.4).  Suppose the initial data sequence
$u_0^\epsilon (x) \in C^k (\R_+^2;K)$ and the bounded set $K$
independent of $\epsilon $.  Then the corresponding $C^k$ solutions
$u^\epsilon (x,t)$ have the following limit behavior.

(i)  Let one of the corresponding initial Riemann invariants
$$
w_{i0}^\epsilon (x) = w_i (u_0^\epsilon (x))
$$
be a highly oscillatory sequence.  Then the initial oscillations
will propagate along the corresponding linearly degenerate field.

(ii)  Let the initial data sequence $u_0^\epsilon(x)$
 be compact in the strong topology of $L^\infty $.
Then there exists a subsequence converging pointwise a.e. to an
$L^\infty $ generalized  solution to the Cauchy problem (4.1)
with the Cauchy data $u_0 (x) = w^* - \lim u_0^\epsilon (x)$.
\end{theorem}

\noindent
{\bf Remark.}
Theorem 4.2 shows that the initial oscillations can propagate along the
corresponding linearly degenerate field, which is not affected by
other fields.  It is consistent with theoretical results of
propagation of the initial oscillations for $2 \times 2$ systems in
\cite{ch2,ch3} and with the evidence of oscillatory solutions 
given in
\cite{s2,b1}.  The phenomenon differs 
from those in \cite{ll,di2,ch1}.

\vspace{.2in}
\noindent
{\bf Proof of Theorem 4.2.}
We first obtain from Theorem 4.1
\begin{equation}
u^\epsilon (x,t) \in K \qquad (K \mbox{ is independent of } \epsilon )
\end{equation}
and
\begin{equation}
\eta_a(u^\epsilon (x,t))_t + q_a (u^\epsilon (x,t))_x = 0,
\end{equation}
for any continuous vector function $a(w)$.
Using Theorem 2.1, we conclude that there exists a family of
Young measures $\nu_{x,t} : \R_+^2 \to \mbox{Prob}(\R^n)$,
$\mbox{supp } \nu_{x,t} \subset K$, uniquely determined by
the approximate solutions $u^\epsilon (x,t)$, satisfying
\begin{equation}
\partial_t <\nu_{x,t} , N_ia_i(w_i)> +  \partial_x<\nu_{x,t},\lambda_i
N_i a(w_i )> = 0 , \qquad 1 \leq i \leq n,
\end{equation}
and, for any continuous vector functions $a(w)$ and $b(w)$,
\begin{equation}
<\nu_{x,t}, \eta_a><\nu_{x,t} ,q_b> - <\nu_{x,t},\eta_b><\nu_{x,t} ,q_a> = 0 .
\end{equation}

We conclude from (4.14) that there exists an $L^\infty$ function
$\bar{\lambda}_i (x,t)$ such that
\begin{equation}
<\nu_{x,t} ,\lambda_iN_i a_i (w_i)> = \bar{\lambda}_i
(x,t)<\nu_{x,t} ,N_ia_i(w_i)>,
\end{equation}
for any continuous vector function $a(w)$.  Therefore, (4.13) can be
written into  the following form:
\begin{equation}
\partial_t<\nu_{x,t},N_ia_i(w_i)> 
+  \partial_x(\bar{\lambda}_i (x,t)< \nu_{x,t} ,N_ia_i(w_i)>) = 0 , 
\qquad 1 \leq i \leq n .
\end{equation}
In particular,
\begin{equation}
\partial_t<\nu_{x,t},N_i> +  \partial_x (\bar{\lambda}_i(x,t)<\nu
_{x,t} ,N_i>) = 0 , \qquad 1 \leq i \leq n ,
\end{equation}
and
$$
<\nu_{x,t} ,N_i> \geq c_0 > 0.
$$

For fixed $i$, $1 \leq i \leq n$, we define a family of probability
measures:
$$
\mu_{x,t}^i = \frac{N_i\nu_{x,t}}{<\nu_{x,t},N_i>} .
$$
Then $\mu_{x,t}^i$ satisfies
\begin{equation}
\partial_t(<\mu_{x,t}^i ,a(w_i)><\nu_{x,t},N_i>) + \partial_x
(<\mu_{x,t}^i ,a(w_i)> \bar{\lambda }_i(x,t)<\nu_{x,t},N_i>) = 0.
\end{equation}

We conclude from (4.17)-(4.18) that the family of probability measures
satisfies Framework (B) and the set $\Lambda$ is the space of  continuous 
functions.
Therefore we have from Theorem 3.5
\begin{equation}
\mu_{x,t}^i = \mu_{T_0^{-1}(p_i(x,t)),0}^i ,
\end{equation}
where $p_i(x,t)$ is determined by
$$
\left\{ \begin{array}{l}
\frac{\partial p_i(x,t)}{\partial x} 
= <\nu_{x,t},N_i> \geq c_0 > 0 , \\
\frac{\partial p_i (x,t)}{\partial t} 
= - \bar{\lambda}_i (x,t) <\nu _ {x,t},N_i>, \\
p_i (x,0) =\int_0^x <\nu_{\xi,0}, N_i> d\xi.
\end{array} \right.
$$

If the composite sequence of approximate initial data $w_{i0}^\epsilon
(x) = w_i (u_0^\epsilon (x))$ is a highly oscillatory function
sequence, then $\mu_{x,0}^i$ is not a point measure; that is,
$\mu_{x,t}^i$ is not a point measure from (4.19).
Therefore $\nu_{x,t}$ is not a point measure.  This fact 
means that
the initial oscillations propagate along the ``average''
eigendirection:
$$
\frac{dx}{dt} = \bar{\lambda}_i(x,t) \equiv \frac
{<\nu_{x,t},\lambda_iN_i>}{<\nu_{x,t},N_i>} .
$$

If the initial data sequence $u_0^\epsilon(x)$ is compact in the strong
topology of $L^\infty$, then there exists a subsequence (still labeled)
$u_0^\epsilon(x)$ such that all
composite sequences $w_{i0}^\epsilon$ converge
strongly to $w_{i0}(x)$, a.e.,
$$
w_{i0}^\epsilon (x) = w_i (u_0^\epsilon (x)) \to w_{i0} (x), \qquad
\mbox{a.e.} \qquad 1 \le i \le n,
$$
and  $\mu_{x,0}^i, 1 \le i \le n$,  is a Dirac measure.
Therefore $\mu_{x,t}^i, 1 \le i \le n$, is a Dirac measure from (4.19).
Finally, we conclude that $\nu_{x,t}$ is a Dirac measure.
We know from Section 2 that there exists a subsequence (still
labeled) $u^\epsilon (x,t)$ converging pointwise a.e. and that the
limit function is a generalized solution of the corresponding
Cauchy problem (4.1) with the Cauchy data $u_0(x) = w^* -
\lim _ {\epsilon \to 0} u_0^\epsilon (x)$.
This completes the proof of Theorem 4.2.

A direct corollary follows.

\begin{corollary}
For the Cauchy problem (4.1) with fully contact fields, the
solution operator $S_t: L^\infty (\R^n) \to L^\infty (\R^n)$ with
$S_t(u^0) = u(\cdot ,t)$ is not compact.
\end{corollary}

As a byproduct, we have the following theorem.

\begin{theorem}
There exists an $L^\infty$ generalized solution $u(x,t) \in K$ to the
Cauchy problem (4.1) with the $L^\infty$ Cauchy data $u_0 (x) \in K$, that
satisfies
\begin{equation}
\eta_a(u(x,t))_t + q_a(u(x,t))_x = 0,
\end{equation}
for any continuous vector function $a(w)$ in the sense of
distribution.
\end{theorem}

\noindent
{\bf Proof.}
Note that, for any $u_0(x) \in L^\infty$, there exists an approximate
sequence $u_0^\epsilon (x) \in C^\infty (\R;K)$ of $u_0(x)$ such that,
for any $N > 0$,
\begin{equation}
\| u_0^\epsilon -u_0 \|_{L^p (-N,N)} \to 0, \qquad \epsilon \to 0.
\end{equation}

In fact, we can take
$$
u_0^\epsilon (x) = (J_\epsilon *u_0 )(x),
$$
where $J_\epsilon $ is a standard compactly supported nonnegative
mollifier:
$$
J_\epsilon = \epsilon J(x/\epsilon ), \quad J(x) \geq 0, \quad
\int J(x)dx = 1, \quad \mbox{supp } J(x) \subset [-1,1].
$$
Then we consider the Cauchy problem (4.1) with the Cauchy initial
data $u_0^\epsilon (x)$.  We obtain from Theorem 4.1 that there
exists a unique solution $u^\epsilon (x,t)$ for this Cauchy problem.
Using Theorem 4.1, we conclude that there exists a subsequence
$u^{\epsilon _ k} (x,t)$ converging a.e. to an $L^\infty$ solution
$u(x,t)$ for the Cauchy problem (4.1) satisfying (4.20).

Finally, we establish a regularity theorem for the generalized solutions of 
the Cauchy problem (4.1) in Theorem 4.3 by using the quasidecouplind
method.

\begin{theorem}
Consider the generalized solutions of the Cauchy problem (4.1) satisfying
(4.20). If the initial data $u_0(x) \in BV (\R^1; K)$, then
the corresponding generalized solution $u(x,t)$ is also a bounded
variation function,
\begin{equation}
u(x,t) \in BV (\R_+^2; K),
\end{equation}
satisfying (4.20).
\end{theorem}

\vspace{.2in}
\noindent
{\bf Proof.}
Suppose that $w_i, 1 \le i \le n$, are Riemann invariants corresponding to
$i$-th characteristic fields, respectively. Choose
$$
P(x,t) = N_i(u(x,t)),  \qquad  \lambda(x,t) = \lambda_i(u(x,t)),
$$
and 
$$
a_0(w - k) = |w_i - k|
$$
in Framework (A). We immediately obtain from Theorem 3.2 that, for almost
all $t\in [0, T)$ and $ x\in (-\infty, \infty)$,
\begin{equation}
w_i(u(x,t)) = w_i(u_0(T_{0i}^{-1}(p_i(x,t)))),
\end{equation}
where the Lipschitz function $T_{0i}^{-1}(p)$ is the inverse map of $T_{0i}$:
$$
x\to T_{0i}(x) = \int_0^x N_i(u_0(\xi))d\xi,
$$
and the Lipschitz function $p_i(x,t), 1 \le i \le n$,  are determined
by the quasidecoupling transformations corresponding to the vector
fields $(N_i(u), \lambda(u)N_i(u))$, respectively.
Therefore, we conclude from (4.23) that
$$
w(u(x,t)) \in BV(\R_+^2; K), \qquad 1\le i \le n,
$$
and (4.22) follows.

\vspace{.3in}
\noindent
{\bf b.  Uniqueness and Stability of Generalized Solutions}

\vspace{.2in}

We first study the stability of $L^\infty$ generalized  solutions for
the Cauchy problem (4.1) established in Subsection 4a.

\begin{theorem}( Stability of $L^\infty$ Generalized Solution).  
Suppose that $L^\infty$ functions
$u(x,t)$ and $v(x,t)$ are generalized solutions of the
Cauchy problem (4.1)  with initial data $u_0(x)$ and $v_0(x)$,
respectively.
Then, for almost all $t\in [0,T)$ and any positive constant $m$,
\begin{equation} 
w_i(u(x,t)) = w_i(u_0(T_{0i}^{-1}(p_i(x,t)))), 
\qquad w_i(v(x,t)) = w_i(v_0(\bar{T}_{0i}^{-1}(q_i(x,t)))), \qquad 
 \mbox{a.e.}, 
\end{equation}
\begin{equation} \left \{\begin{array}{l}
\int_{|x| \leq N} |w_i(u(x,t))
-w_i(v(\bar{T}_{ti}^{-1}T_{ti}(x),t))|^m dx \\ 
\mbox{      }\leq C \int_{|x| \leq K_0(N + Lt)} 
|w(u_0(x))-w(v_0(\bar{T}_{0i}^{-1}T_{0i}(x)))|^m dx , \\
\int_{|x| \leq N} |w_i(u(T^{-1}_{ti}\bar{T}_{ti}(x),t))
 - w_i(v(x,t))|^m dx \\
\mbox{      }\leq C \int_{|x| \leq K_0(N + Lt)} 
|w(u_0(T_{0i}^{-1}\bar{T}_{0i}(x)))-w(v_0(x))|^m dx, 
\end{array} \right.
\end{equation}
\begin{equation} \left \{\begin{array}{l}
\int_{|x| \leq N} |w_i(u(x,t))
-w_i(v(\bar{T}_{ti}^{-1}\bar{T}_{0i}T_{0i}^{-1}T_{ti}(x),t))|^m dx \\
\mbox{      }\leq C \int_{|x| \leq K_0(N + Lt)}
|w(u_0(x))-w(v_0(x))|^m dx, \\
\int_{|x| \le N} |w_i(u(T^{-1}_{ti}T_{0i}\bar{T}_{0i}^{-1}\bar{T}_{ti}(x),t))
- w_i(v(x,t))|^m dx \\
\mbox{       } \leq C \int_{|x| \leq K_0(N + Lt)}
|w(u_0(x))- w(v_0(x))|^m dx,
\end{array} \right.
\end{equation}
where 
$L = \max _ {1\le i \le n} (\sup |\lambda_i (u) | , \sup |\lambda_i (v )|)$;
$C$ is a constant independent of $m$, $L$, and $N$; 
$K_0= \max_{1 \le i \le n} (\sup_{\Pi}N_i(u(x,t)), \sup_{\Pi}N_i(v(x,t)))/c_0$;
$w = (w_1, w_2, ..., w_n)$
with Riemann invariants $w_i, 1 \le i \le n,$  corresponding to the $i$-th
 characteristic field; and $p_i(x,t) = T_{ti}(x)$ and 
$q_i(x,t)= \bar{T}_{ti}(x)$ are determined by the
 quasidecoupling transformations corresponding to the vector fields
$(N_i(u(x,t)), \lambda(u(x,t))N_i(u(x,t)))$
and
$(N_i(v(x,t)), \lambda(v(x,t))N_i(v(x,t)))$.
\end{theorem}

\noindent
{\bf Proof.}
Choose 
$$
\begin{array}{ll}
P_1(x,t) = N_i(u(x,t)),
 \qquad \lambda_1(x,t) =\lambda_i(u(x,t)),\\
P_2(x,t) = N_i(v(x,t)),
 \qquad \lambda_2(x,t) = \lambda_i(v(x,t)),
\end{array} 
$$
and 
$$
a_0(w-k) = |w_i -k|^m
$$
in Framework (A).  We immediately obtain from Theorem 3.2 that, for almost all
$t\in [0, T)$ and $x\in (-\infty, \infty)$,
$$
w_i(u(x,t)) = w_i(u_0(T_{0i}^{-1}(p_i(x,t)))), 
\qquad w_i(v(x,t)) = w_i(v_0(\bar{T}_{0i}^{-1}(q_i(x,t)))),
$$
and then  (4.25)--(4.26)  follows for any $N>0$.

