May  2015, 14(3): 759-792. doi: 10.3934/cpaa.2015.14.759

Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities

1. 

Department of Mathematics, Fuzhou University, Fuzhou 350002

Received  August 2012 Revised  January 2015 Published  March 2015

In the present paper the author investigates the generalized nonlinear initial-boundary Riemann problem with small BV data for general $n\times n$ quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space $\{(t,x)|t\geq 0,x\geq 0\}$, where the Riemann solution only contains shocks and contact discontinuities. Combining the techniques employed by Li-Kong with the modified Glimm's functional, the author obtains the almost global existence and lifespan of classical discontinuous solutions to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw-Rascle model.
Citation: Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759
References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, \emph{SIAM J. Appl. Math.}, 60 (2000), 916. doi: 10.1137/S0036139997332099. Google Scholar

[2]

J. M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace,, \emph{Journ$\acutee$es E.D.P. (Saint Jean de Monts}, (1987). Google Scholar

[3]

A. Bressan, A locally contractive metric for systems of conservation laws,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV}, 22 (1995), 109. Google Scholar

[4]

A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, \emph{Indiana Univ. Math. J.}, 37 (1988), 409. doi: 10.1512/iumj.1988.37.37021. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar

[6]

A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws,, \emph{Mem. Amer. Math. Soc.}, 146 (2000), 1. Google Scholar

[7]

A. Bressan and P. G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws,, \emph{Indiana Univ. Math. J.}, 48 (1999), 43. Google Scholar

[8]

A. Bressan, T. P. Liu and T. Yang, $L^{1}$ stability estimates for $n \times n$ conservation laws,, \emph{Arch. Rational Mech. Anal.}, 149 (1999), 1. Google Scholar

[9]

G. Q. Chen and H. Frid, Asymptotic stability of Riemann waves for conservation laws,, \emph{Z. Angew. Math. Phys.}, 48 (1997), 30. doi: 10.1007/PL00001468. Google Scholar

[10]

G. Q. Chen and H. Frid, Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 1103. doi: 10.1090/S0002-9947-00-02660-X. Google Scholar

[11]

G. Q. Chen, H. Frid and Y. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics,, \emph{Commun. Math. Phys.}, 228 (2002), 201. Google Scholar

[12]

G. Q. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations,, \emph{J. Differential Equations}, 202 (2004), 332. Google Scholar

[13]

C. M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws,, in \emph{Recent Mathematical Methods in Nonlinear Wave Propagation} (Montecatini Terme, (1994), 48. doi: 10.1007/BFb0093706. Google Scholar

[14]

W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,, \emph{J. Differential Equations}, 235 (2007), 127. doi: 10.1016/j.jde.2006.12.020. Google Scholar

[15]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model,, \emph{Comm. Partial Differential Equations}, 31 (2006), 243. doi: 10.1080/03605300500358053. Google Scholar

[16]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, \emph{Comm. Pure Appl. Math.}, 18 (1965), 697. doi: 10.1002/cpa.3160180408. Google Scholar

[17]

L. Hsiao and R. Pan, Nonlinear stability of rarefaction waves for a rate-type viscoelastic system,, \emph{Chinese Ann. Math. Ser. B}, 20 (1999), 223. doi: 10.1142/S0252959999000254. Google Scholar

[18]

L. Hsiao and S. Q. Tang, Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping,, \emph{J. Differential Equations}, 123 (1995), 480. Google Scholar

[19]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, \emph{Adv. Math.}, 219 (2008), 1246. Google Scholar

[20]

F. John, Formation of singularities in one-dimensional nonlinear wave propagation,, \emph{Comm. Pure Appl. Math.}, 27 (1974), 377. Google Scholar

[21]

D. X. Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data,, \emph{Chinese Ann. Math. Ser. B}, 21 (2000), 413. doi: 10.1142/S0252959900000431. Google Scholar

[22]

D. X. Kong, Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities,, \emph{J. Differential Equations}, 188 (2003), 242. doi: 10.1016/S0022-0396(02)00068-2. Google Scholar

[23]

D. X. Kong, Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves,, \emph{J. Differential Equations}, 219 (2005), 421. doi: 10.1016/j.jde.2005.03.001. Google Scholar

[24]

P. D. Lax, Hyperbolic systems of conservation laws II,, \emph{Comm. Pure Appl. Math.}, 10 (1957), 537. Google Scholar

[25]

M. Lewicka, Well-posedness for hyperbolic systems of conservation laws with large BV data,, \emph{Arch. Rational Mech. Anal.}, 173 (2004), 415. doi: 10.1007/s00205-004-0325-6. Google Scholar

