﻿ Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates

Natural Science
Vol.6 No.7(2014), Article ID:45340,10 pages DOI:10.4236/ns.2014.67046

Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates

Jose Francisco Caicedo1, C. Klingenberg2, Yunguang Lu3*, Leonardo Rendon1

1Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

2Institute of Mathematics, University of Würzburg, Würzburg, Germany

3Department of Mathematics, Hangzhou Normal University, Hangzhou, China

Email: *ylu2005@ustc.edu.cn

Received 14 December 2013; revised 14 January 2014; accepted 21 January 2014

ABSTRACT

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).

Keywords:Entropy Solutions, Isentropic Gas Dynamics, Lax Entropy

1. Introduction

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

(1)

where is the density of gas, the velocity and the pressure; the nonlinear hyperbolic system of elasticity

(2)

where denotes the strain, is the stress and the velocity, which describes the balance of mass and linear momentum, and is equivalent to the nonlinear wave equation

(3)

and the system of compressible fluid flow

(4)

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. [1] [2] or the books [3] -[5] ) is still a powerful and unique method until now.

For the polytropic gas where and is an arbitrary positive constant, the Cauchy problem (1) with bounded initial data was completely resolved by many authors (cf. [6] -[11] ). When has the same principal singularity as the -law in the neighborhood of vacuum, a compact framework was first provided in [12] [13] and later, the necessary compactness of weak entropy-entropy flux pairs for general pressure function was completed in [14] .

Under the strictly hyperbolic condition and some linearly degenerate conditions or as, the global existence of weak bounded solutions, or solutions, was obtained by Diperna [15] and Lin [16] , Shearer [17] respectively.

Without the strictly hyperbolic restriction, a preliminary existence result of the nonlinear wave Equation (3) was proved in [18] for the special case under the assumption or.

Using the Glimm’s scheme method (cf. [19] ), Diperna [20] first studied the system (4) in a strictly hyperbolic region. Roughly speaking, for the polytropic case, Diperna’s results cover the case.

Since the solutions for the case of always touch the vacuum, its existence was obtained in [21] by using the compensated compactness method coupled with some basic ideas of the kinetic formulations (cf. [10] [11] ). The existence of the Cauchy problem (7) for more general function was given in [22] under some conditions to ensure the compactness for all smooth entropy-entropy flux pairs.

If all smooth entropy-entropy flux pairs satisfy the compactness, an ideal compactness framework to prove the global existence was provided by Diperna in [15] . For the above three systems (1)-(2) and (4), we can prove the compactness only for half of the entropies (weak or strong entropy).

2. Main New Ideas

In [14] (see also [23] for inhomogeneous system), the author constructed a sequence of regular hyperbolic systems

(5)

to approximate system (1), where in (5) denotes a regular perturbation constant and the perturbation pressure

(6)

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 [14] , the compactness of weak entropy-entropy flux pairs was also proved for general pressure function.

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

respectively. Then by using Murat-Tartar theorem, we have

(7)

for any fixed, where the weak-star limit is denoted by as goes to zero.

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

(8)

are the entropy-entropy flux pairs of system

(9)

or system

(10)

respectively.

If we could prove from the arbitrary of in (7) that

(11)

and

(12)

where denotes the weak-star limit as tend to zero, then we would have more function Equations (12) to reduce the strong convergence of as tend to zero.

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

(13)

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.

3. Main Results

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

(14)

with corresponding right eigenvectors

(15)

and Riemann invariants

(16)

Moreover

(17)

and

(18)

Any entropy-entropy flux pair of system (13) satisfies the additional system

(19)

Eliminating the from (19), we have

(20)

Therefore systems (13) and (2) have the same entropies. From these calculations, we know that system (13) is strictly hyperbolic in the domain or, while it is nonstrictly hyperbolic on the domain since when.

However, from (17) and (18), for each fixed, both characteristic fields of system (13) are genuinely nonlinear in the domain if or in the domain if

. In the first case, we have an a-priori estimate for the solutions of system (13)

(21)

because the region

is an invariant region, where (is given in Theorem 1), and are positive constants depending on the initial date, but being independent of. In the second case, we have the estimate

(22)

because the region

is an invariant region.

In this paper, for fixed, we first establish the existence of entropy solutions for the Cauchy problem (13) with bounded measurable initial data

(23)

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 goes to zero.

