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September  2015, 8(3): 559-585. doi: 10.3934/krm.2015.8.559

## Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system

 1 Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

Received  December 2014 Revised  May 2015 Published  June 2015

In this paper, we consider the time-asymptotic stability of a superposition of shock waves with contact discontinuities for the one dimensional Jin-Xin relaxation system with small initial perturbations, provided that the strengths of waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a delicate decay estimate on the heat kernel in Huang-Li-Matsumura [5].
Citation: Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic and Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559
##### References:
 [1] S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model, Comm. Pure Appl. Math., 59 (2006), 688-753. doi: 10.1002/cpa.20114. [2] G. Q. Chen, C. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602. [3] L. Hsiao and T. Luo, Stability of travelling wave solutions for a rate-type viscoelastic system, in Advances in Nonlinear Partial Differential Equations and Related Areas, World Sci. Publishing, River Edge, NJ, 1998, 166-186. [4] L. Hsiao and R. H. Pan, Nonlinear stability of rarefaction waves for a rate-type viscoelastic system, Chinese Ann. Math. Ser. -B, 20 (1999), 223-232. doi: 10.1142/S0252959999000254. [5] F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. [6] F. M. Huang, R. H. Pan and Y. Wang, Stability of contact discontinuity for Jin-Xin relaxation system, J. Differential Equations, 244 (2008), 1114-1140. doi: 10.1016/j.jde.2007.12.002. [7] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. [8] S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276. doi: 10.1002/cpa.3160480303. [9] P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS Regional Conf. Ser. in Appl. Math., Vol. 11, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. [10] T. Li and H. L. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. doi: 10.4310/CMS.2005.v3.n2.a1. [11] H. L. Liu, The $L^p$ stability of relaxation rarefaction profiles, J. Differential Equations, 171 (2001), 397-411. doi: 10.1006/jdeq.2000.3849. [12] H. L. Liu, Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws, J. Differential Equations, 192 (2003), 285-307. doi: 10.1016/S0022-0396(03)00124-4. [13] 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. [14] T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707. [15] T. P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. [16] T. P. Liu, T. Yang, S. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2006), 333-371. doi: 10.1007/s00205-005-0414-1. [17] T. Luo, Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensionals, J. Differential Equations, 133 (1997), 255-279. doi: 10.1006/jdeq.1996.3214. [18] T. Luo and Z. P. Xin, Nonlinear stability of shock fronts for a relaxation system in several space dimensions, J. Differential Equations, 139 (1997), 365-408. doi: 10.1006/jdeq.1997.3302. [19] C. Mascia and R. Natalini, $L^1$ nonlinear stability of traveling waves for a hyperbolic system with relaxation, J. Differential Equations, 132 (1996), 275-292. doi: 10.1006/jdeq.1996.0180. [20] C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913. doi: 10.1137/S0036141004435844. [21] M. Mei and T. Yang, Convergence rates to travelling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh Sect.-A, 128 (1998), 1053-1068. doi: 10.1017/S0308210500030067. [22] R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis of Systems of Conservation Laws, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 99, Chapman & Hall/CRC, Boca Raton, FL, 1999, 128-198. [23] D. Serre, Relaxation semi-lineaire et cinetique des systemes de lois de conservation, Ann. Inst. Henri Poincaré, 17 (2000), 169-192. doi: 10.1016/S0294-1449(99)00105-5. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] W. C. Wang, Nonlinear stability of centered rarefaction waves of the Jin-Xin relaxation model for $2\times2$ conservation laws, Electron. J. Differential Equations, (2002), 1-20. [26] G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [27] Z. P. Xin, Theory of viscous conservation laws, in Some current Topics on Nonlinear Conservation Laws, AMS/IP Stud. Adv. Math., 15, Amer. Math. Soc., Providence, RI, 2000, 141-193. [28] W. Q. Xu, Relaxation limit for piecewise smooth solutions to systems of conservation laws, J. Differential Equations, 162 (2000), 140-173. doi: 10.1006/jdeq.1999.3653. [29] W. A. Yong, Existence and asymptotic stability of traveling wave solutions of a model system for reacting flow, Nonlinear Anal., 26 (1996), 1791-1809. doi: 10.1016/0362-546X(94)00356-M. [30] S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118. doi: 10.1090/S0894-0347-2010-00671-6. [31] H. H. Zeng, Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws, J. Differential Equations, 246 (2009), 2081-2102. doi: 10.1016/j.jde.2008.07.034. [32] Y. N. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188. [33] H. J. Zhao, Nonlinear stability of strong palnar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions, J. Differential Equations, 163 (2000), 198-220. doi: 10.1006/jdeq.1999.3722. [34] C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063.

