Skip to main content
SpringerLink
Log in

Generalized Riemann problems and exact solutions for p-systems with relaxation

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In this paper a p-system with relaxation is considered. Within the theoretical framework of the differential constraints method, we determine two possible classes of exact solutions of the governing model both parameterized by one arbitrary function. This allows to solve classes of initial value problems of interest in nonlinear wave propagation. In fact a generalized Riemann problem is solved by determining a smooth solution which plays the role of the well known rarefaction wave of the homogeneous case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Chen, G.Q., Levermore, C.D., Liu, T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chern, I.-L.: Long-time effect of relaxation for hyperbolic conservation laws. Commun. Math. Phys. 172, 39–55 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ruggeri, T., Müller, I.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37. Springer, Berlin (1998)

    Google Scholar 

  4. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer International Publishing, Switzerland (2015)

    Book  MATH  Google Scholar 

  5. Carrisi, M.C., Pennisi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases in the presence of dynamic pressure. Ricerche di Matematica 64(2), 403–419 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jin, S., Xin, Z.P.: The relaxing schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 555–563 (1995)

    Article  MathSciNet  Google Scholar 

  7. Kawashima, K., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems of one-dimensional gas motion. Commun. Math. Phys. 101, 97–127 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu, H.L., Woo, C.W., Yang, T.: Decay rate for travelling waves of a relaxation model. J. Differ. Equ. 134, 343–367 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Luo, T., Xin, Z.P.: Nonlinear stability of shock fronts for a relaxation system in several space dimensions. J. Differ. Equ. 139, 365–408 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nishibata, S., Yu, S.H.: The asymptotic behavior of the hyperbolic conservation laws with relaxation on the quarter-plane. SIAM J. Math. Anal. 28, 304–321 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Natalini, R.: Recent Mathematical Results on Hyperbolic Relaxation Problems. Analysis of Systems of Conservation Laws, pp. 128–198. Chapman and Hall/CRC, Boca Raton (1999)

    Google Scholar 

  13. Nishihara, K.: Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J. Differ. Equ. 131, 171–188 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Smoller, J.: Shock Waves and Reaction–Diffusion Equations (A Series of Comprehensive Studies in Mathematics), vol. 258. Springer, Berlin (1983)

    Book  Google Scholar 

  15. Douglis, A.: Existence theorems for hyperbolic systems. Commun. Pure Appl. Math. 5, 119–154 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  16. John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math. 27, 377–405 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lattanzio, C., Marcati, P.: The zero relaxation limit for the hydrodynamic Whitham traffic flow model. J. Differ. Equ. 141, 150–178 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhu, C.: Asymptotic behavior of solutions for \(p\)-system with relaxation. J. Differ. Equ. 180, 272–306 (2002). doi:10.1006/jdeq.2001.4063

    Article  MathSciNet  Google Scholar 

  19. Nishihara, K., Wang, W., Yang, T.: \(L_p\)-Convergence rate to nonlinear diffusion waves for \(p\)-system with damping. J. Differ. Equ. 161, 191–218 (2000). doi:10.1006/jdeq.1999.3703

    Article  MathSciNet  MATH  Google Scholar 

  20. Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143, 599–605 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marcati, P., Mei, M.: Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping. Quat. Appl. Math. LVII(4), 763–784 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Hsiao, L., Liu, T.-P.: Nonlinear diffusion phenomena of nonlinear hyperbolic system. Chin. Ann. Math. Ser. B 14, 465–480 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Hsiao, L., Luo, T.: Nonlinear diffusion phenomena of solutions for the system of compressibile adiabatic flow through porous media. J. Differ. Equ. 125, 329–365 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mei, M.: Best asymptotic profile for hyperbolic \(p\)-system with damping. SIAM J. Math. Anal. Appl. Math. 42(1), 1–23 (2010). doi:10.1137/090756594

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, T., Zhu, C.: Existence and non-existence of global smooth solutions for \(p\)-system with relaxation. J. Differ. Equ. 161, 321–336 (2000). doi:10.1006/jdeq.2000.3710

