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August  2015, 35(8): 3683-3706. doi: 10.3934/dcds.2015.35.3683

## Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type

 1 Clermont Université, Université Blaise Pascal, 63000 Clermont-Ferrand, France 2 Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

Received  February 2014 Revised  December 2014 Published  February 2015

We study linearly degenerate hyperbolic systems of rich type in one space dimension. It is showed that such a system admits exact traveling wave solutions after a finite time, provided that the initial data are Riemann type outside a space interval. We prove the convergence of entropy solutions toward traveling waves in the $L^1$ norm as the time goes to infinity. The traveling waves are determined explicitly in terms of the initial data and the system. We also obtain the stability of entropy solutions in $L^1$. Applications concern physical models such as the generalized extremal surface equations, the Born-Infeld system and augmented Born-Infeld system.
Citation: Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
##### References:
 [1] S. Bianchini, Stability of $L^{\infty}$ solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. Anal., 33 (2001), 959-981. doi: 10.1137/S0036141000377900. [2] F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. [3] Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics, Proceedings of the International Congress of Mathematicians, 3, Higher Education Press, Beijing, (2002), 761-772. [4] Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172 (2004), 65-91. doi: 10.1007/s00205-003-0291-4. [5] A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem, Oxford Lecture Series in Math. and its Applications, 20. Oxford University Press, Oxford, 2000. [6] H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1993. [7] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44 (2003), 6132-6139. doi: 10.1063/1.1621057. [8] G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal., 121 (1992), 131-185. doi: 10.1007/BF00375416. [9] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 695-715. doi: 10.1002/cpa.3160180408. [10] J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws, Amer. Math. Soc., Providence, R.I. 1970. [11] F. John, Nonlinear Waves Equations, Formation of Singularities, Pitcher Lectures in Math. Sciences, Lehigh University, Amer. Math. Soc., 1990. [12] S. N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sbornik (N.S.), 81 (1970), 228-255. [13] D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $\mathbbR^{2+n}$, J. Math. Phys., 47 (2006), 013503, 16 pages. doi: 10.1063/1.2158435. [14] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Part. Diff. Eqs., 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192. [15] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [16] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Appl. Math., 32, Wiely/Masson, 1994. [17] T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568. doi: 10.1016/j.matpur.2009.01.008. [18] T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Equations, 19 (1994), 1263-1317. doi: 10.1080/03605309408821055. [19] H. Lindblad, A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102. doi: 10.1090/S0002-9939-03-07246-0. [20] J. L. Liu and Y. Zhou, Asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500. doi: 10.1002/mma.797. [21] T. P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796. doi: 10.1002/cpa.3160300605. [22] Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems, Appl. Math. Letters, 11 (1998), 75-78. doi: 10.1016/S0893-9659(98)00083-4. [23] Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications, Nonlinearity, 20 (2007), 1927-1953. doi: 10.1088/0951-7715/20/8/007. [24] Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, J. Math. Phys., 52 (2011), 053702 (23 pages). doi: 10.1063/1.3591133. [25] D. Serre, Richness and the classification of quasilinear hyperbolic systems, in IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333. doi: 10.1007/978-1-4613-9121-0_24. [26] D. Serre, Systèmes de Lois de Conservation I-II, Diderot, Paris, 1996. [27] D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations, Arch. Rat. Mech. Anal., 172 (2004), 309-331. doi: 10.1007/s00205-003-0303-4. [28] B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation, Mém. Soc. Math. France, 56 (1994), 125pp. [29] S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type, Dolk. Akad. Nauk SSSR, 282 (1985), 534-537. [30] S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419. [31] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqs., 68 (1987), 118-136. doi: 10.1016/0022-0396(87)90188-4.

show all references

##### References:
 [1] S. Bianchini, Stability of $L^{\infty}$ solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. Anal., 33 (2001), 959-981. doi: 10.1137/S0036141000377900. [2] F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. [3] Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics, Proceedings of the International Congress of Mathematicians, 3, Higher Education Press, Beijing, (2002), 761-772. [4] Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172 (2004), 65-91. doi: 10.1007/s00205-003-0291-4. [5] A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem, Oxford Lecture Series in Math. and its Applications, 20. Oxford University Press, Oxford, 2000. [6] H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1993. [7] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44 (2003), 6132-6139. doi: 10.1063/1.1621057. [8] G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal., 121 (1992), 131-185. doi: 10.1007/BF00375416. [9] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 695-715. doi: 10.1002/cpa.3160180408. [10] J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws, Amer. Math. Soc., Providence, R.I. 1970. [11] F. John, Nonlinear Waves Equations, Formation of Singularities, Pitcher Lectures in Math. Sciences, Lehigh University, Amer. Math. Soc., 1990. [12] S. N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sbornik (N.S.), 81 (1970), 228-255. [13] D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $\mathbbR^{2+n}$, J. Math. Phys., 47 (2006), 013503, 16 pages. doi: 10.1063/1.2158435. [14] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Part. Diff. Eqs., 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192. [15] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [16] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Appl. Math., 32, Wiely/Masson, 1994. [17] T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568. doi: 10.1016/j.matpur.2009.01.008. [18] T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Equations, 19 (1994), 1263-1317. doi: 10.1080/03605309408821055. [19] H. Lindblad, A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102. doi: 10.1090/S0002-9939-03-07246-0. [20] J. L. Liu and Y. Zhou, Asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500. doi: 10.1002/mma.797. [21] T. P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796. doi: 10.1002/cpa.3160300605. [22] Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems, Appl. Math. Letters, 11 (1998), 75-78. doi: 10.1016/S0893-9659(98)00083-4. [23] Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications, Nonlinearity, 20 (2007), 1927-1953. doi: 10.1088/0951-7715/20/8/007. [24] Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, J. Math. Phys., 52 (2011), 053702 (23 pages). doi: 10.1063/1.3591133. [25] D. Serre, Richness and the classification of quasilinear hyperbolic systems, in IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333. doi: 10.1007/978-1-4613-9121-0_24. [26] D. Serre, Systèmes de Lois de Conservation I-II, Diderot, Paris, 1996. [27] D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations, Arch. Rat. Mech. Anal., 172 (2004), 309-331. doi: 10.1007/s00205-003-0303-4. [28] B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation, Mém. Soc. Math. France, 56 (1994), 125pp. [29] S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type, Dolk. Akad. Nauk SSSR, 282 (1985), 534-537. [30] S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419. [31] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqs., 68 (1987), 118-136. doi: 10.1016/0022-0396(87)90188-4.
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