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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

Yue-Jun Peng 1, and  Yong-Fu Yang 2,

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.
Keywords: Entropy solution, long-time behavior, rich system, $L^1$ stability., linearly degeneracy.
Mathematics Subject Classification: Primary: 35L45, 35L65; Secondary: 35B40, 35B3.
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, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
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show all references

References:
[1]

SIAM J. Math. Anal., 33 (2001), 959-981. doi: 10.1137/S0036141000377900.  Google Scholar

[2]

Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1.  Google Scholar

[3]

Proceedings of the International Congress of Mathematicians, 3, Higher Education Press, Beijing, (2002), 761-772.  Google Scholar

[4]

Arch. Rat. Mech. Anal., 172 (2004), 65-91. doi: 10.1007/s00205-003-0291-4.  Google Scholar

[5]

Oxford Lecture Series in Math. and its Applications, 20. Oxford University Press, Oxford, 2000.  Google Scholar

[6]

Masson, Paris, 1993.  Google Scholar

[7]

J. Math. Phys., 44 (2003), 6132-6139. doi: 10.1063/1.1621057.  Google Scholar

[8]

Arch. Rational Mech. Anal., 121 (1992), 131-185. doi: 10.1007/BF00375416.  Google Scholar

[9]

Comm. Pure Appl. Math., 18 (1965), 695-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[10]

Amer. Math. Soc., Providence, R.I. 1970.  Google Scholar

[11]

Pitcher Lectures in Math. Sciences, Lehigh University, Amer. Math. Soc., 1990.  Google Scholar

[12]

Mat. Sbornik (N.S.), 81 (1970), 228-255.  Google Scholar

[13]

J. Math. Phys., 47 (2006), 013503, 16 pages. doi: 10.1063/1.2158435.  Google Scholar

[14]

Comm. Part. Diff. Eqs., 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192.  Google Scholar

[15]

Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[16]

Research in Appl. Math., 32, Wiely/Masson, 1994.  Google Scholar

[17]

J. Math. Pures Appl., 91 (2009), 553-568. doi: 10.1016/j.matpur.2009.01.008.  Google Scholar

[18]

Comm. Partial Diff. Equations, 19 (1994), 1263-1317. doi: 10.1080/03605309408821055.  Google Scholar

[19]

Proc. Am. Math. Soc., 132 (2004), 1095-1102. doi: 10.1090/S0002-9939-03-07246-0.  Google Scholar

[20]

Math. Meth. Appl. Sci., 30 (2007), 479-500. doi: 10.1002/mma.797.  Google Scholar

[21]

Comm. Pure Appl. Math., 30 (1977), 767-796. doi: 10.1002/cpa.3160300605.  Google Scholar

[22]

Appl. Math. Letters, 11 (1998), 75-78. doi: 10.1016/S0893-9659(98)00083-4.  Google Scholar

[23]

Nonlinearity, 20 (2007), 1927-1953. doi: 10.1088/0951-7715/20/8/007.  Google Scholar

[24]

J. Math. Phys., 52 (2011), 053702 (23 pages). doi: 10.1063/1.3591133.  Google Scholar

[25]

in IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333. doi: 10.1007/978-1-4613-9121-0_24.  Google Scholar

[26]

Diderot, Paris, 1996.  Google Scholar

[27]

Arch. Rat. Mech. Anal., 172 (2004), 309-331. doi: 10.1007/s00205-003-0303-4.  Google Scholar

[28]

Mém. Soc. Math. France, 56 (1994), 125pp.  Google Scholar

[29]

Dolk. Akad. Nauk SSSR, 282 (1985), 534-537.  Google Scholar

[30]

Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419.  Google Scholar

[31]

J. Diff. Eqs., 68 (1987), 118-136. doi: 10.1016/0022-0396(87)90188-4.  Google Scholar

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