    Article  MathSciNet  MATH  Google Scholar 

  26. Dafermos, C.: A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys. 46, 294–307 (1995)

    MathSciNet  MATH  Google Scholar 

  27. Chen, S., Huang, D., Han, X.: The generalized Riemann problem for first order quasilinear hyperbolic systems of conservation laws \(I\). Bull. Korean Math. Soc. 46(3), 409–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gu, C., Li T., Hou, Z.: The Cauchy problem of hyperbolic systems with discontinuous initial values. Collections of Scientific and Technological Papers (Shanghai: Mathematics Chemistry Ed.), pp. 55–65 (1960)

  29. Gu, C., Li, T., Hou, Z.: Discontinuous initial value problems for systems of quasilinear hyperbolic equations \(III\). Acta Math. Sinica. 12, 132–143 (1962)

    Google Scholar 

  30. Li, T., Yu, W.: Boundary Value Problems for Quasilinear Hyperbolic Systems (Duke University Mathematics Series). Duke University, Durham (1985)

    Google Scholar 

  31. Li, T., Yu, W.: The problem for quasilinear hyperbolic systems with discontinuous initial values. J. Eng. Math. 4(2), 1–12 (1987)

    Google Scholar 

  32. Shao, Z.: Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws \(II\). Math. Nachr. 281(6), 879–902 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ben-Artzi, M., Li, J.: Hyperbolic balance laws: Riemann invariants and hyperbolic balance laws. Numerische Mathematik 106, 369–425 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. LeFloch, P., Raviart, P.A.: An asymptotic expansion for the solution of the generalized Riemann problem \(I\). Gen. Theory. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(2), 179–207 (1988)

    MathSciNet  Google Scholar 

  35. Ben-Artzi, M.J., Falcovitz, J.: Generalized Riemann Problems in Computational Fluid Dynamics. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  36. Toro, E.F., Castro, C.E.: Solvers for the higher order Riemann problem for hyperbolic balance laws. J. Comp. Phys. 227(4), 2481–2513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Janenko, N.N.: Compatibility theory and methods of integration of systems of nonlinear partial differential equation. Proc. of the Fourth All-Union Math. Cong. (Leningrad), Nauka, Leningrad, pp. 247–252 (1964)

  38. Fusco, D., Manganaro, N.: Riemann invariants-like solutions for a class of rate-type materials. Acta Mechanica 105, 23–32 (1994). doi:10.1007/BF01183939

    Article  MathSciNet  MATH  Google Scholar 

  39. Fusco, D., Manganaro, N.: A method for determining exact solutions to a class of nonlinear models based on introduction of differential constraints. J. Math. Phys. 35(7), 3659–3669 (1994). doi:10.1063/1.530439

    Article  MathSciNet  MATH  Google Scholar 

  40. Fusco, D., Manganaro, N.: A method for finding exact solutions to hyperbolic systems of first order PDEs. IMA J. Appl. Math. 57, 223–242 (1996). doi:10.1093/imamat/57.3.223

    Article  MathSciNet  MATH  Google Scholar 

  41. Fusco, D., Manganaro, N.: A reduction approach for determining generalized simple waves. ZAMP 59, 63–75 (2008). doi:10.1007/s00033-006-5128-1

    Article  MathSciNet  MATH  Google Scholar 

  42. Manganaro, N., Meleshko, S.V.: Reduction procedure and generalized simple waves for systems written in the Riemann variables. Nonlinear Din. 30(1), 87–102 (2002). doi:10.1023/A:1020341610639

    Article  MathSciNet  MATH  Google Scholar 

  43. Meleshko, S.V., Shapeev, V.P.: An application of the differential constraints method for the two-dimensional equations of gas dynamics. Prikl. Matem. Mech. 63(6), 909–916 (1999). (English transl. in J. Appl. Maths Mechs 63(6), 885–891)