Now we consider the deviation between 
$(p_i(x,t), t) \equiv (p_i(x,t; w(u(x,t))), t) = T_i(x,t)$ and 
$(q_i(x,t), t) \equiv (q_i(x,t; w(v(x,t))), t) = \bar{T}_i(x,t)$ for the 
systems consisting of two equations (n=2).

\begin{theorem}  
Suppose that $L^\infty$ functions
$u(x,t)$ and $v(x,t)$ are generalized solutions of the
Cauchy problem (4.1)  with initial data $u_0(x)$ and $v_0(x)$,
respectively.
Then, for almost all $t\in [0,T)$,
\begin{equation}  \begin{array}{l}
\int_{|x| \leq N} |(\bar{T}^{-1}_{ti}T_{ti}- I)(x)|dx \\  
\mbox{   } \leq C \int_{|x| \leq K_0(N+ \bar{L}t)}
 \{|w(u_0(x)) - w(v_0(x))|+|w(u_0(x))-w(v_0(\bar{T}_{0i}^{-1}T_{0i}(x)))|\}dx, 
\qquad i=1,2,
\end{array} 
\end{equation}
where $L = \max _ {i=1,2} (\sup |\lambda_i (u) | , \sup |\lambda_i (v )|)$;
$\bar{L} = L+1 
+ \max_{i = 1, 2}(\sup_{[0, T]}(|\bar{T}_{t_i}^{-1}T_{t_i}(\pm N)|-N)/t)$;
$K_0= \max_{i=1,2}
(\sup_{\Pi}(\lambda_2(u)-\lambda_1(u))^{-1}, 
\sup_{\Pi}(\lambda_2(v)-\lambda_1(v))^{-1})/P_0$;
$C$ is a constant independent of $m$, $L$, and $N$; $w = (w_1, w_2)$
with Riemann invariants $w_i, 1 \le i \le 2,$  corresponding to the $i$-th
 characteristic field; and $p_i(x,t) = T_{ti}(x)$ and 
$q_i(x,t)= \bar{T}_{ti}(x)$ are determined by the
 quasidecoupling transformations corresponding to the vector fields
$((\lambda_2 -\lambda_1)^{-1}, \lambda_i (\lambda_2 - \lambda_1)^{-1})(u(x,t))$
and 
$((\lambda_2 - \lambda_1)^{-1}, \lambda_i(\lambda_2 -\lambda_1)^{-1})(v(x,t))$.
\end{theorem}

\noindent
{\bf Proof.}
Using Theorem 4.5, we know that  the Lipschitz functions $p_i(x,t)$ and 
$q_i(x,t)$, determined by
the quasidecoupling transformations, satisfy the following Cauchy problems:
\begin{equation} \left \{\begin{array}{l}
p_i(x,t)_t + \lambda_i(w_j(u(x,t)))p_i(x,t)_x = 0, \qquad j \ne i, \\
p_i(x,0) =\int_0^x P_1(\xi,0)d\xi, 
\end{array} \right.
\end{equation}
and
\begin{equation} \left \{\begin{array}{l}
q_i(x,t)_t + \lambda_i(w_j(v(x,t)))q_i(x,t)_x = 0, \qquad j \ne i,\\
q_i(x,0) = \int_0^x P_2(\xi,0)d\xi.
\end{array} \right.
\end{equation}

Notice that, under the transformation $T_{i}(x,t) =(p_i(x,t), t)$, 
$w_j(u(T^{-1}_{i}(p_i,t))) (i\ne j)$ satisfies the following 
Cauchy problem:
\begin{equation} \left \{\begin{array}{l}
|w_j -k|_t + |w_j -k|_{p_i} = 0, \\
w_j|_{t=0} = w_j(u_0(T_{0i}^{-1}(p_i))).
\end{array} \right.
\end{equation}
Furthermore, under the transformation $\bar{T}_{i}(x,t) = (q_i(x,t), t)$,
$w_j(v(\bar{T}_i^{-1}(q_i,t))) (i \ne j)$ 
satisfies the following Cauchy problem:
\begin{equation} \left \{\begin{array}{l}
|w_j - k|_t + |w_j -k|_{q_i} = 0, \\
w_j|_{t=0} = w_j(v_0(\bar{T}_{0i}^{-1}(q_i))).
\end{array} \right.
\end{equation}
Therefore, we can deduce that, for almost all $(x,t)\in \Pi_T$,
\begin{equation}
w_j(u(x,t)) = w_j(u_0(T_{0i}^{-1}(p_i(x,t)-t))), 
\qquad w_j(v(x,t)) = w_j(v_0(\bar{T}_{0i}^{-1}(q_i(x,t)-t))).
\end{equation}

Substitute (4.32) into (4.28)-(4.29). We have, for almost
all $t\in (0, T)$,
\begin{equation} \left \{\begin{array}{l}
p_i(x,t)_t + 
\lambda_i(w_j(u_0(T_{0i}^{-1}(p_i(x,t)-t))))p_i(x,t)_x 
= 0, \qquad j \ne i, \\
p_i(x,0) = \int_0^x P_1(\xi,0)d\xi,
\end{array} \right.
\end{equation}
and
\begin{equation} \left \{\begin{array}{l}
q_i(x,t)_t + \lambda_i(w_j(v_0(\bar{T}_{0i}^{-1}(q_i(x,t)-t))))q_i(x,t)_x = 0, 
\qquad j \ne i,\\
q_i(x,0) =\int_0^x P_2(\xi,0)d\xi.
\end{array} \right.
\end{equation}



Notice that
\begin{equation}
\left\{\begin{array}{l}
\int_{|x| \le N + Lt}|p_i(x,0)-q_i(x,0)|dx 
\le C \int_{|x|\le N + Lt}|w(u_0(x)) - w(v_0(x))|dx, \\
\int_{|x|\le \|P_1\|_{L^\infty}N+Lt}|w_j(u_0(T_{0i}^{-1}(x-t)))
-w_j(v_0(\bar{T}_{0i}^{-1}(x-t)))|dx\\
\mbox{       } 
\le C \int_{|x|\le \|P_1\|_{L^\infty}N + (L+1)t} |w_j(u_0(T_{0i}^{-1}(x)))
-w_j(v_0(\bar{T}_{0i}^{-1}(x)))|dx\\
\mbox{       }
\le C \int_{|x|\le K_0(N+(L+1)t)}|w_j(u_0(x))
-w_j(v_0(\bar{T}_{0i}^{-1}T_{0i}(x)))|dx.
\end{array} \right.
\end{equation}
Using Theorem 3.3, and (4.33)-(4.35), we finally obtain
$$
\begin{array}{l}
\int_{|x| \le N}|p_i(x,t) - q_i(x,t)|dx\\ 
\le C \int_{|x| \le K_0(N + (L+1)t)}
(|w(u_0(x)) - w(v_0(x))| + |w(u_0(x))-w(v_0(\bar{T}_{0i}^{-1}T_{0i}(x)))|)dx.
\end{array}
$$
Thus (4.27) follows. This completes the proof of Theorem 4.6.

\vspace{.2in}
\noindent
{\bf Remark.} Theorem 4.5 and Theorem 4.6 show that, 
for the systems with fully contact fields $(n = 2)$,
the generalized solution of the Cauchy problem (4.1) is $L^m$-stable
in the sense of shift, for any positive constant $ m $.

As a corollary, we obtain the following theorem.

\begin{theorem}(Uniqueness).  
The $L^\infty$ generalized solution of the Cauchy problem (4.1) $(n=2)$ 
is unique.
\end{theorem}

Finally, we consider the systems (4.1)-(4.3) consisting of arbitrary
$n$ equations. For simplicity, we assume that 
$$
\sup_{w \in K}\lambda_1(w) < \inf_{w \in K}\lambda_2(w) 
<\sup_{w \in K}\lambda_2(w) < ... 
< \sup_{w \in K}\lambda_{n-1}(w)< \inf_{w \in K}\lambda_n(w),
$$
where $K$ is the parallelipiped $K = \prod_i[\inf w_i(u_0), \sup w_i(u_0)]$.
Then we have the following stability result.

\begin{theorem}(Stability of $BV$ Generalized Solution). Suppose that the
vector functions $u(x,t)$ and $v(x,t)$ are generalized solutions of 
the Cauchy problem (4.1) in $\Pi_T$ with  initial data $u_0(x)$ and $v_0(x)$ 
in $BV (\R_+^2; K)$, respectively. 
If $TV_{-\infty}^{\infty}(u_0(x))$
is  sufficiently small and 
$u_0 - v_0 \in L^1(-\infty, \infty)$, then
\begin{equation}
\int_{-\infty}^{\infty}|u(x,t)-v(x,t)|dx 
\le C\int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx,
\end{equation}
for almost all $t \in (0, T)$,
where the constant 
$C = C(TV_{-\infty}^{\infty}(u_0(x)))$ with $C(s) \in C[0, s_0)$
and $\lim_{s \to s_0}C(s) =+\infty$ for some $s_0 > 0$ only depending on the
parallelipiped $K$.
\end{theorem}

\noindent
{\bf Proof.} 
We first prove that the Lipschitz functions $p_i(x,t)$ and $q_i(x,t)$, 
$1 \le i \le n$, satisfy
\begin{equation}
\sup_{\Pi_T}\sum_{i=1}^n |T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|
\le C \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx,
\end{equation}
where the constant 
$C = C(TV_{-\infty}^{\infty}(u_0(x)))$, with $C(s) \in C[0, s_0)$
and $\lim_{s \to s_0}C(s) = + \infty$ for some $s_0 > 0$ only depending on the
parallepiped $K$;
the functions $p_i(x,t)$ and 
$q_i(x,t)$ are determined by 
the quasidecoupling transformations corresponding to the vector functions 
$u(x,t)$ and $v(x,t)$, and satisfy
\begin{equation} \left \{\begin{array}{l}
p_i(x,t)_t + \lambda_i(u(x,t))p_i(x,t)_x =0, \\
p_i(x,0)= T_{0i}(x) \equiv \int_0^xP_1(\xi, 0)d\xi,
\end{array} \right.
\end{equation}
and
\begin{equation} \left \{\begin{array}{l}
q_i(x,t)_t + \lambda_i(v(x,t))q_i(x,t)_x =0, \\
q_i(x,0)= \bar{T}_{0i}(x) \equiv \int_0^xP_2(\xi, 0)d\xi,
\end{array} \right.
\end{equation}
for almost all $t\in (0, T)$.

Then we have
\begin{equation}
|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i)|_t 
+ \lambda_i(u)|T_{0i}^{-1}(p_i)-\bar{T}_{0i}^{-1}(q_i)|_x
= \mbox{sign}(T_{0i}^{-1}(p_i)-\bar{T}_{0i}^{-1}(q_i))
(\lambda_i(u)-\lambda_i(v))\bar{T}_{0i}^{-1}(q_i)_x.
\end{equation}

Multiply (4.40) by $N_i(u)\phi(p_i)$, for any $\phi(x)\in C_0^1, \phi(x) \ge 0$,
and note that
$$
(N_i(u)\phi(p_i))_t + (\lambda_i(u)N_i(u)\phi(p_i))_x = 0.
$$
We obtain
\begin{equation} \begin{array}{l}
(N_i(u)\phi(p_i)|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i)|)_t 
+ (\lambda_i(u)N_i(u)\phi(p_i)|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i)|)_x \\
=\mbox{sign}(T_{0i}^{-1}(p_i)-\bar{T}_{0i}^{-1}(q_i))
(\lambda_i(u) - \lambda_i(v))N_i(u)\phi(p_i)\bar{T}_{0i}^{-1}(q_i)_x.
\end{array}
\end{equation}

Integrate (4.41) over the domain $(-\infty, \infty) \times [0, t]$. We have
$$
\begin{array}{l}
\int_{-\infty}^{\infty}N_i(u)\phi(p_i)|T_{0i}^{-1}(p_i) 
				      - \bar{T}_{0i}^{-1}(q_i)|dx \\
\le  \int_{-\infty}^{\infty}\int_0^t
|\lambda_i(u)-\lambda_i(v)|N_i(u)\phi(p_i)|\bar{T}_{0i}^{-1}(q_i)_x|dxd\tau,
\end{array}
$$
that is,
\begin{equation}
\int_{-\infty}^{\infty}
   \phi(p_i(x,t))|T_{0i}^{-1}(p_i(x,t)) - \bar{T}_{0i}^{-1}(q_i(x,t))|dx \\
\le C \int_{-\infty}^{\infty}\int_0^t
|\lambda_i(u)-\lambda_i(v)|\phi(p_i)dxd\tau,
\end{equation}
for any $\phi \in C_0^1(-\infty, \infty), \phi(x) \ge 0$.

