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In this paper, we construct a sequence of hyperbolic systems (13) to approximate the general system of one-dimensional nonlinear elasticity in Lagrangian coordinates (2). For each fixed approximation parameter , we establish the existence of entropy solutions for the Cauchy problem (13) with bounded initial data (23).

Three most classical, hyperbolic systems of two equations in one-dimension are the system of isentropic gas dynamics in Eulerian coordinates

where

where

and the system of compressible fluid flow

To obtain the global existence of weak solutions for nonstrictly hyperbolic systems (two eigenvalues are real, but coincide at some points or lines), the compensated compactness theory (cf. [

For the polytropic gas

Under the strictly hyperbolic condition

Without the strictly hyperbolic restriction, a preliminary existence result of the nonlinear wave Equation (3) was proved in [

Using the Glimm’s scheme method (cf. [

Since the solutions for the case of

If all smooth entropy-entropy flux pairs satisfy the

In [

to approximate system (1), where

The most interesting point of this kind approximation is that both systems (5) and (1) have the same entropies (or the same entropy equation). In [

Let the entropy-entropy flux pairs of systems (1) and (5) be

for any fixed

Paying attention to the approximation function (6), we know that

are the entropy-entropy flux pairs of system

or system

respectively.

If we could prove from the arbitrary of

and

where

Between systems (2) and (4), we have the following approximation

which has also the same entropy equation like system (2). If we could prove (11) and (12) from (7), then similarly we could prove the equivalence of systems (2) and (4). Moreover, we have much more information from system (13) to prove the existence of solutions for system (2) or (4).

Systems (13) and (2) have many common basic behaviors, such as the nonstrict hyperbolicity, the same entropy equation, same Riemann invariants and so on.

By simple calculations, two eigenvalues of system (13) are

with corresponding right eigenvectors

and Riemann invariants

Moreover

and

Any entropy-entropy flux pair

Eliminating the

Therefore systems (13) and (2) have the same entropies. From these calculations, we know that system (13) is strictly hyperbolic in the domain

However, from (17) and (18), for each fixed

because the region

is an invariant region, where

because the region

is an invariant region.

In this paper, for fixed

In a further coming paper, we will study the relation between the functions equations (11) and (12), and the convergence of approximated solutions of system (13) as

Theorem 1 Suppose the initial data

where

with the bounded measurable initial data (23) has a global bounded entropy solution.

Note 1. The idea to use the flux perturbation coupled with the vanishing viscosity was well applied by the author in [

Note 2. It is well known that system (2) is equivalent to system (1), but (1) is different from system (4) although the latter can be derived by substituting the first equation in (1) into the second. However, (4) can be considered as the approximation of (2). In fact, let

for some nonlinear function

Note 3. For any fixed

In the next section, we will use the compensated compactness method coupled with the construction of Lax entropies [

In this section, we prove Theorem 1.

Consider the Cauchy problem for the related parabolic system

with the initial data (23).

We multiply (25) by

and

Then the assumptions on

and

if

and

if

If we consider (28) and (29) (or (30) and (31)) as inequalities about the variables

or

is respectively an invariant region. Thus we obtain the estimates given in (21) or (22) respectively.

It is easy to check that system (13) has a strictly convex entropy when

We multiply (4.1) by

in

is bounded in

when

for any fixed

Now we multiply (4.1) by

where

Then for smooth entropy-entropy flux pairs

where

To finish the proof of Theorem 1, it is enough to prove that Young measures given in (37) are Dirac measures.

For applying for the framework given by DiPerna in [

where

uniformly for

In fact, substituting entropies

Let

and

Then

The existence of

Lemma 2 Let

and functions

for some positive functions

for some positive constants

If

with

where

Furthermore, we can use Lemma 2 again to obtain the bound of

By the second equation in (19), an entropy flux

where

if

In a similar way, we can obtain estimates on another three pairs of entropy-entropy flux of Lax type. Hence, Theorem 1 is proved when we use these entropy-entropy flux pairs in (38)-(41) together with the theory of compensated compactness coupled with DiPerna’s framework [

In this paper we have looked at the general system of one-dimensional nonlinear elasticity in Lagrangian coordinates (2).

We construct a hyperbolic approximations to this which are parameterized by

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).