NS  Vol.6 No.7 , April 2014
Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates
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).



Cite this paper
Caicedo, J. , Klingenberg, C. , Lu, Y. and Rendon, L. (2014) Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates. Natural Science, 6, 477-486. doi: 10.4236/ns.2014.67046.
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.

[26]   Kamke, E. (1959) Differential gleichungen, Losungsmethoden und Losungen: 1. Gewohnliche Differentialgleichungen. 6th Edition, Akademische Verlagsanstalt, Leipzig.

 
 
Top