Since $p_i(x,t)_x \ge c_0 >0$, we obtain by changing variable $(x, t) \to
(p_i, t)$ in the integrations,
$$
\begin{array}{l}
\int_{-\infty}^{\infty}
   \phi(p_i)|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i(x(p_i, t),t))|dp_i \\
\le C\{ \int_{-\infty}^{\infty}\int_0^t
|\lambda_i(u(x(p_i, \tau), \tau)) - \lambda_i(v(x(p_i, \tau), \tau))|
\phi(p_i)dp_id\tau \},
\end{array}
$$
for any $\phi \in C_0^1(-\infty, \infty), \phi \ge 0$.
Therefore, we have
\begin{equation}
|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i(x(p_i, t),t))| 
\le C
\int_0^t|\lambda_i(u(x(p_i,\tau),\tau))-\lambda_i(v(x(p_i,\tau),\tau))|d\tau.
\end{equation}

As (4.21), for any $u_0(x) \in L^{\infty} \cap BV$, there exists an approximate
sequence $u_0^\epsilon(x) \in C^\infty(\R; K)$ of $u_0(x)$ such that 
$$
\left\{\begin{array}{l}
\|u_0^\epsilon - u_0\|_{L^1(-\infty, \infty)} 
\le \epsilon C TV_{-\infty}^{\infty}(u_0(x)) \to 0,\\
TV_{-\infty}^{\infty}(u_0^\epsilon(x)) \le TV_{-\infty}^{\infty}(u_0(x)).
\end{array} \right.
$$
Furthermore, we have from Theorem 4.5 that
$$
\left\{\begin{array}{l}
\lambda_i(u(x,t)) = \lambda_i(w_j(u_0(T_{0i}^{-1}(p_i(x,t))))),
\qquad j \ne i, \\
\lambda_i(v(x,t)) = \lambda_i(w_j(v_0(\bar{T}_{0i}^{-1}(q_i(x,t))))),
\qquad j \ne i,
\end{array} \right.
$$
where $\lambda_i(w_j), j \ne i,$ is the compact notation of 
$\lambda_i(w_1,..., w_{i-1}, w_{i+1},...,w_n)$.
Then we have 
$$
\begin{array}{l}
|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i(x(p_i, t),t))| \\
\le C \{ 
\int_0^t|\lambda_i(w_j(u_0^\epsilon(T_{0j}^{-1}(p_j(x(p_i, \tau), \tau))))) -
\lambda_i(w_j(u_0^\epsilon(\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau),\tau)))))|d\tau \\
\qquad +\int_0^t|\lambda_i(w_j(u_0(\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau), \tau)))))
   -\lambda_i(w_j(v_0(\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau),\tau)))))|d\tau \\
\qquad +\int_0^t
   |\lambda_i(w_j(u_0^\epsilon(T_{0j}^{-1}(p_j(x(p_i, \tau), \tau)))))
   -\lambda_i(w_j(u_0(T_{0j}^{-1}(p_j(x(p_i, \tau),\tau)))))|d\tau \\
\qquad +\int_0^t
   |\lambda_i(w_j(u_0^\epsilon(\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau), \tau)))))
   -\lambda_i(w_j(u_0(\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau),\tau)))))|d\tau\} \\
\le C \sum_{j \ne i}\{
\int_0^t\int_0^1|u_{0x}^\epsilon(s_j(p_i, \tau; \theta))| 
     |T_{0j}^{-1}(p_j(x(p_i, \tau), \tau))
       -\bar{T}_{0j}^{-1}(q_j(x(p_i, \tau), \tau))|d\tau d\theta \\
 \qquad + \int_0^t(|u_0^\epsilon(\bar{s}_j(p_i, \tau))
	  -u_0(\bar{s}_j(p_i, \tau))| 
           +|u_0^\epsilon(\tilde{s}_j(p_i, \tau))
	  -u_0(\tilde{s}_j(p_i, \tau))|)d\tau  \\
 \qquad  + \int_0^t|u_0(\tilde{s}_j(p_i, \tau))
	  -v_0(\tilde{s}_j(p_i, \tau))|d\tau\}
\end{array}
$$
for $j \ne i$, where
$$
\left\{\begin{array}{l}
s_j(p_i, t; \theta) =  \theta T_{0j}^{-1}(p_j(x(p_i,t), t))
			+(1-\theta)T_{0j}^{-1}(q_j(x(p_i,t), t)),\\
\bar{s}_j(p_i, t)= T_{0j}^{-1}(p_j(x(p_i, t), t)), \\
\tilde{s}_j(p_i, t)= \bar{T}_{0j}^{-1}(q_j(x(p_i, t), t)).
\end{array} \right.
$$

Since 
$$
\left\{\begin{array}{l}
|T_{0i}^{-1}(p_i(x,0)) - \bar{T}_{0i}^{-1}(q_i(x,0))| =0, \\
|\partial_t|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|| 
\le M_0 < \infty,
\end{array} \right.
$$
for all $i$, $1\le i \le n$, we conclude that the functions
$|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|, 1 \le i \le n$,
are uniformly bounded,
$$
|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))| \le M_0T, 
\qquad 1 \le i \le n.
$$
Moreover, one can check that, for $j \ne i$,  
$$
\begin{array}{l}
|s_j(p_i, t)_t| \ge c_0 >0 ,\\
|\bar{s}_j(p_i, t)_t| \ge c_0 > 0, \\
|\tilde{s}_j(p_i, t)_t| \ge c_0 > 0,
\end{array}
$$
and the transformations
$$
\begin{array}{l}
(p_i, t) \to (p_i, s_j),\\
(p_i, t) \to (p_i, \bar{s}_j), \\
(p_i, t) \to (p_i, \tilde{s}_j),
\end{array}
$$
are one-to-one for $j \ne i$, respectively.

Make these transformations in the corresponding integrations, we can 
immediately obtain
\begin{equation}
\begin{array}{l}
|T_{0i}^{-1}(p_i) - \bar{T}_{0i}^{-1}(q_i(x(p_i, t),t))| \\
\le C \{ 
\sup_{\Pi_T}\sum_{j \ne i}|T_{0j}^{-1}(p_j(x,t))-\bar{T}_{0j}^{-1}(q_j(x,t))|
	 \int_{-\infty}^\infty|u_{0x}^\epsilon(x)|dx \\
\qquad \ \ + \int_{-\infty}^{\infty}|u_0^\epsilon(x)-u_0(x)|dx
	+\int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx\} \\ 
\le C \{ TV_{-\infty}^{\infty}(u_0(x))\sup_{\Pi_T}
      \sum_{j \ne i}|T_{0j}^{-1}(p_j(x,t))-\bar{T}_{0j}^{-1}(q_j(x,t))| 
      +\int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx + \epsilon\}.
\end{array} 
\end{equation}
Set $\epsilon \to 0$ and then substitute $p_j(x,t)$ into $p_j$ in (4.44).
We obtain
$$
\begin{array}{l}
|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))| \\ 
\le C\{ \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx +
      TV_{-\infty}^{\infty}(u_0(x))\sup_{\Pi_T}
      \sum_{j \ne i}|T_{0j}^{-1}(p_j(x,t))-\bar{T}_{0j}^{-1}(q_j(x,t))|\}.
\end{array}
$$
This implies
\begin{equation}
\begin{array}{l}
\sup_{\Pi_T}\sum_{i=1}^{n}|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|\\
\le C \{ \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx
	+ TV_{-\infty}^{\infty}(u_0(x))\sup_{\Pi_T}
\sum_{i=1}^{n}|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|\}.
\end{array} 
\end{equation}

If $TV_{-\infty}^\infty(u_0(x))$ is sufficiently small such that
$C TV_{-\infty}^{\infty}(u_0(x)) < 1$, we obtain 
\begin{equation}
\sup_{\Pi_T}\sum_{i=1}^n |T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|
\le C \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx,
\end{equation}

Notice that 
$$
w_i(u(x,t))=w_i(u_0(T_{0i}^{-1}(p_i(x,t)))),
\qquad w_i(v(x,t))=w_i(v_0(\bar{T}_{0i}^{-1}(q_i(x,t)))).
$$
Similarly, we have
$$
\begin{array}{l}
\int_{-\infty}^{\infty}|u(x,t)-v(x,t)|dx \\
\le C \int_{-\infty}^{\infty}\sum_{i=1}^n|w_i(u(x,t))-w_i(v(x,t))|dx \\
\le C \int_{-\infty}^{\infty}
\sum_{i=1}^n|u_0(T_{0i}^{-1}(p_i(x,t)))- v_0(\bar{T}_{0i}^{-1}(q_i(x,t)))|dx\\
\le C
\int_{-\infty}^{\infty}\sum_{i=1}^n\{
|u_0(T_{0i}^{-1}(p_i(x,t)))- u_0(\bar{T}_{0i}^{-1}(q_i(x,t)))|
+ |u_0(\bar{T}_{0i}^{-1}(q_i(x,t)))- v_0(\bar{T}_{0i}^{-1}(q_i(x,t)))|\}dx\\
\le C \{TV_{-\infty}^{\infty}(u_0(x))\sup_{\Pi_T}
\sum_{i=1}^n|T_{0i}^{-1}(p_i(x,t))-\bar{T}_{0i}^{-1}(q_i(x,t))|
+ \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx\} \\
\le C \int_{-\infty}^{\infty}|u_0(x)-v_0(x)|dx.
\end{array}
$$
This completes the proof of Theorem 4.8.

As a corollary, we obtain the following uniqueness theorem.
\begin{theorem} (Uniqueness). The  generalized solution of the Cauchy 
problem (4.1) with the initial data $u_0(x) \in BV(\R^1; K)$ is unique 
provided that $TV_{-\infty}^{\infty}(u_0(x))$ is sufficiently small.
\end{theorem}


\newpage




\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{lemma}{0}
\setcounter{corollary}{0}
\section{Systems with One Contact Field and One Line Field}

In this section,
with the aid of the quasidecoupling method,  
we discuss systems with one contact field
and one line field, for which the shock wave curves and rarefaction wave 
curves coincide (see {\sc Temple} \cite{te1}). 
Such systems arise in the fields of elasticity theory (cf. \cite{ke2,bh}),
enhanced oil recovery (cf. \cite{is,te2}), and magnetohydrodynamics 
(cf. \cite{fr,bh}).

We note that a $2 \times 2$ system of conservation laws
\begin{equation}
u_t + f(u)_x = 0,
\end{equation}
has one contact field and one line field if and only if the flux function
$$
f(u) = u \phi (u),
$$
for some smooth function $\phi$ of $u$, where $\phi$ with $\phi_u \neq
0$ is both the wave speed ( i.e., $\phi = \lambda_1$ is an integral curve of 
$R_1$) and the
Riemann invariant for the contact field.
The Riemann invariant for the line field is
$$
w = \frac{u_2}{u_1};
$$
that is, $w$ = $const.$, is a line of $w = \frac{du_2}{du_1}$.

Using the polar coordinates $( r,\theta )$, we have that the eigenvalues
of the system (5.1) are
$$
\left\{ \begin{array}{l}
\lambda_1 = \phi, \\
\lambda_2 = ( r\phi )_r,
\end{array}
\right.
$$
respectively.
The corresponding Riemann invariants are
$$
\left\{ \begin{array}{l}
w = \theta , \\
z = \phi ( r, \theta ),
\end{array}
\right.
$$
respectively.

There are two families of entropy pairs---entropy pairs $(E_a,F_a)$:
\begin{equation}
\left\{ \begin{array}{l}
E_a = \Phi_0a(w) = ra(\theta), \\
F_a = \lambda _1 \Phi_0a(w) = r\phi (r,\theta )a(\theta),
\end{array} \right.
\end{equation}
and entropy pairs $(G_b,H_b)$:
\begin{equation}
\left\{ \begin{array}{l}
G_b = \Phi_0 \int_{\bar \phi}^{\phi ( r, \theta )} 
\frac{b'(\eta)}{(\lambda_2-\lambda_1)(\theta, \eta)} 
\exp (-\int_{\bar{\phi}}^{\eta} 
\frac {1}{\lambda_2 (\theta, \xi) - \lambda_1 (\xi)} d\xi) d\eta,\\
\mbox{     }  =  r \phi_r(r(\bar{\phi}, \theta), \theta)
 \int_{\bar{\phi}}^{\phi(r, \theta)}b'(\eta)
  \exp(-\int_{\bar{\phi}}^{\eta}\frac{\lambda_{2\phi}(\theta, \xi)}
  {\lambda_2(\theta, \xi) - \lambda_1(\xi)} d\xi) d\eta, \\
H_b = \lambda _1 G_b + b(\phi ( r , \theta )) ,
\end{array} \right.
\end{equation}
where $a$ and $b$ are two arbitrary continuous functions.

$(E_a,F_a)$ is a convex entropy pair if and only if $a''(\theta ) + a(
\theta ) \geq 0$.  
Furthermore, if $ |r\phi_r| \geq c_0 > 0$ and $\|u\|_{L^\infty} \le C$,
then there exists a family of $C^2$  entropy pairs $(G_{b_k}, H_{b_k})$
that are convex with respect to $\phi$, with $b_k(\phi) = e^{k\phi}$, $k \gg 1$.

