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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1137/S0036141000377900.  Google Scholar [2] F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited,, Trans. Amer. Math. Soc., 350 (1998), 2847.  doi: 10.1090/S0002-9947-98-02204-1.  Google Scholar [3] Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics,, Proceedings of the International Congress of Mathematicians, 3 (2002), 761.   Google Scholar [4] Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations,, Arch. Rat. Mech. Anal., 172 (2004), 65.  doi: 10.1007/s00205-003-0291-4.  Google Scholar [5] A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem,, Oxford Lecture Series in Math. and its Applications, (2000).   Google Scholar [6] H. Brézis, Analyse Fonctionnelle,, Masson, (1993).   Google Scholar [7] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations,, J. Math. Phys., 44 (2003), 6132.  doi: 10.1063/1.1621057.  Google Scholar [8] G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 131.  doi: 10.1007/BF00375416.  Google Scholar [9] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 695.  doi: 10.1002/cpa.3160180408.  Google Scholar [10] J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws,, Amer. Math. Soc., (1970).   Google Scholar [11] F. John, Nonlinear Waves Equations, Formation of Singularities,, Pitcher Lectures in Math. Sciences, (1990).   Google Scholar [12] S. N. Kruzkov, First order quasilinear equations in several independent variables,, Mat. Sbornik (N.S.), 81 (1970), 228.   Google Scholar [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).  doi: 10.1063/1.2158435.  Google Scholar [14] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Part. Diff. Eqs., 28 (2003), 1203.  doi: 10.1081/PDE-120021192.  Google Scholar [15] P. D. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar [16] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Appl. Math., 32 (1994).   Google Scholar [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.  doi: 10.1016/j.matpur.2009.01.008.  Google Scholar [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.  doi: 10.1080/03605309408821055.  Google Scholar [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.  doi: 10.1090/S0002-9939-03-07246-0.  Google Scholar [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.  doi: 10.1002/mma.797.  Google Scholar [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.  doi: 10.1002/cpa.3160300605.  Google Scholar [22] Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems,, Appl. Math. Letters, 11 (1998), 75.  doi: 10.1016/S0893-9659(98)00083-4.  Google Scholar [23] Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications,, Nonlinearity, 20 (2007), 1927.  doi: 10.1088/0951-7715/20/8/007.  Google Scholar [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).  doi: 10.1063/1.3591133.  Google Scholar [25] D. Serre, Richness and the classification of quasilinear hyperbolic systems,, in IMA Vol. Math. Appl., 29 (1991), 315.  doi: 10.1007/978-1-4613-9121-0_24.  Google Scholar [26] D. Serre, Systèmes de Lois de Conservation I-II,, Diderot, (1996).   Google Scholar [27] D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations,, Arch. Rat. Mech. Anal., 172 (2004), 309.  doi: 10.1007/s00205-003-0303-4.  Google Scholar [28] B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation,, Mém. Soc. Math. France, 56 (1994).   Google Scholar [29] S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type,, Dolk. Akad. Nauk SSSR, 282 (1985), 534.   Google Scholar [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.   Google Scholar [31] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions,, J. Diff. Eqs., 68 (1987), 118.  doi: 10.1016/0022-0396(87)90188-4.  Google Scholar

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.  doi: 10.1137/S0036141000377900.  Google Scholar [2] F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited,, Trans. Amer. Math. Soc., 350 (1998), 2847.  doi: 10.1090/S0002-9947-98-02204-1.  Google Scholar [3] Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics,, Proceedings of the International Congress of Mathematicians, 3 (2002), 761.   Google Scholar [4] Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations,, Arch. Rat. Mech. Anal., 172 (2004), 65.  doi: 10.1007/s00205-003-0291-4.  Google Scholar [5] A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem,, Oxford Lecture Series in Math. and its Applications, (2000).   Google Scholar [6] H. Brézis, Analyse Fonctionnelle,, Masson, (1993).   Google Scholar [7] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations,, J. Math. Phys., 44 (2003), 6132.  doi: 10.1063/1.1621057.  Google Scholar [8] G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 131.  doi: 10.1007/BF00375416.  Google Scholar [9] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 695.  doi: 10.1002/cpa.3160180408.  Google Scholar [10] J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws,, Amer. Math. Soc., (1970).   Google Scholar [11] F. John, Nonlinear Waves Equations, Formation of Singularities,, Pitcher Lectures in Math. Sciences, (1990).   Google Scholar [12] S. N. Kruzkov, First order quasilinear equations in several independent variables,, Mat. Sbornik (N.S.), 81 (1970), 228.   Google Scholar [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).  doi: 10.1063/1.2158435.  Google Scholar [14] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Part. Diff. Eqs., 28 (2003), 1203.  doi: 10.1081/PDE-120021192.  Google Scholar [15] P. D. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar [16] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Appl. Math., 32 (1994).   Google Scholar [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.  doi: 10.1016/j.matpur.2009.01.008.  Google Scholar [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.  doi: 10.1080/03605309408821055.  Google Scholar [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.  doi: 10.1090/S0002-9939-03-07246-0.  Google Scholar [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.  doi: 10.1002/mma.797.  Google Scholar [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.  doi: 10.1002/cpa.3160300605.  Google Scholar [22] Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems,, Appl. Math. Letters, 11 (1998), 75.  doi: 10.1016/S0893-9659(98)00083-4.  Google Scholar [23] Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications,, Nonlinearity, 20 (2007), 1927.  doi: 10.1088/0951-7715/20/8/007.  Google Scholar [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).  doi: 10.1063/1.3591133.  Google Scholar [25] D. Serre, Richness and the classification of quasilinear hyperbolic systems,, in IMA Vol. Math. Appl., 29 (1991), 315.  doi: 10.1007/978-1-4613-9121-0_24.  Google Scholar [26] D. Serre, Systèmes de Lois de Conservation I-II,, Diderot, (1996).   Google Scholar [27] D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations,, Arch. Rat. Mech. Anal., 172 (2004), 309.  doi: 10.1007/s00205-003-0303-4.  Google Scholar [28] B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation,, Mém. Soc. Math. France, 56 (1994).   Google Scholar [29] S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type,, Dolk. Akad. Nauk SSSR, 282 (1985), 534.   Google Scholar [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.   Google Scholar [31] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions,, J. Diff. Eqs., 68 (1987), 118.  doi: 10.1016/0022-0396(87)90188-4.  Google Scholar
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