Theorem 1 Suppose the initial data be bounded measurable. Let (I):

where is a positive constant, or (II):. Then the Cauchy problem (13)

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 [24] to control the super-line, source terms and to obtain the estimate for the nonhomogeneous system of isentropic gas dynamics.

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 in (13). Then (13) is rewritten to the form

(24)

for some nonlinear function.

Note 3. For any fixed, the invariant region above is bounded, so the vacuum is avoided. However, the limit of, as goes to zero, is the original invariant region of system (2) because could be infinity from the estimates in (21).

In the next section, we will use the compensated compactness method coupled with the construction of Lax entropies [25] to prove Theorem 1.

4. Proof of Theorem 1

In this section, we prove Theorem 1.

Consider the Cauchy problem for the related parabolic system

(25)

with the initial data (23).

We multiply (25) by and, respectively, to obtain

(26)

and

(27)

Then the assumptions on yield

(28)

and

(29)

if; or

(30)

and

(31)

if

If we consider (28) and (29) (or (30) and (31)) as inequalities about the variables and, then we can get the estimates by applying the maximum principle to (28) and (29) (or

by applying the maximum principle to (30) and (31)). Then, using the first equation in (25), we get or depending on the conditions on. Therefore, the region

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 or

(32)

We multiply (4.1) by to obtain the boundedness of

(33)

in. Then it follows that

(34)

is bounded in. Since for some bounded constants

when or, we get the boundedness of

(35)

for any fixed.

Now we multiply (4.1) by, where is any smooth entropy of system (13), to obtain

(36)

where is the entropy-flux corresponding to. Then using the estimate given in (35), we know that the first term in the right-hand side of (36) is compact in, and the second is bounded in

. Thus the term in the left-hand side of (36) is compact in.

Then for smooth entropy-entropy flux pairs of system (13), the following measure equations or the communicate relations are satisfied

(37)

where is the family of positive probability measures with respect to the viscosity solutions of the Cauchy problem (25) and (23).

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 [5] to prove that Young measures are Dirac ones, we construct four families of entropy-entropy flux pairs of Lax’s type in the following special form:

(38)

(39)

(40)

(41)

where are the Riemann invariants of system (13) given by (16). Notice that all the unknown functions are only of a single variable. This special simple construction yields an ordinary differential equation of second order with a singular coefficient before the term of the second order derivative. Then the following necessary estimates for functions are obtained by the use of the singular perturbation theory of ordinary differential equations:

(42)

(43)

uniformly for or, where and is a positive constant independent of.

In fact, substituting entropies into (20), we obtain that

(44)

Let

(45)

and

(46)

Then

(47)

The existence of and its uniform bound on or with respect to can be obtained by the following lemma (cf. [26] ) (also see Lemma 10.2.1 in [15] ):

Lemma 2 Let be the solution of the equation

and functions be continuous on the regions

for some positive functions and. In addition,

for some positive constants and.

If is a solution of the following ordinary differential equation of second order:

with and being arbitrary, then for sufficiently small and , exists for all and satisfies

where

Furthermore, we can use Lemma 2 again to obtain the bound of with respect to if we differentiate Equation (46) with respect to.

By the second equation in (19), an entropy flux corresponding to is provided by

(48)

where

(49)

if or, and both are bounded uniformly on or.

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 [15] .

5. Conclusions

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. They all have the same entropies as the original system. Under suitable assumptions we are able to establish uniform compactness estimates, and then obtain the existence of entropy solutions for the Cauchy problem.

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).