[26]

T. Li and D. X. Kong, Global classical discontinuous solutions to a class of generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws,, \emph{Comm. Partial Differential Equations}, 24 (1999), 801. doi: 10.1080/03605309908821447. Google Scholar

[27]

T. Li and L. Wang, The generalized nonlinear initial-boundary Riemann problem for quasilinear hyperbolic systems of conservation laws,, \emph{Nonlinear Anal.}, 62 (2005), 1091. Google Scholar

[28]

T. Li and L. Wang, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 59. doi: 10.3934/dcds.2005.12.59. Google Scholar

[29]

T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985). Google Scholar

[30]

P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 599. Google Scholar

[31]

J. Liu and Z. Xin, Nonlinear stability of discrete shocks for systems of conservation laws,, \emph{Arch. Rational Mech. Anal.}, 125 (1993), 217. doi: 10.1007/BF00383220. Google Scholar

[32]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, \emph{Mem. Amer. Math. Soc.}, 56 (1985), 1. doi: 10.1090/memo/0328. Google Scholar

[33]

T. P. Liu, Nonlinear stability and instability of overcompressive shock waves,, in \emph{Shock Induced Transitions and Phase Structures in General Media} (eds. J. E. Dunn, (1993), 159. Google Scholar

[34]

T. P. Liu and Z. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, \emph{Comm. Math. Phys.}, 118 (1988), 451. Google Scholar

[35]

T. P. Liu and T. Yang, $L^{1}$ stability of conservation laws with coinciding Hugoniot and characteristic curves,, \emph{Indiana Univ. Math. J.}, 48 (1999), 237. doi: 10.1512/iumj.1999.48.1601. Google Scholar

[36]

T. P. Liu and T. Yang, $L^{1}$ stability for $2\times 2$ systems of hyperbolic conservation laws,, \emph{J. Amer. Math. Soc.}, 12 (1999), 729. doi: 10.1090/S0894-0347-99-00292-1. Google Scholar

[37]

T. P. Liu and K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation,, \emph{Comm. Math. Phys.}, 168 (1995), 163. Google Scholar

[38]

T. Luo and Z. Xin, Nonlinear stability of shock fronts for a relaxation systemin several space dimensions,, \emph{J. Differential Equations}, 139 (1997), 365. doi: 10.1006/jdeq.1997.3302. Google Scholar

[39]

A. Majda, The stability of multidimensional shock fronts,, \emph{Mem. Amer. Math. Soc.}, 41 (1983), 1. doi: 10.1090/memo/0275. Google Scholar

[40]

M. Sablé-Tougeron, Méthode de Glimm et problème mixte,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 10 (1993), 423. Google Scholar

[41]

M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem,, \emph{Indiana Univ. Math. J.}, 34 (1985), 533. doi: 10.1512/iumj.1985.34.34030. Google Scholar

[42]

M. Schatzman, The Geometry of Continuous Glimm Functionals,, in \emph{Nonlinear systems of partial differential equations in applied mathematics, (1984), 417. Google Scholar

[43]

S. Schochet, Sufficient condition for local existence via Glimm's scheme for large BV data,, \emph{J. Differential Equations}, 89 (1991), 317. doi: 10.1016/0022-0396(91)90124-R. Google Scholar

[44]

Z. Q. Shao, Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary,, \emph{J. Math. Anal. Appl.}, 330 (2007), 511. doi: 10.1016/j.jmaa.2006.07.078. Google Scholar

[45]

Z. Q. Shao, Global structure stability of Riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary,, \emph{Nonlinear Anal.}, 69 (2008), 2651. doi: 10.1016/j.na.2007.07.059. Google Scholar

[46]

Z. Q. Shao, The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws,, \emph{J. Math. Anal. Appl.}, 379 (2011), 589. Google Scholar

[47]

Z. Q. Shao, Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: shocks and contact discontinuities,, \emph{J. Math. Anal. Appl.}, 387 (2012), 698. Google Scholar

[48]

Z. Q. Shao, Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves,, \emph{J. Math. Anal. Appl.}, 409 (2014), 1066. Google Scholar

[49]

J. A. Smoller, J. B. Temple and Z. Xin, Instability of rarefaction shocks in systems of conservation laws,, \emph{Arch. Rational Mech. Anal.}, 112 (1990), 63. Google Scholar

[50]

Z. Xin, On nonlinear stability of contact discontinuities,, in \emph{Hyperbolic Problems: Theory, (1994), 249. Google Scholar

[51]