One can verify that any  admissible solution to the system
(5.1) (see \cite{ch2}) satisfies
\begin{equation}
E_a(u)_t + F_a(u)_x = 0,
\end{equation}
for an arbitrary continuous function $a(\theta )$.
Moreover, if the system (5.1) is strictly hyperbolic on the solution domain
under consideration, then
on the contact discontinuity $C = (x(t), t)$,
\begin{equation}
\left \{\begin{array}{l}
\phi(u(x(t)-0, t)) = \phi(u(x(t)+0,t)), \\
\phi \ (G_{b_k}(u(x(t)+0))-G_{b_k}(u(x(t)-0,t)))
  = H_{b_k}(u(x(t)+0,t))-H_{b_k}(u(x(t)-0,t)),
\end{array} \right.
\end{equation}
and, on the second kind of 
shock wave $S = (x(t), t)$ with the slope $s$,
\begin{equation} \left \{\begin{array}{l}
\theta(u(x(t)-0,t))=\theta(u(x(t)+0,t)), \\
s(G_{b_k}(u(x(t)+0,t))-G_{b_k}(u(x(t)-0,t))) 
-(H_{b_k}(u(x(t)+0,t))-H_{b_k}(u(x(t)-0,t))) \geq 0,
\end{array} \right.
\end{equation}
for almost all $t \in (0, \infty)$.

We consider the Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
u_t + (u\phi(u))_x = 0, \\
u|_{t=0} = u_0(x).
\end{array} \right.
\end{equation}

\noindent
{\bf Definition 5.1.}
A $BV$ vector function $u(x,t)$, $r(u(x,t)) \ge r_0 > 0$ is called a 
generalized solution
for the Cauchy problem (5.7) in $\Pi_T$ if

\vspace{.2in}
\noindent
(a)  $u(x,t)$ is a weak solution of the Cauchy problem (5.7).

\vspace{.2in}
\noindent
(b)  $u(x,t)$ satisfies the following entropy conditions:

(i)  For any function $a \in C_{\rm{loc}} (-\infty , \infty)$ and
$\psi \in C_0^1 (\Pi_T),$
\begin{equation}
{\int \int}_{\Pi_T} (ra(\theta ) \psi_t + r\phi a(\theta )
\psi_x )dxdt + \int_{-\infty}^\infty r_0 (x)a(\theta_0 (x)) \psi
(x,0)dx = 0 .
\end{equation}

(ii)  On the discontinuity wave curves with $\theta = const.$, and
with left state $(r_-, \theta) = (r(\phi_-, \theta), \theta)$ and right state 
$(r_+, \theta)=(r(\phi_+, \theta), \theta)$,
\begin{equation}
s(r_+, r_- ; \theta ) \le s(r, r_- ; \theta), \qquad r \mbox{ between }
r_- \mbox{ and } r_+ ,
\end{equation}
where
$$
s(r_+ , r_- ; \theta )
= \frac{r_+ \phi (r_+ , \theta )- r_- \phi (r_- , \theta )}{r _+ - r_-} .
$$

(iii) On the discontinuity wave curves with $\phi(r, \theta) = const.$, and 
with left state $(r_-, \theta_-) = (r(\phi, \theta_-), \theta_-)$ and 
right state $(r_+, \theta_+) = (r(\phi, \theta_+), \theta_+)$,
\begin{equation}
(r(\phi, \theta_-) - r(l, \theta_-))(r(\phi, \theta_+)-r(l, \theta_+)) \ge 0,
\end{equation}
where $l$ is any constant and $r(l, \theta)$ is any branch such that 
the curve $\phi(r, \theta) = l $
passes both $(r(l, \theta_-), \theta_-)$ and $(r(l, \theta_+), \theta_+)$.


\noindent
{\bf Remark 1.}
The condition (5.9) is equivalent to
\begin{equation} \begin{array}{l} 
s(|r_-  - k| - |r_+ - k|) \\
- (r_-\phi(r_- , \theta )-k\phi(k, \theta ))
\mbox{sign}(r_- - k) +
(r_+\phi(r_-, \theta )-k \phi(k, \theta ))
\mbox{sign}(r_+ - k) \leq 0,
\end{array} 
\end{equation} 
for any constants $k$ and $\theta$; equivalently,
\begin{equation} \begin{array}{l} 
s(|r(\phi_-,\theta)- r(l, \theta )| - | r(\phi _+ , \theta )- r(l, \theta )|)
- (\phi_- r(\phi_- , \theta )-l r(l, \theta ))
\mbox{sign}(r(\phi_-, \theta)-r(l, \theta)) \\
+ (\phi_+ r(\phi_+, \theta )-l r(l, \theta ))
\mbox{sign}(r(\phi_+, \theta)- l r(l, \theta)) \leq 0,
\end{array} 
\end{equation} 
for the constants $l$ and $\theta$ such that 
$r(l, \theta) = k$, for any constants $k$ and $\theta$.

If the system (5.1) is strictly hyperbolic, $|r\phi_r| \ge c_0 > 0$,
then the condition (5.10) naturally
holds. For some nonstrictly hyperbolic systems, for example, systems arising
in magnetohydrodynamics and elasticity (see Subsection 5.b) and a system 
arising in oil recovery (see Subsection 5.c), the condition (5.10), together 
with (5.8) and (5.9), rules out the unstable solutions for the Riemann 
problem.




\noindent
{\bf Remark 2.}
If the function $\phi$ satisfies
$$
\left\{ \begin{array}{l}
|r(r\phi )_{rr} | \geq c_0 > 0 , \\
|r \phi _r| \geq c_0 > 0 ,
\end{array} \right.
$$
then the condition (5.9) or (5.11) is equivalent to the Lax shock
criterion:
\begin{equation}
\lambda_2 (+) < s(+,-) < \lambda_2 (-) ,
\end{equation}
where (+,-) denotes $(u_+,u_-)$, which is a pair of the right state
and left state of the second kind of shock waves.
Furthermore, the condition (5.9) is also equivalent to the entropy
criterion:  for some  entropy $(G_{b_0},H_{b_0} )$ strictly convex 
with respect to  $\phi$,
\begin{equation}
G_{b_0} (u)_t + H_{b_0} (u)_x \leq 0 ,
\end{equation}
in the sense of distribution by using (5.8).

\noindent
{\bf Remark 3.}
The condition $r(u(x,t)) \ge r_0 > 0$ in Definition 5.1 can be removed
by following the process we shall introduce and Wagner's 
arguments \cite{w}.

We recall that there exist  generalized solutions to the Cauchy
problem (5.7) for the cases of physical significance (see \cite{ch2,te2}).
Now we study the uniqueness and stability of the generalized
solutions for the Cauchy problem (5.7) with the aid of the
quasidecoupling method introduced in Section 3.

\newpage
\noindent
{\bf a.  Uniqueness and Stability of Generalized Solutions}


\begin{theorem}(Stability).  Suppose that $BV$ functions $u(x,t)$ and 
$v(x,t)$ are generalized
solutions of the Cauchy problem (5.7) in $\Pi_T$ with initial
data $u_0(x)$ and $v_0(x)$, respectively.  Then, for almost all
$t \in [0,T)$
and for any positive constant m, N,
\begin{equation}
\mbox{(i)  } \left\{ \begin{array}{l}
\theta(u(x,t)) = \theta(u_0(T_0^{-1}(p(x,t)))), 
\qquad \theta(v(x,t)) = \theta(v_0(\bar{T}_0^{-1}(q(x,t)))), \\
\int_{|x| \leq N} | \theta (u(x,t)) 
 - \theta(v(\bar{T}_t^{-1}T_t(x),t))|^mdx \\
\mbox{       }\leq C \int_{|x| \leq K_0(N + Lt)} 
|\theta(u_0(x)) - \theta(v_0(\bar{T}_0^{-1}T_0(x)))|^mdx,  \\
\int_{|x| \leq N} |\theta(u(x,t)) 
 - \theta(v(\bar{T}_t^{-1}\bar{T}_0T_0^{-1}T_t(x),t))|^mdx \\
\mbox{       }\leq C \int_{|x| \leq K_0(N + Lt)}
|\theta(u_0(x)) - \theta(v_0(x))|^mdx,
\end{array} \right.
\end{equation}
\begin{equation}
\mbox{(ii)  } \left\{ \begin{array}{l} 
\int_{|x| \leq N}|r(u(x,t))-r(v(\bar{T}_t^{-1}T_t(x),t))|dx 
\leq C \int_{|x| \leq K_0(N+Lt)} |u_0(x)- v_0(\bar{T}_0^{-1}T_0(x))|dx, \\ 
\int_{|x| \leq N} |(\bar{T}_t^{-1}T_t - I)(x)| dx 
\leq C \int_{|x| \leq K_0(N+\bar{L}t)} (|u_0(x)- v_0(x)|
+|u_0(x)- v_0(\bar{T}_0^{-1}T_0(x))|) dx, \end{array} \right.
\end{equation}
provided that
$$
\theta (u_0(T_0^{-1}(x))) = \theta (v_0(\bar{T}_0^{-1}(x))), \qquad \mbox{a.e.,}
$$
where  
$$
\begin{array}{l}
L = \max  \left(\sup _ {\Pi_T}(|\phi(u)| + |(r^{-2}\phi_r)(u)|),
\sup _ {\Pi_T}(|\phi(v)| + |(r^{-2}\phi_r)(v)|)\right),\\
\bar{L} = L + \max(\sup_{[0,T]}( |\bar{T}_t^{-1}T_t(\pm N)|-N)/t),\\ 
K_0 = \max(\sup_{\Pi_T}|r(u(x,t))|, \sup_{\Pi}|r(v(x,t))|)/r_0,

\end{array}
$$
$C$ is a constant independent of $m$, and 
$(p,t)= (T_t(x),t)$ and $(q,t)= (T_t(x), t)$ are the quasidecoupling 
transformations with
\begin{equation} \left \{ \begin{array}{l}
\frac{\partial p}{\partial x} =r(u(x,t)), \\
\frac{\partial p}{\partial t} 
= -r(u(x,t))\phi(u(x,t)), \\
p(x,0) = \int_0^x r(u_0(\xi))d\xi, 
\end{array} \right.
\end{equation}
and
\begin{equation} \left \{ \begin{array}{l}
\frac{\partial q}{\partial x} = r(v(x,t)), \\
\frac{\partial q}{\partial t} 
=-r(v(x,t))\phi(v(x,t)), \\
q(x,0) = \int_0^x r(v_0(\xi))d\xi, 
\end{array} \right.
\end{equation}
respectively.
\end{theorem}

The proof consists of several lemmas.  Using the quasidecoupling method,
we immediately obtain the following lemma.

\begin{lemma}
For the generalized solutions $u(x,t)$ and $v(x,t)$ in Theorem 5.1, the
functions $\theta (u(x,t))$ and $\theta(v(x,t))$ satisfy (5.15).
\end{lemma}

\noindent
{\bf Proof.}
Since $u(x,t)$ and $v(x,t)$ are the generalized solutions with initial
data $u_0(x)$ and $v_0(x)$,  we have, for any positive constant $m$, 
$$
\left\{ \begin{array}{l}
r(u(x,t))_t + ((r\phi )(u(x,t)))_x = 0, \\
(r(u(x,t)) | \theta (u(x,t)) - k|^m )_t + 
((r\phi )(u(x,t))|\theta(u(x,t)) - k|^m )_x = 0, 
\quad \mbox{for any } k \in (-\infty, \infty ),
\end{array} \right.
$$
and
$$
\left\{ \begin{array}{l}
r(v(x,t))_t + ((r\phi )(v(x,t)))_x = 0, \\
(r(v(x,t)) | \theta (v(x,t)) - l|^m )_t 
+ ((r\phi )(v(x,t))|\theta(v(x,t)) - l|^m )_x = 0, 
\quad \mbox{for any } l \in (-\infty, \infty ).
\end{array} \right.
$$

Using Theorem 3.2 (Framework (A)), we immediately obtain (5.15).

\vspace{.3in}
\noindent
{\bf Remark.}
This lemma  holds for  the $L^\infty$ weak solutions $u(x,t)$ and 
$v(x,t)$ of the Cauchy problem (5.7) satisfying (5.8)
with initial
data $u_0(x)$ and $v_0(x)$, respectively.
We did not use the assumption that both $u(x,t)$ and
$v(x,t)$ are $BV$ functions.

