A New Application of the Flux Approximation Method on Hyperbolic Conservation Systems ()
1. Introduction
It is well known that no classical solution exists for the following initial value problem
(1.1)
with bounded measurable initial data
(1.2)
where 
is the unknown vector function standing for the density of physical quantities and 
is a given vector function denoting the conservative term. These equations are commonly called conservation laws.
Since, in general, the discontinuity or the shock waves will appear in the solution to the Cauchy problem (1.1)- (1.2), there are two standard methods to obtain a weak solution or a generalized solution 
for given hyperbolic conservation laws. One is to construct a sequence of smooth functions to approximate
. For example, to add a small parabolic perturbation term to the right-hand side of (1.1):
(1.3)
where 
is a constant. For each fixed
, we have a classical solution 
of (1.3)-(1.2), then we try to prove that the limit 
of 
as 
goes to zero is the solution of (1.1)-(1.2), where the compactness could be obtained by the compensated compactness arguments [1,2] when the functions have only the uniform boundedness in a suitable Banach space or the technique given in [3] when the functions are of total bounded variation estimates; another is the finite difference method [4]. We construct a sequence of simple functions by choosing a suitable difference scheme which is based on the given hyperbolic conservation laws and then prove the compactness of the sequence of functions. Normally, in the second method, we know that the sequence of simple functions is of total bounded variation estimates.
However, the third front tracking method [5], here we just call it the flux approximation method, is also used in many different cases.
In [6], Dafermos first introduced this method to the scalar conservation law
(1.4)
where 
is a scalar function, and 
is a locally Lipschitz continuous function. He constructed a sequence of piecewise linear functions 
and a sequence of step functions 
to approximate 
and the initial date 
respectively. Let the solutions of the following Cauchy problem be
:
(1.5)
with the initial data
(1.6)
For each fixed
, since the simplicity of the flux function 
and the initial date
, the sequence of solutions 
can be easily obtained first. Then by using the standard compactness argument by Oleinik, the convergence of 
can be proved as 
goes to zero.
Later, the above idea was used to study the existence of Riemann solutions for some special systems of two equations. For example, in [7], the author first studied the Riemann solution for the Cauchy problem of the following system
(1.7)
with initial data
(1.8)
The more details about the Front Tracking method for systems of hyperbolic conservation laws can be found in the books [5,8] and the references cited therein.
In [9], Keyfitz introduced a different way to approximate the nonlinear flux function
. Consider the Cauchy problem
(1.9)
with the Riemann initial data, where 
since the system is hyperbolic or 
as required in
[9]. For each fixed
, System (1.9) is strictly hyperbolic and Riemann solution 
could be easily obtained. Then a Riemann solution of system (1.7) follows since it is the limit of 
as 
goes to zero.
The method of flux approximation was applied by the first author of this paper to study the existence of weak solutions [10,11], the existence of global Lipschitz solutions [12], 
compactness for weak entropy-entropy flux pairs of the isentropic gas dynamics [11], 
estimate for isentropic gas dynamics with a superline source [13], the global 
solutions of Aw-Rascle traffic flow model [14] (or the nonsymmetric systems of Keyfitz-Kranzer type) with negative adiabatic exponent and so on, which we shall introduce below. A new application of this method related to the LeRoux system is introduced in Theorem 1, Section 2.
2. A New Application of Flux Approximation Method
In this section, we introduce a new application of the flux approximation method. We found two hyperbolic conservation systems of Temple’s type [15], and the global weak solution of each system could be obtained by the limit of the linear combination of two systems.
Consider the hyperbolic systems
(2.1)
and
(2.2)
By simple calculations, two eigenvalues of system (2.1) are
(2.3)
where
, with corresponding right eigenvectors
(2.4)
and
(2.5)
The Riemann invariants of (2.1) are
(2.6)
Thus, the curves 
are straight lines on the 
-plane.
