# American Institute of Mathematical Sciences

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:
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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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##### 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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