Denote $\tau_1(x,t) = r(u(x,t))^{-1}$ and $\tau_2(x,t) = r(v(x,t))^{-1}$ .
We have the following lemma.

\begin{lemma} 
If $\theta(u_0(T_0^{-1}(x))) = \theta(v_0(\bar{T}_0^{-1}(x))) 
\equiv \theta_0(x)$, 
a.e., then
\begin{equation} \left \{\begin{array}{l}
\tau_1(x,t) = \tau_1(p(x,t), t; r(u_0(T_0^{-1}(p(x,t))))^{-1}), \\
\tau_2(x,t) = \tau_2(q(x,t), t; r(v_0(\bar{T}_0^{-1}(q(x,t))))^{-1}),
\end{array} \right.
\end{equation}
where both $\tau_i(y,t; \tau_0(y)), i=1,2,$ are solutions of the Cauchy
problem
\begin{equation} \left \{\begin{array}{l}
\tau_t - \phi(\tau, \theta_0(y))_y =0, \\
\tau|_{t=0} = \tau_0(y),
\end{array} \right.
\end{equation}
satisfying the following entropy condition: On the discontinuity wave 
curve with $\theta_0(y) = const.,$ and  with left state
$\tau_-= \tau(\phi_-, \theta_0(y))$ and right state 
$\tau_+=\tau(\phi_+, \theta_0(y))$, 
\begin{equation}
\sigma(\tau_+, \tau_-; \theta_0(y)) \le \sigma(\tau, \tau_-; \theta_0(y)),
\qquad \mbox{ for any } \tau \mbox{ between } \tau_- \mbox{ and } \tau_+,
\end{equation}
and on the discontinuity wave curve with $\phi = const.,$ and 
with left state
$\tau_-=\tau(\phi, \theta_-)$ and right state
$\tau_+=\tau(\phi, \theta_+)$, 
\begin{equation}
(\tau(\phi, \theta_-) - \tau(l, \theta_-))
(\tau(\phi, \theta_+)-\tau(l, \theta_+)) \ge 0,
\end{equation}
where
$l$ is any constant and $\tau(l, \theta)$ is any branch such that 
the curve $\phi(r, \theta) = l$
passes both $(r(l, \theta_-), \theta_-)$ and $(r(l, \theta_+), \theta_+)$,
and
$$
\sigma(\tau_+, \tau_-; \theta_0(y)) 
= -\frac{\phi(\tau_+, \theta_0(y))-\phi(\tau_-, \theta_0(y))}{\tau_+ - \tau_-}.
$$
\end{lemma}

\noindent
{\bf Proof.} Using the quasidecoupling transformations (5.17) and (5.18),
we can immediately verify that both $\tau_1(T^{-1}(p,t))$ and 
$\tau_2(\bar{T}^{-1}(q,t))$ are weak solutions of the Cauchy problem 
(5.20) with 
the initial data $r(u_0(T_0^{-1}(p)))^{-1}$ and 
$r(v_0(\bar{T}_0^{-1}(q)))^{-1}$, respectively.


Now we prove the entropy condition (5.21). For concreteness, we prove only 
that $\tau_1(x,t) = \tau_1(T^{-1}(p,t))$ satisfies (5.21).  In fact, on the 
shock wave curve of $u(x,t)$, i.e., 
$\theta = \theta(u_0(T_0^{-1}(p))) = const.$, 
$$ 
s(r_+, r_-; \theta(u_0(T_0^{-1}(p)))) 
\le s(r, r_-; \theta(u_0(T_0^{-1}(p)))), 
$$ 
for any $r$ between $r_-$ and $r_+$, where 
$$ s(r_+, r_-; \theta(u_0(T_0^{-1}(p)))) 
= \frac{r_+\phi(r_+, \theta(u_0(T_0^{-1}(p)))) 
 - r_-\phi(r_-, \theta(u_0(T_0^{-1}(p))))}{r_+ -r_-}.  
$$ 
The quasidecoupling transformation $T_1: (x,t) \longmapsto (p,t)$ transforms 
the shock wave $S = (x(t; r_+, r_-), t)$ with 
$\theta(u_0(T_0^{-1}(p))) = const.$ and  the slope 
$s(r_+, r_-; \phi(u_0(T_0^{-1}(p))))$ in the (x, t)-plane to the shock 
wave $S^\prime = (p(t; \tau_+, \tau_-), t)$ with 
$\theta(u_0(T_0^{-1}(p))) = const$.  and the slope 
\begin{eqnarray*}
\sigma(\tau_+, \tau_-; \theta(u_0(T_0^{-1}(p)))) 
&= & r_+(s(r_+, r_-; \theta(u_0(T_0^{-1}(p))))-\phi_+)\\ 
&= &r_-(s(r_+, r_-; \theta(u_0(T_0^{-1}(p)))) - \phi_-).
\end{eqnarray*}
Therefore, on the shock wave curve,
$$
\left \{ \begin{array}{l}
\sigma(\tau - \tau_-) =\phi(\tau_-, \theta(u_0(T_0^{-1}(p))))
-\phi(\tau, \theta(u_0(T_0^{-1}(p)))),
\\
\mbox{  }\theta(u_0(T_0^{-1}(p))) = const.,
\end{array} \right.
$$
equivalently,
$$
\left \{\begin{array}{l}
s(r-r_-) = r\phi(r, \theta(u_0(T_0^{-1}(p))))
-r_-\phi(r_-, \theta(u_0(T_0^{-1}(p)))), \\
\mbox{  }\theta(u_0(T_0^{-1}(p))) = const.,
\end{array} \right.
$$
we have
\begin{eqnarray*}
 \sigma(\tau_+, \tau_-; \theta(u_0(T_0^{-1}(p))))
&  =  & r_-(s(r_+, r_-; \theta(u_0(T_0^{-1}(p))))-\phi_-) \\ 
& \le & r_-(s(r, r_-; \theta(u_0(T_0^{-1}(p)))) - \phi_-) \\
&  =  & r(s(r, r_-; \theta(u_0(T_0^{-1}(p)))) -\phi) \\ 
&  =  & \sigma(\tau, \tau_-; \theta(u_0(T_0(p)))).
\end{eqnarray*}
Similarly, we can prove that 
$\tau_2(x,t) = \tau_2(\bar{T}^{-1}(q,t))$ satisfies
(5.21).
The entropy condition (5.22) can directly be verified from (5.10).

\begin{lemma} For any positive constant $N_1$,
$$
\int_{|y| \le N_1} |\tau(y,t; \tau_0(y))- \tau(y,t; \bar{\tau}_0(y))|dy
\le C \int_{|y|\le N_1 + L_1t}|\tau_0(y)-\bar{\tau}_0(y)| dy,
$$
where 
$L_1 = \sup _ {\{|\tau|\le sup _ {\Pi_T} |\tau|, y\in R^1 \}}
|\phi_{\tau}(\tau, \theta_0(y))|$.
\end{lemma}

\noindent
{\bf Proof.} First we note that the entropy condition (5.21) is equivalent
to the condition:
\begin{equation} \begin{array}{l} 
 \sigma(|\tau_- - k|
 -|\tau_+ - k|) \\ 
 + (\phi(\tau_-, \theta)- \phi(k, \theta))\mbox{sign}(\tau_- - k)
- (\phi(\tau_+, \theta)- \phi(k, \theta))\mbox{sign}(\tau_+ - k) \le 0,
\end{array}
\end{equation}
for any constants $k$ and $\theta$; equivalently,
\begin{equation} \begin{array}{l} 
 \sigma(|\tau(\phi_-, \theta) - \tau(l, \theta)|
 -|\tau(\phi_+, \theta) - \tau(l, \theta)|) \\ 
 + (\phi_- - l)\mbox{sign}(\tau(\phi_-, \theta)-\tau(l, \theta))
- (\phi_+ - l)\mbox{sign}(\tau(\phi_+, \theta)-\tau(l, \theta)) \le 0,
\end{array}
\end{equation}
for the  constants $l$ and $\theta$ such that $\tau(l, \theta) = k$.

Next we consider the deviation between $\tau(y,t) \equiv \tau(y,t; \tau_0(y))$
and $\bar{\tau}(y,t) \equiv \tau(y,t; \bar{\tau}_0(y))$
in the norm of $L^1$.
For this purpose, we associate with the pair 
$(\tau, \bar{\tau})$ a Radon measure $B = B_{(\tau, \bar{\tau})}$ 
defined
on $\Pi_T$ as follows:  $B$ is the divergence of the  vector field
$(|\tau - \bar{\tau}|, 
(\phi(\tau, \theta_0) - \phi(\bar{\tau}, \theta_0))
\mbox{sign}(\tau - \bar{\tau}))$: 
\begin{equation} 
B = |\tau - \bar{\tau}|_t -((\phi(\tau, \theta_0)- \phi(\bar{\tau}, \theta_0))
\mbox{sign}(\tau- \bar{\tau}))_y.  
\end{equation} 

We recall that with each function $\tau$ in $BV$, there is associated a
set of points $\Gamma (\tau)$ at which $\tau$ experiences an approximate jump
discontinuity \cite{v}.  We refer to $\Gamma(\tau)$, $\Gamma(\bar{\tau})$, 
$\Gamma ( \phi )$, $\Gamma (\bar{\phi})$,
and $\Gamma (\theta_0)$ as the jump set of the functions $\tau$, $\bar{\tau}$, 
$\phi$, $\bar{\phi}$, 
and $\theta_0$, respectively.
Note that if $\tau(y,t)$, $\bar{\tau}(y,t)$ and $\theta_0(y)$ are continuous 
functions, then
$$
B_{(\tau, \bar{\tau})} (\Pi_T) = 0.
$$
Then we can conclude that the measure $B_{(\tau, \bar{\tau})}$ is concentrated 
on
$\Gamma = \Gamma (\phi) \cup \Gamma (\bar{\phi}) \cup \Gamma (\theta_0)$:
\begin{equation}
B(E) = B(E \cap \Gamma ),
\end{equation}
for any Borel set $E \subset \Pi_T$.

