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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 |
References:
[1] |
S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model,, Comm. Pure Appl. Math., 59 (2006), 688.
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.
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, (1998), 166.
|
[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.
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.
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.
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.
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.
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., (1973).
|
[10] |
T. Li and H. L. Liu, Stability of a traffic flow model with nonconvex relaxation,, Commun. Math. Sci., 3 (2005), 101.
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.
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.
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.
doi: 10.1002/cpa.3160300605. |
[14] |
T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153.
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.
|
[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.
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.
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.
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.
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.
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.
doi: 10.1017/S0308210500030067. |
[22] |
R. Natalini, Recent results on hyperbolic relaxation problems,, in Analysis of Systems of Conservation Laws, (1999), 128.
|
[23] |
D. Serre, Relaxation semi-lineaire et cinetique des systemes de lois de conservation,, Ann. Inst. Henri Poincaré, 17 (2000), 169.
doi: 10.1016/S0294-1449(99)00105-5. |
[24] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (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.
|
[26] |
G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974).
|
[27] |
Z. P. Xin, Theory of viscous conservation laws,, in Some current Topics on Nonlinear Conservation Laws, (2000), 141.
|
[28] |
W. Q. Xu, Relaxation limit for piecewise smooth solutions to systems of conservation laws,, J. Differential Equations, 162 (2000), 140.
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.
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.
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.
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.
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.
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.
doi: 10.1006/jdeq.2001.4063. |
show all references
References:
[1] |
S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model,, Comm. Pure Appl. Math., 59 (2006), 688.
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.
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, (1998), 166.
|
[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.
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.
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.
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.
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.
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., (1973).
|
[10] |
T. Li and H. L. Liu, Stability of a traffic flow model with nonconvex relaxation,, Commun. Math. Sci., 3 (2005), 101.
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.
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.
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.
doi: 10.1002/cpa.3160300605. |
[14] |
T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153.
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.
|
[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.
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.
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.
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.
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.
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.
doi: 10.1017/S0308210500030067. |
[22] |
R. Natalini, Recent results on hyperbolic relaxation problems,, in Analysis of Systems of Conservation Laws, (1999), 128.
|
[23] |
D. Serre, Relaxation semi-lineaire et cinetique des systemes de lois de conservation,, Ann. Inst. Henri Poincaré, 17 (2000), 169.
doi: 10.1016/S0294-1449(99)00105-5. |
[24] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (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.
|
[26] |
G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974).
|
[27] |
Z. P. Xin, Theory of viscous conservation laws,, in Some current Topics on Nonlinear Conservation Laws, (2000), 141.
|
[28] |
W. Q. Xu, Relaxation limit for piecewise smooth solutions to systems of conservation laws,, J. Differential Equations, 162 (2000), 140.
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.
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.
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.
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.
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.
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.
doi: 10.1006/jdeq.2001.4063. |
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