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

Zhi-Qiang Shao

doi:10.3934/cpaa.2015.14.759

Article Contents

2015, Volume 14, Issue 3: 759-792. Doi: 10.3934/cpaa.2015.14.759

This issue Previous Article General types of spherical mean operators and $K$-functionals of fractional orders Next Article Differential Harnack estimates for backward heat equations with potentials under geometric flows

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

Zhi-Qiang Shao^1,

1.
Department of Mathematics, Fuzhou University, Fuzhou 350002

Received: August 2012

Revised: January 2015

Published: May 2015

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Generalized nonlinear initial-boundary Riemann problem,
quasilinear hyperbolic system of conservation laws,
piecewise $c^1$ solution,
shock wave,
contact discontinuity,
lifespan.

Mathematics Subject Classification: Primary: 35L65, 35L45; Secondary: 35L67.

Citation:

Related Papers

Cited by

References

[1]	A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.doi: 10.1137/S0036139997332099.
[2]	J. M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace, Journées E.D.P. (Saint Jean de Monts, 1987), Exp. No. XVI, 10 pp., École Polytechnique, Palaiseau, 1987.
[3]	A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV, 22 (1995), 109-135.
[4]	A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J., 37 (1988), 409-421.doi: 10.1512/iumj.1988.37.37021.
[5]	A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000.
[6]	A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc., 146 (2000), 1-134.
[7]	A. Bressan and P. G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J., 48 (1999), 43-84.
[8]	A. Bressan, T. P. Liu and T. Yang, $L^{1}$ stability estimates for $n \times n$ conservation laws, Arch. Rational Mech. Anal., 149 (1999) 1-22.
[9]	G. Q. Chen and H. Frid, Asymptotic stability of Riemann waves for conservation laws, Z. Angew. Math. Phys., 48 (1997), 30-44.doi: 10.1007/PL00001468.
[10]	G. Q. Chen and H. Frid, Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations, Trans. Amer. Math. Soc., 353 (2001), 1103-1117.doi: 10.1090/S0002-9947-00-02660-X.
[11]	G. Q. Chen, H. Frid and Y. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Commun. Math. Phys., 228 (2002), 201-217.
[12]	G. Q. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations, J. Differential Equations, 202 (2004), 332-353.
[13]	C. M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, in Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994) (T. Ruggeri Ed.), Lecture Notes in Mathematics, vol. 1640, Springer, Berlin, 1996, pp. 48-69.doi: 10.1007/BFb0093706.
[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, J. Differential Equations, 235 (2007), 127-165.doi: 10.1016/j.jde.2006.12.020.
[15]	M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.doi: 10.1080/03605300500358053.
[16]	J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.doi: 10.1002/cpa.3160180408.
[17]	L. Hsiao and R. 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.
[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, J. Differential Equations, 123 (1995), 480-503.
[19]	F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.
[20]	F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27 (1974), 377-405.
[21]	D. X. Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. Ser. B, 21 (2000), 413-440.doi: 10.1142/S0252959900000431.
[22]	D. X. Kong, Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities, J. Differential Equations, 188 (2003), 242-271.doi: 10.1016/S0022-0396(02)00068-2.
[23]	D. X. Kong, Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, J. Differential Equations, 219 (2005), 421-450.doi: 10.1016/j.jde.2005.03.001.
[24]	P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.
[25]	M. Lewicka, Well-posedness for hyperbolic systems of conservation laws with large BV data, Arch. Rational Mech. Anal., 173 (2004), 415-445.doi: 10.1007/s00205-004-0325-6.
[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, Comm. Partial Differential Equations, 24 (1999), 801-820.doi: 10.1080/03605309908821447.
[27]	T. Li and L. Wang, The generalized nonlinear initial-boundary Riemann problem for quasilinear hyperbolic systems of conservation laws, Nonlinear Anal., 62 (2005), 1091-1107.
[28]	T. Li and L. Wang, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete Contin. Dyn. Syst., 12 (2005), 59-78.doi: 10.3934/dcds.2005.12.59.
[29]	T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, Duke University, Durham, 1985.
[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, Comm. Pure Appl. Math., 49 (1996), 599-638.
[31]	J. Liu and Z. Xin, Nonlinear stability of discrete shocks for systems of conservation laws, Arch. Rational Mech. Anal., 125 (1993), 217-256.doi: 10.1007/BF00383220.
[32]	T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.doi: 10.1090/memo/0328.
[33]	T. P. Liu, Nonlinear stability and instability of overcompressive shock waves, in Shock Induced Transitions and Phase Structures in General Media (eds. J. E. Dunn, R. Posdick and M. Slemrod), IMA Volumes in Mathematical Applications, vol. 52, Springer, New York, 1993, pp. 159-167.
[34]	T. P. Liu and Z. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
[35]	T. P. Liu and T. Yang, $L^{1}$ stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J., 48 (1999), 237-247.doi: 10.1512/iumj.1999.48.1601.
[36]	T. P. Liu and T. Yang, $L^{1}$ stability for $2\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc., 12 (1999), 729-774.doi: 10.1090/S0894-0347-99-00292-1.
[37]	T. P. Liu and K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys., 168 (1995), 163-186.
[38]	T. Luo and Z. Xin, Nonlinear stability of shock fronts for a relaxation systemin several space dimensions, J. Differential Equations, 139 (1997), 365-408.doi: 10.1006/jdeq.1997.3302.
[39]	A. Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc., 41 (1983), 1-95.doi: 10.1090/memo/0275.
[40]	M. Sablé-Tougeron, Méthode de Glimm et problème mixte, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 423-443.
[41]	M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana Univ. Math. J., 34 (1985), 533-589.doi: 10.1512/iumj.1985.34.34030.
[42]	M. Schatzman, The Geometry of Continuous Glimm Functionals, in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Applied Mathematics, vol. 23, American Mathematical Society, Providence, RI, 1986, pp. 417-439.
[43]	S. Schochet, Sufficient condition for local existence via Glimm's scheme for large BV data, J. Differential Equations, 89 (1991), 317-354.doi: 10.1016/0022-0396(91)90124-R.
[44]	Z. Q. Shao, Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary, J. Math. Anal. Appl., 330 (2007), 511-540.doi: 10.1016/j.jmaa.2006.07.078.
[45]	Z. Q. Shao, Global structure stability of Riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary, Nonlinear Anal., 69 (2008), 2651-2676.doi: 10.1016/j.na.2007.07.059.
[46]	Z. Q. Shao, The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws, J. Math. Anal. Appl., 379 (2011), 589-615.
[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, J. Math. Anal. Appl., 387 (2012), 698-720.
[48]	Z. Q. Shao, Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves, J. Math. Anal. Appl., 409 (2014), 1066-1083.
[49]	J. A. Smoller, J. B. Temple and Z. Xin, Instability of rarefaction shocks in systems of conservation laws, Arch. Rational Mech. Anal., 112 (1990), 63-81.
[50]	Z. Xin, On nonlinear stability of contact discontinuities, in Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), World Scientific Publishing, River Edge, NJ, 1996, pp. 249-257.
[51]	Z. Xin, Theory of viscous conservation laws, in Some Current Topics on Nonlinear Conservation Laws (Eds. L. Hsiao and Z. Xin), AMS/IP Studies in Advanced Mathematics, vol. 15, Amer. Math. Soc, Providence, RI, 2000, pp. 141-193.
[52]	Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56.doi: 10.1142/S0252959904000044.