According to the generalized Green's theorem for measures \cite{f},
the $B$-measure is expressed by
\begin{equation}
B_{(\tau, \bar{\tau})} (E) 
= \int_{E \cap \Gamma} (n_t[|\tau - \bar{\tau}|]
 - n_y[(\phi(\tau, \theta_0) - \phi(\bar{\tau}, \theta_0))
 \mbox{sign}(\tau-\bar{\tau})])dH_1 .
\end{equation}
Here $H_1$ denotes the one-dimensional Hausdorff measure, 
$ \vec n = (n_y, n_t)$ denotes
the unit normal to $E \cap \Gamma$ in the sense of {\sc Federer} \cite{f}, and
the bracket denotes the jump of the enclosed quantity across $E$ in the
direction of $\vec n$:
$$
[|\tau - \bar{\tau} |] = [|\tau - \bar{\tau}|](P) =
|\tau_+ - \bar{\tau}_+ | - |\tau_- - \bar{\tau}_- | ,
$$
where $(\tau_\pm ,\bar{\tau}_\pm )(P)
=(\tau(\phi_\pm, \theta_{0\pm}), \tau(\bar{\phi}_\pm, \theta_{0\pm})) $ denote 
the approximate limits of 
$(\tau, \bar{\tau})$
at $P \in E \cap \Gamma$ with the half-planes
$$
H_{\pm \vec n}(P) = \{ Q: <Q-P, \pm \vec n> \geq 0, n_y < 0 \} .
$$
We note that the set $E_0$ of irregular points and points on which
$n_y \neq 0$ for $(\tau, \bar{\tau}) \in BV(\Pi_T)$ is the set of zero 
one-dimensional Hansdorff  measure.
Therefore we can write (5.27) as
\begin{equation}
B_{(\tau, \bar{\tau})} (E) = \int_{E \cap E_0^c \cap \Gamma } 
(\sigma [|\tau -\bar{\tau}|] +
[(\phi- \bar{\phi})\mbox{sign}(\tau -\bar{\tau})])dt ,
\end{equation}
where $\sigma \equiv -n_t(P)/n_y(P)$ denotes the speed of propagation
of $\Gamma$ at $P \in E \cap E_0^c \cap \Gamma$ and $dt = - n_y dH_1$.
Then
for any Borel set $E$, the Borel measure
$$
B_{(\tau, \bar{\tau})}(E) \leq 0.
$$
In fact, if 
$E \subset (\Gamma (\phi) \cap \Gamma (\bar{\phi})\cap \Gamma(\theta_0))^c$, 
we use
(5.23) and (5.28) and immediately obtain
$$
B_{(\tau, \bar{\tau})} (E) \leq 0;
$$
If $E \subset \Gamma(\phi)\cap (\Gamma(\bar{\phi})\cap \Gamma(\theta_0))^c$,
or $\Gamma(\bar{\phi})\cap(\Gamma(\phi)\cap \Gamma(\theta_0))^c$, or
$\Gamma(\theta_0)\cap(\Gamma(\phi)\cap \Gamma(\bar{\phi}))^c$,
we use (5.22), (5.23) and (5.28), and also deduce
$$
B_{(\tau, \bar{\tau})} (E) \leq 0;
$$
if  $E \subset \Gamma (\phi) \cap \Gamma (\bar{\phi})\cap \Gamma(\theta_0)^c$, 
then for any
point $P \in E \cap E_0^c$, we have from (5.24) that
$$ 
\begin{array}{l}
\sigma(|\tau(\phi_-, \theta_0) - \tau(\bar{\phi}_-, \theta_0)| 
- | \tau(\phi_+ , \theta_0)-  \tau(\bar{\phi}_-, \theta_0)|) \\
+ (\phi_- - \bar{\phi}_-)
\mbox{sign}(\tau(\phi_-, \theta_0)-\tau(\bar{\phi}_- , \theta_0))
-(\phi_+ - \bar{\phi}_-)
\mbox{sign}(\tau(\phi_+, \theta_0)- \tau(\bar{\phi}_-, \theta_0)) \leq 0,
\end{array} 
$$
and
$$
\begin{array}{l}
\sigma(|\tau(\bar{\phi}_-, \theta_0) - \tau(\phi_+, \theta_0) | 
- |\tau(\bar{\phi}_+, \theta_0) - \tau(\phi_+, \theta_0) |) \\
+ (\bar{\phi}_- - \phi_+)
\mbox{sign}(\tau(\bar{\phi}_-, \theta_0)- \tau(\phi_+, \theta_0)) -
(\bar{\phi}_+ - \phi_+)
\mbox{sign}( \tau(\bar{\phi}_+, \theta_0) - \tau(\phi_+, \theta_0))
\leq 0,
\end{array} 
$$
and, therefore, we obtain
$$
\begin{array}{l}
\sigma(|\tau(\phi_-, \theta_0) - \tau(\bar{\phi}_- , \theta_0)| 
- | \tau(\phi_+, \theta_0) - \tau(\bar{\phi}_+, \theta_0) | )\\
+ (\phi_- - \bar{\phi}_-)
\mbox{sign}( \tau(\phi_-, \theta_0)- \tau( \bar{\phi}_-, \theta_0)) -
(\phi_+ - \bar{\phi}_+)
\mbox{sign}(\tau(\phi_+, \theta_0)- \tau( \bar{\phi}_+,\theta_0)) \leq 0,
\end{array}  
$$
and, equivalently,
$$
\begin{array}{l}
\sigma(|\tau_- - \bar{\tau}_-| - |\tau_+ - \bar{\tau}_+|)\\
+ (\phi(\tau_-, \theta_0)  - \phi(\bar{\tau}_-, \theta_0))
\mbox{sign}( \tau_- - \bar{\tau}_-) -
(\phi(\tau_+ , \theta_0) - \phi(\bar{\tau}_+, \theta_0))
\mbox{sign}(\tau_+ - \bar{\tau}_+) \leq 0,
\end{array}  
$$
which means from (5.28) that
$$
B_{(\tau, \bar{\tau})}(E) \leq 0;
$$
If $E \subset \Gamma(\phi) \cap \Gamma(\theta_0)\cap \Gamma(\bar{\phi})^c$,
or $\Gamma(\bar{\phi}) \cap \Gamma(\theta_0)\cap \Gamma(\phi)^c$, for example,
the former case, then
$$
\left \{\begin{array}{l} 
\sigma = 0, \\
(\phi_- - \bar{\phi})
\mbox{sign}(\tau(\phi_-, \theta_{0-})-\tau(\bar{\phi}, \theta_{0-}))
-(\phi_+ - \bar{\phi})
\mbox{sign}(\tau(\phi_+, \theta_{0-})-\tau(\bar{\phi}, \theta_{0-})) \le 0, \\
(\tau(\phi_\pm, \theta_-)- \tau(\bar{\phi}, \theta_-))
(\tau(\phi_\pm, \theta_+)- \tau(\bar{\phi}, \theta_+)) \ge 0,
\end{array} \right. 
$$
and, therefore, we have
$$
\left \{\begin{array}{l}
\sigma = 0, \\
(\phi_- - \bar{\phi})
\mbox{sign}(\tau(\phi_-, \theta_{0-}) -\tau(\bar{\phi}, \theta_{0-}))
-(\phi_+ - \bar{\phi})
\mbox{sign}(\tau(\phi_+, \theta_{0+})-\tau(\bar{\phi}, \theta_{0+})) \le 0,
\end{array} \right. 
$$
and, equivalently,
$$
\left \{\begin{array}{l}
\sigma = 0, \\
(\phi(\tau_-, \theta_{0-})  - \phi(\bar{\tau}_-, \theta_{0-}))
\mbox{sign}(\tau_- - \bar{\tau}_-)
-(\phi(\tau_+, \theta_{0+}) - \phi(\bar{\tau}_+, \theta_{0+}))
\mbox{sign}(\tau_+ - \bar{\tau}_+) \le 0,
\end{array} \right. 
$$
which means from (5.28) that
$$
B_{(\tau, \bar{\tau})}(E) \leq 0;
$$
If $E \subset \Gamma(\phi)\cap \Gamma(\bar{\phi}) \cap \Gamma(\theta_0)$,
then
$$
\left \{\begin{array}{l}
\sigma = 0, \\
(\phi_- -\bar{\phi}_-)
\mbox{sign}(\tau(\phi_-, \theta_{0-})- \tau(\bar{\phi}_-, \theta_{0-}))
-(\phi_+ - \bar{\phi}_-) \mbox{sign}(\tau(\phi_+, \theta_{0-})
-\tau(\bar{\phi}_-, \theta_{0-})) \le 0,\\ 
(\phi_+ - \bar{\phi}_-) \mbox{sign}(\tau(\phi_+, \theta_{0-})
- \tau(\bar{\phi}_-, \theta_{0-}))
-(\phi_+ - \bar{\phi}_+)
\mbox{sign}(\tau(\phi_+, \theta_{0-})-\tau(\bar{\phi}_+, \theta_{0-})) \le 0,\\
(\tau(\phi_\pm, \theta_{0-})-\tau(\bar{\phi}_\pm, \theta_{0-}))
(\tau(\phi_\pm, \theta_{0+})-\tau(\bar{\phi}_\pm, \theta_{0+})) \ge 0,
\end{array} \right. 
$$
and, therefore, we obtain
$$
\left \{\begin{array}{l}
\sigma = 0, \\
(\phi_- - \bar{\phi}_-)
\mbox{sign}(\tau(\phi_-, \theta_{0-})- \tau(\bar{\phi}_-, \theta_{0-}))
-(\phi_+ - \bar{\phi}_+)
\mbox{sign}(\tau(\phi_+, \theta_{0+})- \tau(\bar{\phi}_+, \theta_{0+}))\le 0,
\end{array} \right. 
$$
and, equivalently,
$$
\left \{\begin{array}{l}
\sigma = 0, \\
(\phi(\tau_-, \theta_{0-}) - \phi(\bar{\tau}_-, \theta_{0-}))
\mbox{sign}(\tau_- - \bar{\tau}_-)
-(\phi(\tau_+ , \theta_{0+}) - \phi(\bar{\tau}_+, \theta_{0+}))
\mbox{sign}(\tau_+ - \bar{\tau}_+)\le 0,
\end{array} \right. 
$$
which means from (5.28) that
$$
B_{(\tau, \bar{\tau})}(E) \leq 0.
$$

Fix $N > 0$, and consider domains of the form
$$
D = \{ (y,s)\mid |y| \leq N + L(T-s), t_1 \leq s \leq t \}, \quad 0 <
t_1 < t \leq T,
$$
$$
S_s = \{ y \mid |y| \leq N + L(T-s)\}, \quad t_1 \leq s \leq t ,
$$
where the constant $L$ will be chosen below.

It follows from the generalized Green's theorem for measures that
\begin{eqnarray*}
B_{(\tau, \bar{\tau})}(D)  
& = & \int_{\partial D} \{n_t |\tau - \bar{\tau}|
- n_y (\phi(\tau, \theta_0) - \phi(\bar{\tau}, \theta_0))
\mbox{sign}(\tau - \bar{\tau})\}dH_1  \\
& = & \int_\gamma \{ n_t |\tau - \bar{\tau}|
- n_y
(\phi(\tau, \theta_0) - \phi(\bar{\tau}, \theta_0))
\mbox{sign}(\tau - \bar{\tau})\}dH_1 \\
&& + \int_{S_t} |\tau -\bar{\tau}|dx - \int_{S_{t_1}} |\tau -\bar{\tau}|dx ,
\end{eqnarray*}
where $\gamma$ is the lateral boundary.

Note that, on $\gamma$,
$$
\begin{array}{l}
n_t|\tau - \bar{\tau}| - n_y (\phi(\tau, \theta_0)- 
\phi(\bar{\tau}, \theta_0))
\mbox{sign}(\tau - \bar{\tau}) \\
= |\tau - \bar{\tau} | 
(n_t - \frac{(\phi(\tau, \theta_0)- \phi(\bar{\tau}, \theta_0))\mbox{sign}
(\tau - \bar{\tau})}
{|\tau- \bar{\tau}|} n_y )  \\
\geq  |\tau - \bar{\tau} | (L - \max (\sup _ {\Pi_T} 
|\phi_{\tau}(\tau, \theta_0)|, 
\sup _ {\Pi_T} |\phi_{\tau}(\bar{\tau}, \theta_0)|)) = 0,
\end{array}
$$
provided 
$$
L = \max (\sup _ {\Pi}|\phi_{\tau}(\tau, \theta_0)|,
\sup _ {\Pi_T} |\phi_{\tau}(\bar{\tau}, \theta_0)|).
$$
Therefore, we obtain from Lemma 5.3,
$$
\int_{S_t} |\tau (y,t) - \bar{\tau}(y,t)|dy \leq \int_{S_{t_1}}
|\tau (y,t_1)- \bar{\tau} (y,t_1)|dy.
$$
Notice
that $\tau (x,t)$ and $\bar{\tau}(y,t)$ are $BV$ functions.
Letting $t_1 \to 0$, we obtain
$$
\int_{S_t} |\tau (y,t) - \bar{\tau}(y,t)|dy \leq \int_{S_0}
|\tau_0(y) - \bar{\tau}_0(y)|dy .
$$
This completes the proof of Lemma 5.3.

Therefore, we have the following lemma.

\begin{lemma} The solution of the Cauchy problem (5.20) with (5.21) and
(5.22) is
unique. In particular, 
$$
\tau_1(y, t; \tau_0(y)) = \tau_2(y, t; \tau_0(y)), \qquad \mbox{a.e.}
$$
\end{lemma}

\begin{lemma} For the generalized solutions $u(x,t)$ and $v(x,t)$ with
$\theta(u_0(T_0^{-1}(x))) = \theta(v_0(\bar{T}_0^{-1}(x)))$ in Theorem 5.1, 
the functions $r(u(x,t))$ and $r(v(x,t))$ satisfy (5.16).
\end{lemma}

\noindent
{\bf Proof}. We conclude from Lemma 5.2 and Lemma 5.3 that
$$ \left \{\begin{array}{l}

\tau(u(x,t)) = \tau(u(T_t^{-1}(p),t)) 
= \tau(p, t; r(u_0(T_0^{-1}(p)))^{-1}),\\
\tau(v(x,t)) = \tau(v(\bar{T}_t^{-1}(q),t)) 
= \tau(q,t; r(v_0(\bar{T}_0^{-1}(q)))^{-1}),
\end{array} \right.
$$
and both $\tau(u(x,t)) = \tau(u(T_t^{-1}(p),t))$ and 
$\tau(v(\bar{T}_t^{-1}T_t(x),t)) 
= \tau(v(\bar{T}_t^{-1}(p),t))$ satisfy the Cauchy
problem (5.19) and (5.20) with initial data $r(u_0(T_0^{-1}(p)))^{-1}$ and
$r(v_0(\bar{T}_0^{-1}(p)))^{-1}$, respectively.
Using Lemma 5.3, we obtain

$$
\int_{|p| \le N_1}|\tau(u(T_t^{-1}(p),t))
- \tau(v(\bar{T}_t^{-1}(p),t))|dp
\le C \int_{|p| \le N_1 + L_1t}
|r(u_0(T_0^{-1}(p)))^{-1}-r(v_0(\bar{T}_0^{-1}(p)))^{-1}|dp,
$$
where $L_1 = \max (\mbox{sup}_{\Pi_T} |(r^{-2}\phi_r)(u(x,t))|, 
\mbox{sup}_{\Pi_T}|(r^{-2}\phi_r)(v(x,t))|)$.

Therefore, for any positive constant N, we have

$$
\int_{|x| \le N}|r(u(x,t))
- r(v(\bar{T}_t^{-1}T_t(x),t))| dx
\le C\int_{|x| \le K_0(N+Lt)}|r(u_0(x))-r(v_0(\bar{T}_0^{-1}T_0(x)))| dx.
$$

\begin{lemma} If $\theta(u_0(T_0^{-1}(x))) = \theta(v_0(\bar{T}_0^{-1}(x)))$, 
a.e., then there exists
a constant $C > 0$ such that, for almost all $t \in [0, T]$,
$$
\int_{|x| \le N} |(\bar{T}_t^{-1}T_t - I)(x)| dx 
\le C \int_{|x| \le K_0(N + \bar{L}t)}
(|r(u_0(x))-r(v_0(x))| + |r(u_0(x))-r(v_0(\bar{T}_0^{-1}T_0(x)))|)dx,
$$
for any constant $N > 0$,
where $\bar{L} = L + \max(\sup_{[0, T]}(|\bar{T}_t^{-1}T_t(\pm N)|-N)/t)$,
and  $L = \max(\sup_{\Pi_T}(|\phi(u)| + |(r^{-2}\phi_r)(u)|),
\sup_{\Pi_T}(|\phi(v)| + |(r^{-2}\phi_r)(v)|))$.
\end{lemma}

\noindent
{\bf Proof}. It suffices to prove that, for any constant $N > 0$, there exists
a  constant $C > 0$ such that the Lipschitz functions $p(x,t)$
and $q(x,t)$, determined by (5.17)-(5.18), satisfy
$$
\int_{|x| \le N}|p(x,t)-q(x,t)| dx \le C\int_{|x| \le K_0(N + Lt)}
(|r(u_0(x))-r(v_0(x))| + |r(u_0(x))-r(v_0(\bar{T}_0^{-1}T_0(x)))|)dx.
$$
In fact, $p(x,t)$ and $q(x,t)$ satisfy
$$ \left \{\begin{array}{l}
p_t + \phi(\tau(p,t; u_0(T_0^{-1}(p))), \theta(u_0(T_0^{-1}(p))))p_x = 0,
 \qquad \mbox{ a.e. }\\
p|_{t=0} = \int_0^x r(u_0(\xi))d\xi,
\end{array} \right.
$$
and
$$
\left \{\begin{array}{l}
q_t + \phi(\tau(q,t; v_0(\bar{T}_0^{-1}(q))), 
\theta(v_0(\bar{T}_0^{-1}(q))))q_x = 0,
 \qquad \mbox{ a.e.} \\
q|_{t=0} =\int_0^x r(v_0(\xi))d\xi.
\end{array} \right.
$$

Using Theorem 3.3 and Lemma 5.3, we obtain

$$
\begin{array}{l}
\int_{|x| \le N}|p(x,t)-q(x,t)| dx\\
\le C {\int_{|x| \le N}|\tau(q,t;u_0(T_0^{-1}(q))) 
-\tau(q,t;v_0(\bar{T}_0^{-1}(q)))|  dx 
+\int_{|x| \le N+Lt} |r(u_0(x))- r(v_0(x))|dx}\\
\le   \int_{|x| \le K_0(N + Lt)}(|u_0(x) - v_0(x)| 
+ |u_0(x) - v_0(\bar{T}_0^{-1}T_0(x))|)dx.
\end{array}
$$
This completes the proof of Lemma 5.6.