Similarly, two eigenvalues of system (2.2) are
(2.7)
with the corresponding right eigenvectors (2.4) and
(2.8)
The Riemann invariants of (2.2) are also given by (2.6)
Therefore if we consider the bounded solution in the region:
, it follows from (2.5) (or (2.8)) that both characteristic fields of system (2.1) (or system (2.2)) are genuinely nonlinear in the sense of Lax [16].
Now we prove that both systems (2.1) and (2.2) have the same entropies.
Let
. Then for smooth solutions, (2.2) is equivalent to the following system:
(2.9)
Considering the entropy-entropy flux pair 
of system (2.2) as functions of variables
, we have
(2.10)
Eliminating the 
from (2.10), we have
(2.11)
Similarly, for smooth solutions, (2.1) is equivalent to the following system:
(2.12)
For the entropy-entropy flux pair 
of system (2.1), we have
(2.13)
Eliminating the 
from (2.13), we have also the same entropy Equation (2.11).
Using the compensated compactness arguments, we may easily obtain the global existence of weak solutions for the Cauchy problem of system (2.2) in the upper 
-plane 
or system (2.1) in the region 
for a suitable constant
, which could be guaranteed since the curves 
are straight lines, where 
are four suitable constants. The details could be found in Chapter 7 of [17] or the original paper by Diperna [18].
Now we consider the linear combination of systems (2.1) and (2.2):
(2.14)
where 
are two positive flux approximation perturbations.
The eigenvalues of system (2.14) are solutions of the following characteristic equation:
(2.15)
Two roots of Equation (2.15) are
(2.16)
with the corresponding right eigenvectors (2.4) and the Riemann invariants (2.6). Moreover,
(2.17)
Therefore both characteristic fields of system (2.14) are genuinely nonlinear in the region:
.
Now we consider the Cauchy problem of system (2.14) with initial data
(2.18)
and have the main results in the following theorem
Theorem 1. Suppose the initial data 
be bounded measurable and 
for a suitable constant
. Then for any fixed
, the global weak solution 
of the Cauchy problem (2.14) and (2.18) exists. Moreover, for fixed 
(or
), there exists a subsequence 
(or
) of
, which piontwisely converges, as 
(or
) goes to zero, to the solution of the Cauchy problem of system (2.1) (or (2.2)) with the initial data (2.18).
The proof of Theorem 1: The proof of Theorem 1 can be obtained by the standard vanishing artificial viscosity method coupled with the compensated compactness argument and the famous framework of DiPerna [18] for strictly hyperbolic, genuinely nonlinear systems of two equations. We add the viscosity terms to the right hand side of (2.14) and consider the following parabolic system
(2.19)
with the initial data (2.18). According to the calculations given in (2.3) and (2.7), we know that the two eigenvalues of system (2.14) are
(2.20)
with the corresponding right eigenvectors (2.4) and the Riemann invariants (2.6).
For any constant
, the curves 
or 
is a straight line on the 
-plane, then we may choose suitable constants 
such that 
forms a bounded invariant region. Moreover, in this region, 
for a suitable constant
. Since system (2.14) is strictly hyperbolic and genuinely nonlinear, and the viscosity solutions 
of system (2.19) are uniformly bounded, then the famous compactness framework of DiPerna [18] gives us the convergence of
(2.21)
where the limit 
is a weak solution of system (2.14) or satisfies (2.14) in the sense of distributions. For fixed 
(or
), and for the generalized functions
, we may rewrite system (2.14) as
(2.22)
Since the left hand side of (2.22) or system (2.1) is also strictly hyperbolic and genuinely nonlinear, and the functions 
are uniformly bounded, independent of
, so the DiPerna’s result [18] reduces the following convergence
(2.23)
where the limit 
is a weak solution of system (2.1) or satisfies (2.1) in the sense of distributions, which ends the proof of Theorem 1.
3. Acknowledgements
This work was partially supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. LY12A01030 and Grant No. LZ13A010002) and the National Natural Science Foundation of China (Grant No. 11271105).
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