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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

Hualin Zheng 1,

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].
Keywords: large time stability., contact discontinuity, superposition, Jin-Xin relaxation system, shock wave.
Mathematics Subject Classification: Primary: 35F25, 35L65; Secondary: 35L6.
Citation: Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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