We can immediately obtain Theorem 5.1 from Lemma 5.1, Lemma 5.4, and Lemma 5.6.
As a corollary, we have the following theorem.

\begin{theorem}
(Uniqueness).  The $BV$ generalized solution of the Cauchy problem (5.7) 
is unique.
\end{theorem}

Propagation and cancellation of  initial oscillations for such
systems (5.1) have been studied by  {\sc Chen} \cite{ch2,ch3}.

Now we discuss some applications of Theorem 5.1 and Theorem 5.2 to
several important systems arising in the field of elasticity theory,
enhanced oil recovery, and magnetohydrodynamics.

\newpage
\noindent
{\bf b.  Systems Arising in Magnetohydrodynamics and Elasticity}

\vspace{.2in}
\noindent
Consider the $2 \times 2$ systems with symmetry that are discussed
in \cite{ch2}.  We assume that $\phi(u)$ satisfies the following symmetry
assumptions:

$S_1$:  $\phi (u) > 0$.

$S_2$:  $\phi (u) = \mbox{const.} > 0$ is a simple closed curve
enclosing the origin $u=0$.

This class of systems contains the elastic string model and
the magnetohydrodynamics model \cite{ke2,li3},
$$
\left\{ \begin{array}{l}
\phi = \phi(r) > 0, \\
\phi '' (r) > 0 .
\end {array} \right.
$$
In particular, $\phi = \phi (r) = 1 + \delta \frac
{(r-1)^2}{r}$, $\delta > 0$ constant, models a nonlinear
stress-strain relation.  On the region $K \supset \{ u: r(u)=1\}$, the
system is nonstrictly hyperbolic.

\begin{theorem}
If the function $\phi(u)$ satisfies
$$
\left\{
\begin{array}{l}
r\phi_r \geq c_0 > 0, \\
r(r\phi )_{rr} \geq c_0 > 0,
\end{array} \right.
$$
besides the assumptions $S_1 - S_2$, and the initial data
$u_0(x) \in \{ u: \phi_1 \leq \phi (u) \leq \phi_2 < \infty \}$,
$\phi_1 > \phi(0)$, then there exists a unique $BV$ generalized solution
$u(x,t)$ of the Cauchy problem (5.7)
satisfying (5.8) and (5.14). 
\end{theorem}

\noindent
{\bf Proof.}
Following \cite{ch2}, we first obtain a global $BV$ weak solution $u(x,t)$
satisfying (5.8).  It is easy to check that, for any $C^2$ strictly convex entropy $(G_b, H_b)$ on $K$, the solution $u(x,t)$ also
satisfies (5.14), which is equivalent to (5.9) for such systems.
The result follows. We omit the details.

Furthermore, we have the following theorem.

\begin{theorem}
If the function $\phi(u) = \phi (r)$ satisfies $ mes \{ r: \phi''(r)= 0 \} = 0$,
and the initial data
$u_0(x) \in \{u: 0 < r_1 \le r(u) \le r_2 \le \infty \}$,
then there exists a unique $L^{\infty}$ solution $u(x,t)$ of the Cauchy 
problem (5.7) satisfying the following entropy conditions:

(i) For any function $ a \in C_{loc}(-\infty, \infty)$ and
    $\psi \in C_0^1(\Pi_T)$,
\begin{equation}
\int \int _ {\Pi_T}(ra(\theta)\psi_t + r\phi a(\theta)\psi_x)dxdt
+ \int_{-\infty}^{\infty} r_0(x)a(\theta_0(x))\psi(x,0)dx = 0.
\end{equation}

(ii) For any constant $ k \in R^1 $ and any nonnegative function $\psi
\in C_0^1(\Pi_T -\{t=0\})$,
\begin{equation} 
\int \int _ {\Pi_T}(|r-k|\psi_t 
+ (r\phi(r) - k\phi(k))\mbox{sign}(r - k)\psi_x) dxdt \le 0.
\end{equation}
\end{theorem}

\noindent
{\bf Proof.}
Following \cite{ch2}, we first obtain a global $BV$ weak solutions $u(x,t)$
satisfying (5.29)-(5.30). Using the framework established in 
Theorem 3.1 
in \cite{ch3}, we conclude that there exists a global $L^{\infty}$ solution
of the Cauchy problem (5.7) satisfying (5.29)-(5.30). Then using Kruzkov's
theorem, we immediately conclude the uniqueness of $r$. Finally, 
using the Theorem 5.1, we obtain the uniqueness of $\theta$. 
This completes the proof of Theorem 5.4.  

\vspace{.3in} 
\noindent
{\bf c.  A System Arising in Oil Recovery}

\vspace{.2in}
\noindent

Consider
\begin{equation}
\phi(u_1,u_2) = \frac{a(u_1, \frac{u_2}{u_1})}{u_1} ,
\end{equation}
where $u_1 \in [0,1]$, $u_2 \in [0,u_1]$, are the concentration of water
and the overall concentration of a polymer at any $x$ and $t$,
respectively, and $a(u_1,c)$ is a smooth function such that $a(u_1,c)$
increases from zero to one with one inflection point where $c$ is
constant and such that $a(u_1,c)$ decreases with increasing $c$ for
fixed $u_1$ (see \cite{te2}).  The essential feature of the system is
that strictly hyperbolicity fails along a certain curve in state space.

{\sc Temple} \cite{te2} obtained a global weak solution to the system (5.1) 
and (5.31) 
with the Cauchy data $u_0 (x) \in BV$.  One can check that Temple's
solution satisfy (5.8).

\begin{theorem}  Suppose that $BV$ functions $u(x,t)$ and 
$v(x,t)$ are solutions of the Cauchy problem (5.7) satisfying (5.8)
in $\Pi_T$
with $BV$ initial data $u_0(x) \geq c_0 >0$ and $v_0(x)$, respectively.
Then for almost all $t \in (0,T)$,
$$ \left \{\begin{array}{l}
\int_{|x| \leq N} | \theta (u(x,t))
- \theta(v(\bar{T}_t^{-1}T_t(x),t))|dx 
\leq C \int_{|x| \leq K_0(N + Lt)} |\theta (u_0
(x)) - \theta (v_0(\bar{T}_0^{-1}T_0(x)))|dx, \\
\int_{|x| \leq N} |\theta(u(x,t)) 
- \theta(v(\bar{T}_t^{-1}\bar{T}_0T_0^{-1}T_t(x),t))|dx
\leq C \int_{|x| \leq K_0(N + Lt)} |\theta(u_0(x)) - \theta(v_0(x))| dx,
\end{array} \right.
$$
and
$$ \left \{\begin{array}{l}
\int_{|x| \leq N} | r(u(x,t))
- r(v(\bar{T}_t^{-1}T_t(x),t)) |dx 
\leq
\int_{|x| \leq K_0(N + Lt)} | r(u_0(x)) - r(v_0 (\bar{T}_0^{-1}T_0(x)))|dx, \\
\int_{|x| \leq N} |(\bar{T}_t^{-1}T_t -I)(x)|dx  \\
\mbox{      } \le C\int_{|x| \le K_0(N + \bar{L}t)}
(|r(u_0(x)) - r(v_0(x))|+ |r(u_0(x)) - r(v_0(\bar{T}_0^{-1}T_0(x)))|)dx,
\end{array} \right.
$$
provided that (5.9)-(5.10) and 
$\theta (u_0(\bar{T}_0^{-1}(x))) = \theta (v_0(T_0^{-1}(x)))$ 
a.e. also hold, 
where $T(x,t) = (T_t(x), t)$ and $\bar{T}(x,t)= (\bar{T}_t(x),t)$ 
are the quasidecoupling transformations corresponding to the vector fields
$(r(u(x,t)), (r\phi)(u(x,t)))$ and $(r(v(x,t)), (r\phi)(v(x,t)))$.
Furthermore,
the $BV$ generalized solution of the Cauchy problem (5.7) satisfying 
(5.8)-(5.10) is unique in $\Pi_T$.
\end{theorem}

In fact, for this system, the condition (5.10) is equivalent to the condition
that the left state and the right state of any  contact discontinuity lie
in the same side of the degenerate curve $\{(u, v): \phi_r = 0\}$,
which is used to ensure the uniqueness of  solutions for the Riemann
problem as in \cite{is} and \cite{te2}.
It would be interesting to study the regularity of Temple's solution
$u(x,t)$ of the Cauchy problem (5.7) and (5.31) with the Cauchy data
$u_0(x) \in BV$, especially with $BV$ regularity. 
In this connection we refer the work of {\sc Tveito and Winther} \cite{tw} for
a stability result within the solution space
$ B^0 = \{ (u_1, u_2): (u_1, (\frac{u_2}{u_1})_x) \in BV, 
 \frac{u_2}{u_1} \in Lip \cap BV\}$.



\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{lemma}{0}
\setcounter{corollary}{0}
\section{The System of Electromagnetic Plane Waves}

In this section we present an example of how to reduce important
problems for large systems with linearly degenerate fields into the
problems for corresponding smaller systems by using the quasidecoupling
method.
We are concerned with the Maxwell equations 
\begin{equation}
\left\{
\begin{array}{l}
B_t + \mbox{curl } E = 0 , \\
D_t - \mbox{curl } H = 0 ,
\end{array} \right.
\end{equation}
where $B$, $D$, $E$, and $H$ are the magnetic induction, the electric
induction, the electric intensity, and the magnetic intensity in
$\R^3$, respectively.
The constitutive laws can be of the following form in the
electromagnetic field (see {\sc Coleman and Dill} \cite{cod}):
\begin{equation}
E = \frac{\partial W}{\partial D} , \quad H = \frac{\partial W}
{\partial B} .
\end{equation}
Here $W(B,D)$ is a convex electromagnetic energy density, i.e.,
$\nabla ^2 W \geq 0$, satisfying
\begin{equation}
W_t + \rm {div} (E \times H) = 0,
\end{equation}
for smooth electromagnetic fields.

As described in \cite{s3}, the system is translationally
invariant and admits plane waves depending only on $t$ and one
space variable $x = x_3$.  Then the components $B_3$ and
$D_3$ can be constant, and (6.1) reduces to four equations:
\begin{equation}
\left\{ \begin{array}{l}
\frac{\partial u_1}{\partial t} + \frac{\partial}{\partial x}
(\phi u_4) = 0 , \\
\frac{\partial u_2}{\partial t} - \frac{\partial}{\partial x}
(\phi u_3 ) = 0 , \\
\frac{\partial u_3}{\partial t} - \frac{\partial}{\partial x}
(\phi u_2) = 0 , \\
\frac{\partial u_4}{\partial t} + \frac{\partial}{\partial x}
(\phi u_1) = 0,
\end{array} \right.
\end{equation}
with the axisymmetry assumption
$$
W(u) = W(r), \qquad  r = \| u \| ,
$$
the notation $(u_1,u_2,u_3,u_4) = (D_1,D_2,B_1,B_2)$, and
\begin{equation}
\phi (r) = \frac{W'(r)}{r} .
\end{equation}
The system (6.4) is nonstrictly hyperbolic because two of four
eigenvalues coincide:  $\lambda _1 = \lambda _2 = \phi (r)$.
The eigenvalues $\lambda _1 = \lambda _2$ are linearly degenerate,
and the other two eigenvalues $\lambda_3,
\lambda_4$ are genuinely nonlinear.

We shall first focus on  how to reduce the
uniqueness problem for the system (6.4) to the uniqueness problem
for a $2 \times 2$ system with the aid of the quasidecoupling method.

Introduce a nonlinear transformation $u \to (\rho, \sigma, \alpha,
\theta ) \in \R_+^2 \times [0,2 \pi ]^2$:
\begin{equation}
\left\{ \begin{array}{l}
u_1 = \frac{1}{2} (\rho \cos \alpha + \sigma \cos \theta ), \\
u_2 = \frac{1}{2} (\rho \cos \alpha - \sigma \cos \theta ), \\
u_3 = \frac{1}{2} (\rho \sin \alpha + \sigma \sin \theta ), \\
u_4 = \frac{1}{2} (\rho \sin \alpha - \sigma \sin \theta ).
\end{array} \right.
\end{equation}
Then one can verify (see \cite{s3}) that there exists an $L^\infty$
global solution $u(x,t)$, $(\rho(u), \sigma (u)) \geq 0$, $\|u\|_
{L^\infty} \leq C( \|u_0 \|_{L^\infty} )$, of the Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
(6.4) , \\
u|_{t=0} = u_0 (x) ,
\end{array} \right.
\end{equation}
satisfying, for any continuous functions $a(\alpha )$ and $b(\theta)$,
\begin{equation}
\left\{ \begin{array}{l}
(\rho a (\alpha ))_t + (\rho \phi a(\alpha ))_x = 0 , \\
(\sigma b (\theta ))_t - (\sigma \phi b(\theta ))_x = 0 ,
\end{array} \right.
\end{equation}
and
\begin{equation}
2W_t + ((\rho^2 - \sigma ^2 ) \phi^2 )_x \leq 0,
\end{equation}
in the generalized sense of distribution under certain physical
conditions on $\phi$ or $W$:
\begin{equation}
W'(r) > 0, \quad \left( \frac{W'(r)}{r} \right) ' > 0 , \quad W'''(r) > 0, 
\quad \left( \frac{rW''(r)}{W'(r)}
\right) ' > 0, \qquad \forall r> 0,
\end{equation}
where $r = \sqrt { \rho^2 + \sigma^2}$.