  44. Raspopov, V.E., Shapeev, V.P., Yanenko, N.N.: The application of the method of differential constraints to one-dimensional gas dynamics equations. Izvestiya V. U. Z. Matematica 11, 69–74 (1974)

    Google Scholar 

  45. Curró, C., Fusco, D., Manganaro, N.: Hodograph transformation and differential constraints for wave solutions to \(2 \times 2\) quasilinear hyperbolic nonhomogeneous systems. J. Phys. A Math. Theor. 45(19), 195207 (2012). doi:10.1088/1751-8113/45/19/195207

    Article  MathSciNet  MATH  Google Scholar 

  46. Curró, C., Fusco, D., Manganaro, N.: An exact description of nonlinear wave interaction processes ruled by \(2 \times 2\) hyperbolic systems. ZAMP 64(4), 1227–1248 (2013). doi:10.1007/s00033-012-0282-0

    MathSciNet  MATH  Google Scholar 

  47. Curró, C., Fusco, D., Manganaro, N.: Exact description of simple wave interactions in multicomponent chromatography. J. Phys. A Math. Theor. 48, 015201 (2015). doi:10.1088/1751-8113/48/1/015201

    Article  MathSciNet  MATH  Google Scholar 

  48. Curró, C., Fusco, D., Manganaro, N.: A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines. J. Phys. A Math. Theor. 44(33), 335205 (2011). doi:10.1088/1751-8113/44/33/335205

    Article  MathSciNet  MATH  Google Scholar 

  49. Curró, C., Fusco, D., Manganaro, N.: Differential constraints and exact solution to Riemann problems for a traffic flow model. Acta Appl Math. 122(1), 167–178 (2012). doi:10.1007/s10440-012-9735-x

    MathSciNet  MATH  Google Scholar 

  50. Curró, C., Manganaro, N.: Riemann problems and exact solutions to a traffic flow model. J. Math. Phys. 54(17), 071503 (2013). doi:10.1063/1.4813473

    Article  MathSciNet  MATH  Google Scholar 

  51. Manganaro, N., Pavlov, M.V.: The constant astigmatism equation. New exact solution. J. Phys. A Math. Theor. 47(7), 075203 (2014). doi:10.1088/1751-8113/47/7/075203

    Article  MathSciNet  MATH  Google Scholar 

  52. Curró, C., Fusco, D., Manganaro, N.: Exact solutions in idela chromatography via differential constraints method. AAPP Atti della Accademia Peloritana dei Pericolanti Classe di Scienze Fisiche Matematiche e Naturali 93(1), A2 (2015). doi:10.1478/AAPP.931A2

    Google Scholar 

  53. Yao, Z., Zhu, C.: \(L^P\)-convergence rate to diffusion waves for \(p\)-system with relaxation. J. Math. Anal Appl. 276, 497–515 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  54. Luo, T., Natalini, R., Yang, T.: Global BV solutions to a \(p\)-system with relaxation. J. Differ. Equ. 162, 174–198 (2000). doi:10.1006/jdeq.1999.3697

    Article  MathSciNet  MATH  Google Scholar 

  55. Yang, T., Zhao, H., Zhu, C.: Asymptotic behavior of solutions to a hyperbolic system with relaxation and boundary effect. J. Differ. Equ. 163, 348–380 (2000). doi:10.1006/jdeq.1999.3741

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by Italian National Group of Mathematical Physics (GNFM-INdAM).

Author information

Authors and Affiliations

  1. MIFT, University of Messina, viale F. Stagno D’Alcontres 31, 98166, Messina, Italy

    Carmela Curró & Natale Manganaro

Authors
  1. Carmela Curró
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natale Manganaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natale Manganaro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Curró, C., Manganaro, N. Generalized Riemann problems and exact solutions for p-systems with relaxation. Ricerche mat 65, 549–562 (2016). https://doi.org/10.1007/s11587-016-0274-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-016-0274-z

Keywords

Mathematics Subject Classification

Search

Navigation