Z. Xin, Theory of viscous conservation laws,, in \emph{Some Current Topics on Nonlinear Conservation Laws} (Eds. L. Hsiao and Z. Xin), (2000), 141. Google Scholar

[52]

Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, \emph{Chinese Ann. Math. Ser. B}, 25 (2004), 37. doi: 10.1142/S0252959904000044. Google Scholar

show all references

References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, \emph{SIAM J. Appl. Math.}, 60 (2000), 916. doi: 10.1137/S0036139997332099. Google Scholar

[2]

J. M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace,, \emph{Journ$\acutee$es E.D.P. (Saint Jean de Monts}, (1987). Google Scholar

[3]

A. Bressan, A locally contractive metric for systems of conservation laws,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV}, 22 (1995), 109. Google Scholar

[4]

A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, \emph{Indiana Univ. Math. J.}, 37 (1988), 409. doi: 10.1512/iumj.1988.37.37021. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar

[6]

A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws,, \emph{Mem. Amer. Math. Soc.}, 146 (2000), 1. Google Scholar

[7]

A. Bressan and P. G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws,, \emph{Indiana Univ. Math. J.}, 48 (1999), 43. Google Scholar

[8]

A. Bressan, T. P. Liu and T. Yang, $L^{1}$ stability estimates for $n \times n$ conservation laws,, \emph{Arch. Rational Mech. Anal.}, 149 (1999), 1. Google Scholar

[9]

G. Q. Chen and H. Frid, Asymptotic stability of Riemann waves for conservation laws,, \emph{Z. Angew. Math. Phys.}, 48 (1997), 30. doi: 10.1007/PL00001468. Google Scholar

[10]

G. Q. Chen and H. Frid, Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 1103. doi: 10.1090/S0002-9947-00-02660-X. Google Scholar

[11]

G. Q. Chen, H. Frid and Y. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics,, \emph{Commun. Math. Phys.}, 228 (2002), 201. Google Scholar

[12]

G. Q. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations,, \emph{J. Differential Equations}, 202 (2004), 332. Google Scholar

[13]

C. M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws,, in \emph{Recent Mathematical Methods in Nonlinear Wave Propagation} (Montecatini Terme, (1994), 48. doi: 10.1007/BFb0093706. Google Scholar

[14]

W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,, \emph{J. Differential Equations}, 235 (2007), 127. doi: 10.1016/j.jde.2006.12.020. Google Scholar

[15]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model,, \emph{Comm. Partial Differential Equations}, 31 (2006), 243. doi: 10.1080/03605300500358053. Google Scholar

[16]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, \emph{Comm. Pure Appl. Math.}, 18 (1965), 697. doi: 10.1002/cpa.3160180408. Google Scholar

[17]

L. Hsiao and R. Pan, Nonlinear stability of rarefaction waves for a rate-type viscoelastic system,, \emph{Chinese Ann. Math. Ser. B}, 20 (1999), 223. doi: 10.1142/S0252959999000254. Google Scholar

[18]

L. Hsiao and S. Q. Tang, Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping,, \emph{J. Differential Equations}, 123 (1995), 480. Google Scholar

[19]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, \emph{Adv. Math.}, 219 (2008), 1246. Google Scholar

[20]

F. John, Formation of singularities in one-dimensional nonlinear wave propagation,, \emph{Comm. Pure Appl. Math.}, 27 (1974), 377. Google Scholar

[21]

D. X. Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data,, \emph{Chinese Ann. Math. Ser. B}, 21 (2000), 413. doi: 10.1142/S0252959900000431. Google Scholar

[22]

D. X. Kong, Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities,, \emph{J. Differential Equations}, 188 (2003), 242. doi: 10.1016/S0022-0396(02)00068-2. Google Scholar

[23]

D. X. Kong, Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves,, \emph{J. Differential Equations}, 219 (2005), 421. doi: 10.1016/j.jde.2005.03.001. Google Scholar

[24]

P. D. Lax, Hyperbolic systems of conservation laws II,, \emph{Comm. Pure Appl. Math.}, 10 (1957), 537. Google Scholar

[25]

M. Lewicka, Well-posedness for hyperbolic systems of conservation laws with large BV data,, \emph{Arch. Rational Mech. Anal.}, 173 (2004), 415. doi: 10.1007/s00205-004-0325-6. Google Scholar

[26]

T. Li and D. X. Kong, Global classical discontinuous solutions to a class of generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws,, \emph{Comm. Partial Differential Equations}, 24 (1999), 801. doi: 10.1080/03605309908821447. Google Scholar

[27]