Using Lemma 3.1 and Corollary 3.3, we can immediately conclude the following
stability behavior of the solutions.

\begin{theorem}(Stability).  Suppose that $u(x,t)$ and $v(x,t)$ are 
$L^\infty$ weak
solutions satisfying (6.8) of the Cauchy problem (6.7) with
initial data $u_0(x)$ and $v_0(x)$ in $\Pi_T$, respectively, and
\begin{equation}
c_0 =\min(\inf_{\R^1} \rho (u_0(x)), \inf_{\R^1}\sigma (u_0(x)),
\inf_{\R^1} \rho (v_0(x)), \inf_{\R^1}\sigma (v_0(x)))>0.
\end{equation}
Then for almost all $t \in (0,T)$ and any positive constant $m$,
\begin{equation}
\left\{ \begin{array}{l}
\int_{|x| \leq N} |\alpha (u(x,t)) 
- \alpha (v(\bar{T}_{t1}^{-1}\bar{T}_{01}T_{01}^{-1}T_{t1}(x),t))|^m 
 \rho (u(x,t))
\rho (v(\bar{T}_{t1}^{-1}\bar{T}_{01}T_{01}^{-1}T_{t1}(x),t))dx \\
\mbox{      } \leq  \|\rho\|^2_{L^\infty} \int_{|x| \leq K_0(N + Lt)} 
|\alpha (u_0(x)) - \alpha (v_0(x))|^mdx , \\
\int_{|x| \leq N} |\theta (u(x,t))
-\theta (v(\bar{T}_{t2}^{-1}\bar{T}_{02}T_{02}^{-1}T_{t2}(x),t))|^m 
 \sigma (u(x,t))
\sigma(v(\bar{T}_{t2}^{-1}\bar{T}_{02}T_{02}^{-1}T_{t2}(x),t))dx \\
 \mbox{      }\leq \|\sigma\|^2_{L^\infty}
\int_{|x| \leq K_0(N+Lt)} |\theta(u_0(x)) - \theta (v_0(x))|^mdx ,
\end{array} \right.
\end{equation}
and
\begin{equation}
\left\{\begin{array}{l}
\int_{|x| \leq N} 
|\alpha (u(T_{t1}^{-1}T_{01}\bar{T}_{01}^{-1}\bar{T}_{t1}(x),t))
- \alpha (v(x,t))|^m
\rho(u(T_{t1}^{-1}T_{01}\bar{T}_{01}^{-1}\bar{T}_{t1}(x),t))\rho(v(x,t))dx \\ 
\leq \|\rho\|^2_{L^\infty} \int_{|x| \leq K_0(N + Lt)}
|\alpha(u_0(x)) - \alpha (v_0(x))|^mdx, \\
|\int_{|x| \leq N}|
\theta(u(T_{t2}^{-1}T_{02}\bar{T}_{02}^{-1}\bar{T}_{t2}(x),t)) 
- \theta(v(x,t))|^m
\sigma (u(T_{t2}^{-1}T_{02}\bar{T}_{02}^{-1}\bar{T}_{t2}(x),t))dx \\
\leq \|\sigma\|^2_{L^\infty}
\int_{|x| \leq K_0(N + Lt)} | \theta(u_0(x)) - \theta (v_0(x))|^mdx,
\end{array} \right.
\end{equation}
where  $T_1(x,t)=(T_{t1}(x), t), \bar{T}_1(x,t)=(\bar{T}_{t1}(x),t),
T_2(x,t) = (T_{t2}(x),t)$, and $ \bar{T}_2(x,t) = (\bar{T}_{t2}(x),t)$
are the quasidecoupling transformations corresponding
to the vector fields \\
$(\rho(u), (\rho \phi)(u))$, 
$(\rho(v),(\rho \phi)(v))$, $ (\sigma(u), (\sigma \phi)(u))$
and $(\sigma(v), (\sigma \phi)(v))$, respectively, and

$$
\begin{array}{l} 
L = \max (\sup_{\Pi_T} |\phi (u(x,t))| , \sup_{\Pi_T}|\phi (v(x,t))|),\\
K_0 = \max(\sup_{\Pi_T}|\sigma(u)|, \sup_{\Pi_T}|\sigma(v)|, \sup|\rho(u)|,
\sup|\rho(v)|)/c_0.

\end{array}
$$
\end{theorem}

In fact, we can choose in Framework (A)
$$
\left\{ \begin{array}{l}
a_0(w) 
= |\alpha (u)-k|^m, \quad (P_1,\lambda_1P_1 )=(\rho(u), \rho(u)\phi(u)), \\
a_0(z) = |\alpha (v)-l|^m, \quad (P_2,\lambda_2P_2)
= (\rho (v), \rho(v)\phi(v)),
\end{array} \right.
$$
and
$$
\left\{
\begin{array}{l}
a_0(w) 
= |\theta (u)-k|^m, \quad (P_1,\lambda_1P_1 )=(\sigma(u), \sigma(u)\phi(u)), \\
a_0(z) = |\theta (v)-l|^m, \quad (P_2,\lambda_2P_2) 
= (\sigma (v), \sigma(v)\phi(v)),
\end{array} \right.
$$
for any $0 < m < \infty$ and any constants $k$, $l \in (-\infty, \infty)$.
The result (6.12) follows.

Then we have the following theorem.

\begin{theorem}
The $L^\infty$ weak solution of the Cauchy problem (6.7) and (6.11)
satisfying (6.8)-(6.9) is unique if and only if the weak solution
of the Cauchy problem
\begin{equation}
\left\{ \begin{array}{l}
\rho_t + (\rho \phi (r))_x = 0 , \\
\sigma_t - (\sigma \phi (r))_x = 0 ,
\end{array} \right.
\end{equation}
\begin{equation}
(\rho, \sigma )|_{t=0} = (\rho (u_0(x)), \sigma (u_0(x))),
\end{equation}
satisfying (6.9) is unique.
\end{theorem}

\noindent
{\bf Proof.} Suppose that $u(x,t)$ and $v(x,t)$ are two solutions of the
Cauchy problem (6.7) and (6.11) satisfying (6.8)-(6.9).
If the weak solution of the Cauchy problem (6.14)-(6.15)
satisfying (6.9) is unique, then 
$$
\rho(u) = \rho(v),  \qquad  \sigma(u) = \sigma(v),
$$
and both $ ( \alpha(u), \theta(u))$ and $(\alpha(v), \theta(v))$
satisfy (6.12). Corresponding quasidecoupling transformations
$(T_1(x,t); T_2(x,t)) = ((p(x,t),t); (q(x,t),t))
= ((p(x,t; u),t); (q(x,t; u),t))$ 
and $(\bar{T}_1(x,t); \bar{T}_2(x,t))=( (\bar{p}(x,t),t); ( \bar{q}(x,t),t))
=((p(x,t; v),t) ; (q(x,t; v),t))$ satisfy

$$ \left \{\begin{array}{l}

p_t + \phi(x, t)p_x = 0, \\
\bar{p}_t + \phi(x, t)\bar{p}_x = 0, \\
p|_{t=0} = \bar{p}|_{t=0},
\end{array} \right.
$$
and

$$ \left \{\begin{array}{l}
q_t + \phi(x, t)q_x = 0, \\
\bar{q}_t + \phi(x, t)\bar{q}_x = 0, \\
q|_{t=0} = \bar{q}|_{t=0},
\end{array} \right.
$$
where $\phi(x,t) = \phi(r(x,t)), 
 r = \sqrt{\rho(u)^2 + \sigma(u)^2} = \sqrt{\rho(v)^2 + \sigma(v)^2}$.

Notice that all $ p, \bar{p}, q$, and $\bar{q}$ are Lipschitz functions.
We immediately conclude that 
$$
p(x,t) = \bar{p}(x,t), \qquad q(x,t) = \bar{q}(x,t), \qquad \mbox{a.e.}
$$
Then  we complete the proof by using Theorem 6.1.



Thus we can transfer the uniqueness problem of Cauchy problem
(6.7) to the uniqueness problem of the Cauchy problem
(6.14)-(6.15) for the $2 \times 2$ system (6.14) with the
aid of the quasidecoupling method.

Let $PL = PL\{\Pi_T\}$ be the class of piecewise Lipschitz functions;
that is, for each $t$ in $[0,T)$ there exists a set of isolated
points $x_j = x_j (t)$ such that the restriction $u(\cdot,t)$ of
$u$ to each interval $(x_j,x_{j+1})$ is a Lipschitz
function of $x$; the dependence of the Lipschitz constant
on the interval $(x_j, x_{j+1})$ as well as the dependence of the
partition points $x_j$ on $t$ is arbitrary.
Note that the system (6.14) is genuinely nonlinear in the
sense of {\sc Lax} \cite{la2}.  
Then combining Theorem 6.2 with DiPerna's theorem
\cite{di1}, we can conclude the following corollary.

\begin{corollary}
Suppose that $u(x,t)$ and $v(x,t)$ are $L^\infty$ weak solutions of
the Cauchy problem (6.7) and (6.11) satisfying (6.8)-(6.9) in
$\Pi_T$ such that
$$
\begin{array}{l}
(\alpha(u),\theta(u),\rho(u),\sigma(u)) \in (L^\infty )^2 \times (BV)
^2 \{ \Pi_T \} , \\
(\alpha (v), \theta (v), \rho (v), \sigma (v)) \in (L^\infty )^2 \times
(PL)^2 \{ \Pi_T \} ,
\end{array}
$$
and
$$
u(x,0) = v(x,0), \qquad \mbox{ for almost all } x.
$$
Then
$$
u(x,t) = v(x,t),
$$
for almost all $(x,t)$ in $\prod_T$.
\end{corollary}

Finally, we conclude the following theorem
by using Lemma 3.1 and Theorem 3.5, and following the proof
of Theorem 4.2 for systems with fully contact fields.  
\begin{theorem}
The initial oscillations will propagate along the linearly
degenerate fields for the system (6.4).
\end{theorem}

A direct corollary follows.

\begin{corollary}
For the Cauchy problem (6.7), the solution operator $S_t: L^\infty
(\R^4) \to L_{\rm {loc}}^1 (\R^4)$ with $S_t(u^0) = u(\cdot ,t)$ is
not compact.
\end{corollary}

We omit the details of the proof of Theorem 6.3.




\section{Remarks}

The quasidecoupling method and ideas developed in the previous sections can 
be applied to solving some nonlinear problems for more general systems.  

{\bf a. Propagation and Cancellation of Initial Oscillations}. 
Using the same arguments as the proof of Theorem 4.2 with the aid of 
Framework (B), we can extend the results virtually without modification to
strictly hyperbolic systems (1.1) endowed with (1.6) and (2.5).
If there exists a linearly degenerate characteristic field for such systems,
then the  entropy-entropy fluxes (2.17) of the same form can be obtained,
which corresponds to this field.
In fact, Framework (B) is useful to solve oscillation problems for
systems of conservation laws endowed with the entropy-entropy fluxes of
the form (2.17).

{\bf b. Generalized Characteristics and the Structure of Solutions}.
Theorem 3.1 and Framework (A)-(B) in Section 3 should be 
applicable to clarify the
generalized characteristics  and 
the structure of solutions to systems
of conservation laws endowed with the entropy-entropy fluxes of the
form (2.17).  In particular,  they can provide insight into how
the interaction between large linear waves and large nonlinear waves is, 
and provide
a useful approach to understand
large time asymptotic behavior of solutions for systems of conservation laws 
with such mixing linear characteristic fields and nonlinear 
characteristic fields.
In this connection we refer {\sc Glimm and Lax} \cite{gll}, 
{\sc Dafermos} \cite{da4} and the references cited
therein on the generalized
characteristics and the structure of solutions for conservation laws. 


{\bf c. Uniqueness  and Existence of Generalized Solutions}. 
From discussions in the previous sections,
it should be evident that Framework (A) and ideas developed here should
be useful to solve the uniqueness  and existence problems for more general 
systems by combining them with other techniques. 

{\bf d. Equivalence of the Euler and Lagrangian Equations of Gas Dynamics for
Entropy Solutions}. {\sc Wagner} \cite{w} verified  the equivalent of existence
of entropy discontinuous solutions between the Euler equations and 
the Lagrange equations
of gas dynamics. As a corollary of Theorem 3.3, we can conclude the
equivalent of \underline{uniqueness} of entropy discontinuous solutions between 
the Euler equations
and the Lagrange equations of gas dynamics because of the uniqueness
of the transformation between the Eulerian coordinates and the Lagrangian
coordinates with the same initial data from Theorem 3.3.








\vspace{.2in}
{\bf Acknowledgments}. 
The author would like to thank Professors  C. Dafermos,  L. C. Evans,
T.-P. Liu, D. Serre and B. Temple for valuable discussions and remarks. 
He is grateful to Professors P. D. Lax, J. Glimm and C. Morawetz for their
encouragement.
This research was supported in part by NSF Grant
DMS-8505550 from MSRI at Berkeley, by a NSF Grant from IMA at Minnesota,
and by the Argonne-University of
Chicago Fellowship funded by the Applied Mathematical Sciences subprogram
of the Office of Energy Research, U.S. Department of Energy, under Contract
W-31-109-Eng-38.



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