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##### References:
 [1] S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model, Comm. Pure Appl. Math., 59 (2006), 688-753. doi: 10.1002/cpa.20114. [2] G. Q. Chen, C. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602. [3] L. Hsiao and T. Luo, Stability of travelling wave solutions for a rate-type viscoelastic system, in Advances in Nonlinear Partial Differential Equations and Related Areas, World Sci. Publishing, River Edge, NJ, 1998, 166-186. [4] L. Hsiao and R. H. Pan, Nonlinear stability of rarefaction waves for a rate-type viscoelastic system, Chinese Ann. Math. Ser. -B, 20 (1999), 223-232. doi: 10.1142/S0252959999000254. [5] F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. [6] F. M. Huang, R. H. Pan and Y. Wang, Stability of contact discontinuity for Jin-Xin relaxation system, J. Differential Equations, 244 (2008), 1114-1140. doi: 10.1016/j.jde.2007.12.002. [7] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. [8] S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276. doi: 10.1002/cpa.3160480303. [9] P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS Regional Conf. Ser. in Appl. Math., Vol. 11, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. [10] T. Li and H. L. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. doi: 10.4310/CMS.2005.v3.n2.a1. [11] H. L. Liu, The $L^p$ stability of relaxation rarefaction profiles, J. Differential Equations, 171 (2001), 397-411. doi: 10.1006/jdeq.2000.3849. [12] H. L. Liu, Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws, J. Differential Equations, 192 (2003), 285-307. doi: 10.1016/S0022-0396(03)00124-4. [13] 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. [14] T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707. [15] T. P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. [16] T. P. Liu, T. Yang, S. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2006), 333-371. doi: 10.1007/s00205-005-0414-1. [17] T. Luo, Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensionals, J. Differential Equations, 133 (1997), 255-279. doi: 10.1006/jdeq.1996.3214. [18] T. Luo and Z. P. Xin, Nonlinear stability of shock fronts for a relaxation system in several space dimensions, J. Differential Equations, 139 (1997), 365-408. doi: 10.1006/jdeq.1997.3302. [19] C. Mascia and R. Natalini, $L^1$ nonlinear stability of traveling waves for a hyperbolic system with relaxation, J. Differential Equations, 132 (1996), 275-292. doi: 10.1006/jdeq.1996.0180. [20] C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913. doi: 10.1137/S0036141004435844. [21] M. Mei and T. Yang, Convergence rates to travelling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh Sect.-A, 128 (1998), 1053-1068. doi: 10.1017/S0308210500030067. [22] R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis of Systems of Conservation Laws, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 99, Chapman & Hall/CRC, Boca Raton, FL, 1999, 128-198. [23] D. Serre, Relaxation semi-lineaire et cinetique des systemes de lois de conservation, Ann. Inst. Henri Poincaré, 17 (2000), 169-192. doi: 10.1016/S0294-1449(99)00105-5. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] W. C. Wang, Nonlinear stability of centered rarefaction waves of the Jin-Xin relaxation model for $2\times2$ conservation laws, Electron. J. Differential Equations, (2002), 1-20. [26] G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [27] Z. P. Xin, Theory of viscous conservation laws, in Some current Topics on Nonlinear Conservation Laws, AMS/IP Stud. Adv. Math., 15, Amer. Math. Soc., Providence, RI, 2000, 141-193. [28] W. Q. Xu, Relaxation limit for piecewise smooth solutions to systems of conservation laws, J. Differential Equations, 162 (2000), 140-173. doi: 10.1006/jdeq.1999.3653. [29] W. A. Yong, Existence and asymptotic stability of traveling wave solutions of a model system for reacting flow, Nonlinear Anal., 26 (1996), 1791-1809. doi: 10.1016/0362-546X(94)00356-M. [30] S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118. doi: 10.1090/S0894-0347-2010-00671-6. [31] H. H. Zeng, Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws, J. Differential Equations, 246 (2009), 2081-2102. doi: 10.1016/j.jde.2008.07.034. [32] Y. N. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188. [33] H. J. Zhao, Nonlinear stability of strong palnar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions, J. Differential Equations, 163 (2000), 198-220. doi: 10.1006/jdeq.1999.3722. [34] C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063.
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