T. Li and L. Wang, The generalized nonlinear initial-boundary Riemann problem for quasilinear hyperbolic systems of conservation laws,, \emph{Nonlinear Anal.}, 62 (2005), 1091. Google Scholar

[28]

T. Li and L. Wang, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 59. doi: 10.3934/dcds.2005.12.59. Google Scholar

[29]

T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985). Google Scholar

[30]

P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 599. Google Scholar

[31]

J. Liu and Z. Xin, Nonlinear stability of discrete shocks for systems of conservation laws,, \emph{Arch. Rational Mech. Anal.}, 125 (1993), 217. doi: 10.1007/BF00383220. Google Scholar

[32]

T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, \emph{Mem. Amer. Math. Soc.}, 56 (1985), 1. doi: 10.1090/memo/0328. Google Scholar

[33]

T. P. Liu, Nonlinear stability and instability of overcompressive shock waves,, in \emph{Shock Induced Transitions and Phase Structures in General Media} (eds. J. E. Dunn, (1993), 159. Google Scholar

[34]

T. P. Liu and Z. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, \emph{Comm. Math. Phys.}, 118 (1988), 451. Google Scholar

[35]

T. P. Liu and T. Yang, $L^{1}$ stability of conservation laws with coinciding Hugoniot and characteristic curves,, \emph{Indiana Univ. Math. J.}, 48 (1999), 237. doi: 10.1512/iumj.1999.48.1601. Google Scholar

[36]

T. P. Liu and T. Yang, $L^{1}$ stability for $2\times 2$ systems of hyperbolic conservation laws,, \emph{J. Amer. Math. Soc.}, 12 (1999), 729. doi: 10.1090/S0894-0347-99-00292-1. Google Scholar

[37]

T. P. Liu and K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation,, \emph{Comm. Math. Phys.}, 168 (1995), 163. Google Scholar

[38]

T. Luo and Z. Xin, Nonlinear stability of shock fronts for a relaxation systemin several space dimensions,, \emph{J. Differential Equations}, 139 (1997), 365. doi: 10.1006/jdeq.1997.3302. Google Scholar

[39]

A. Majda, The stability of multidimensional shock fronts,, \emph{Mem. Amer. Math. Soc.}, 41 (1983), 1. doi: 10.1090/memo/0275. Google Scholar

[40]

M. Sablé-Tougeron, Méthode de Glimm et problème mixte,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 10 (1993), 423. Google Scholar

[41]

M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem,, \emph{Indiana Univ. Math. J.}, 34 (1985), 533. doi: 10.1512/iumj.1985.34.34030. Google Scholar

[42]

M. Schatzman, The Geometry of Continuous Glimm Functionals,, in \emph{Nonlinear systems of partial differential equations in applied mathematics, (1984), 417. Google Scholar

[43]

S. Schochet, Sufficient condition for local existence via Glimm's scheme for large BV data,, \emph{J. Differential Equations}, 89 (1991), 317. doi: 10.1016/0022-0396(91)90124-R. Google Scholar

[44]

Z. Q. Shao, Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary,, \emph{J. Math. Anal. Appl.}, 330 (2007), 511. doi: 10.1016/j.jmaa.2006.07.078. Google Scholar

[45]

Z. Q. Shao, Global structure stability of Riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary,, \emph{Nonlinear Anal.}, 69 (2008), 2651. doi: 10.1016/j.na.2007.07.059. Google Scholar

[46]

Z. Q. Shao, The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws,, \emph{J. Math. Anal. Appl.}, 379 (2011), 589. Google Scholar

[47]

Z. Q. Shao, Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: shocks and contact discontinuities,, \emph{J. Math. Anal. Appl.}, 387 (2012), 698. Google Scholar

[48]

Z. Q. Shao, Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves,, \emph{J. Math. Anal. Appl.}, 409 (2014), 1066. Google Scholar

[49]

J. A. Smoller, J. B. Temple and Z. Xin, Instability of rarefaction shocks in systems of conservation laws,, \emph{Arch. Rational Mech. Anal.}, 112 (1990), 63. Google Scholar

[50]

Z. Xin, On nonlinear stability of contact discontinuities,, in \emph{Hyperbolic Problems: Theory, (1994), 249. Google Scholar

[51]

Z. Xin, Theory of viscous conservation laws,, in \emph{Some Current Topics on Nonlinear Conservation Laws} (Eds. L. Hsiao and Z. Xin), (2000), 141. Google Scholar

[52]

Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, \emph{Chinese Ann. Math. Ser. B}, 25 (2004), 37. doi: 10.1142/S0252959904000044. Google Scholar

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