References

1. Murat, F. (1978) Compacité par compensation. Annali della Scuola Normale Superiore di Pisa, 5, 489-507.
2. Tartar, T. (1979) Compensated Compactness and Applications to Partial Differential Equations. In: Knops, R.J., Ed., Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Pitman Press, London.
3. Lu, Y.-G. (2002) Hyperboilc Conservation Laws and the Compensated Compactness Method. Chapman and Hall, CRC Press, New York. http://dx.doi.org/10.1201/9781420035575
4. Perthame, B. (2002) Kinetic Formulations. Oxford University Press, Oxford.
5. Serre, D. (1999) Systems of Conservation Laws. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511612374
6. Chen, G.-Q. (1986) Convergence of the Lax-Friedrichs Scheme for Isentropic Gas Dynamics. Acta Mathematica Scientia, 6, 75-120.
7. Ding, X.-X., Chen, G.-Q. and Luo, P.-Z. (1985) Convergence of the Lax-Friedrichs Schemes for the Isentropic Gas Dynamics I-II. Acta Mathematica Scientia, 5, 415-472.
8. DiPerna, R.J. (1983) Convergence of the Viscosity Method for Isentropic Gas Dynamics. Communications in Mathematical Physics, 91, 1-30. http://dx.doi.org/10.1007/BF01206047
9. Huang, F.-M. and Wang, Z. (2003) Convergence of Viscosity Solutions for Isentropic Gas Dynamics. SIAM Journal on Mathematical Analysis, 34, 595-610. http://dx.doi.org/10.1137/S0036141002405819
10. Lions, P.L., Perthame, B. and Souganidis, P.E. (1996) Existence and Stability of Entropy Solutions for the Hyperbolic Systems of Isentropic Gas Dynamics in Eulerian and Lagrangian Coordinates. Communications on Pure and Applied Mathematics, 49, 599-638. http://dx.doi.org/10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5
11. Lions, P.L., Perthame, B. and Tadmor, E. (1994) Kinetic Formulation of the Isentropic Gas Dynamics and p-System. Communications in Mathematical Physics, 163, 415-431. http://dx.doi.org/10.1007/BF02102014
12. Chen, G.-Q. and LeFloch, P. (2003) Existence Theory for the Isentropic Euler Equations. Archive for Rational Mechanics and Analysis, 166, 81-98. http://dx.doi.org/10.1007/s00205-002-0229-2
13. Chen, G.-Q. and LeFloch, P. (2000) Compressible Euler Equations with General Pressure Law and Related Equations. Archive for Rational Mechanics and Analysis, 153, 221-259. http://dx.doi.org/10.1007/s002050000091
14. Lu, Y.-G. (2007) Some Results on General System of Isentropic Gas Dynamics. Differential Equations, 43, 130-138. http://dx.doi.org/10.1134/S0012266107010132
15. DiPerna, R.J. (1983) Convergence of Approximate Solutions to Conservation Laws. Archive for Rational Mechanics and Analysis, 82, 27-70. http://dx.doi.org/10.1007/BF00251724
16. Lin, P.-X. (1992) Young Measures and an Application of Compensated Compactness to One-Dimensional Nonlinear Elastodynamics. Transactions of the American Mathematical Society, 329, 377-413. http://dx.doi.org/10.1090/S0002-9947-1992-1049615-0
17. Shearer, J. (1994) Global Existence and Compactness in for the Quasilinear Wave Equation. Communications in Partial Differential Equations, 19, 1829-1877.
18. Lu, Y.-G. (2007) Nonlinearly Degenerate Wave Equation. Revista de la Academia Colombiana de Ciencias, 119, 275-283.
19. Glimm, J. (1965) Solutions in the Large for Nonlinear Hyperbolic Systems of Equations. Communications on Pure and Applied Mathematics, 18, 95-105. http://dx.doi.org/10.1002/cpa.3160180408
20. DiPerna, R.J. (1973) Global Solutions to a Class of Nonlinear Hyperbolic Systems of Equations. Communications on Pure and Applied Mathematics, 26, 1-28. http://dx.doi.org/10.1002/cpa.3160260102
21. Lu, Y.-G. (2005) Existence of Global Entropy Solutions to a Nonstrictly Hyperbolic System. Archive for Rational Mechanics and Analysis, 178, 287-299. http://dx.doi.org/10.1007/s00205-005-0379-0
22. Lu, Y.-G. (1994) Convergence of the Viscosity Method for Some Nonlinear Hyperbolic Systems. Proceedings of the Royal Society of Edinburgh, 124A, 341-352.
23. Lu, Y.-G., Peng, Y.-J. and Klingenberg, C. (2010) Existence of Global Solutions to Isentropic Gas Dynamics Equations with a Source Term. Science China, 53, 115-124. http://dx.doi.org/10.1007/s11425-010-0003-0
24. Lu, Y.-G. (2011) Global Existence of Solutions to Resonant System of Isentropic Gas Dynamics. Nonlinear Analysis, Real World Applications, 12, 2802-2810. http://dx.doi.org/10.1016/j.nonrwa.2011.04.005
25. Lax, P.D. (1971) Shock Waves and Entropy. In: Zarantonello, E., Contributions to Nonlinear Functional Analysis, Academia Press, New York, 603-634.

NOTES

*Corresponding author.