Skip to main content
SpringerLink
Log in

Formation of \(\delta \)-shock waves in isentropic fluids

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper studies a Riemann problem for the isentropic Euler equations and addresses two gaps in the literature concerning positive pressures. Such study is made using a product of distributions and a solution concept that extends the classical solution concept. Under certain conditions, even for positive pressures, it is shown that the Riemann problem has solutions, which are \(\delta \)-shock waves. As particular cases, this work examines polytropic gases, pressureless gases, and Chaplygin gases.

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

Similar content being viewed by others

References

  1. Ambrosio, L., Crippa, G., Figalli, A., Spinolo, L.A.: Some new well-posedness results for continuity and transport equations and applications to the chromatography system. SIAM J. Math. Anal. 41(5), 1890–1920 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Bento, M.C., Bertolami, O., Sen, A.A.: Generalized Chaplygin gas, accelerated expansion, and dark-energy-matter unification. Phys. Rev. D 66(4), 043507 (2002)

    Google Scholar 

  3. Brenier, Y., Grenier, E.: Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35(6), 2317–2328 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Bressan, A., Rampazzo, F.: On differential systems with vector valued impulsive controls. Bull. Un. Mat. Ital. 7, 641–656 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Caprino, S., Esposito, R., Marra, R., Pulvirenti, M.: Hydrodynamic limits of the Vlasov equation. Commun. Partial. Differ. Equ. 18(5–6), 805–820 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Chaplygin, S.: On gas jets. Sci. Mem. Mosc. Univ. Math. Phys. 21, 1–121 (1904)

    Google Scholar 

  7. Chen, G.-Q., Liu, H.: Formation of \(\delta \)-shocks and vacuum states in the vanishing pressure limit of solutions to Euler equations for isentropic fluid. SIAM J. Math. Anal. 34(3), 925–938 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Cheng, H.: Delta shock waves for a linearly degenerate hyperbolic systems of conservation laws of Keyfitz-Kranzer type. Adv. Math. Phys. 2013 (2013)

  9. Cheng, H., Yang, H.: Delta shock waves in chromatography equations. J. Math. Anal. Appl. 380(2), 475–485 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  11. Colombeau, J.F.: Multiplication of Distributions: A Tool in Mathematics, Numerical Engineering and Theoretical Physics. Springer, Berlin (1992)

    MATH  Google Scholar 

  12. Colombeau, J.F., Le Roux, A.: Multiplication of distributions in elasticity and hydrodynamics. J. Math. Phys. 29(2), 315–319 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Dal Maso, G., Le Flock, P., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Danilov, V.G., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245(12), 3704–3734 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Danilov, V.G., Shelkovich, V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63(3), 401–427 (2005)

    MathSciNet  Google Scholar 

  16. Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of \(\delta \)-shock waves in conservation law systems. J. Differ. Equ. 211(2), 333–381 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Daw, D.A.E., Nedeljkov, M.: Shadow waves for pressureless gas balance laws. Appl. Math. Lett. 57, 54–59 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Earnshaw, S.: On the mathematical theory of sound. Philos. Trans. 150, 1150–1154 (1858)

    Google Scholar 

  19. François, B., François, J.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Commun. Partial Differ. Equ. 24(11–12), 2173–2190 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Guo, L., Li, T., Yin, G.: The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term. J. Math. Anal. Appl. 455(1), 127–140 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Guo, L., Pan, L., Yin, G.: The perturbed Riemann problem and delta contact discontinuity in chromatography equations. Nonlinear Anal. TMA 106, 110–123 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Hilden, S.T., Nilsen, H.M., Raynaud, X.: Study of the well-posedness of models for the inaccessible pore volume in polymer flooding. Transp. Porous Med. 114(1), 65–86 (2016)

    MathSciNet  Google Scholar 

  23. Huang, F.: Weak solutions to pressureless type system. Commun. Partial Differ. Equ. 30(3), 283–304 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Kalisch, H., Mitrovic, D.: Singular solutions for the shallow-water equations. IMA J. Appl. Math. 77(3), 340–350 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Kalisch, H., Mitrovic, D.: Singular solutions of a fully nonlinear \(2\times 2\) system of conservation laws. Proc. Edinb. Math. Soc. 55(3), 711–729 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Kalisch, H., Mitrovic, D., Teyekpiti, V.: Delta shock waves in shallow water flow. Phys. Lett. A 381(13), 1138–1144 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Kalisch, H., Teyekpiti, V.: A shallow-water system with vanishing buoyancy. Appl. Anal., 1–15 (2018)

  28. Kamenshchik, A., Moschella, U., Pasquier, V.: An alternative to quintessence. Phys. Lett. B 511(2–4), 265–268 (2001)

    MATH  Google Scholar 

  29. Karman, T.: Compressibility effects in aerodynamics. J. Aeronaut. Sci. 8(9), 337–365 (1941)

    MathSciNet  MATH  Google Scholar 

  30. Keyfitz, B.L., Kranzer, H.C.: Spaces of weighted measures for conservation laws with singular shock solutions. J. Differ. Equ. 118(2), 420–451 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34(4), 481–524 (1981)

    MathSciNet  MATH  Google Scholar 

  32. König, H.: Neue Begründung der Theorie der “Distributionen” von L. Schwartz. Math. Nachr. 9(3), 129–148 (1953)

    MATH  Google Scholar 

  33. Korchinski, D.J.: Solution of a Riemann problem for a \(2\times 2\)system of conservation laws possessing no classical weak solution. PhD thesis, Adelphi University (1977)

  34. Lax, P.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)

    MathSciNet  MATH  Google Scholar 

  35. Lu, Y.G.: Global weak solutions for the chromatography system. Isr. J. Math. 225(2), 721–741 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Mazzotti, M., Tarafder, A., Cornel, J., Gritti, F., Guiochon, G.: Experimental evidence of a delta-shock in nonlinear chromatography. J. Chromatogr. A 1217(13), 2002–2012 (2010)

    Google Scholar 

  37. Mitrovic, D., Bojkovic, V., Danilov, V.G.: Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process. Math. Methods Appl. Sci. 33(7), 904–921 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Mitrovic, D., Nedeljkov, M.: Delta shock waves as a limit of shock waves. J. Hyberbolic Differ. Equ. 4(4), 629–653 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Nedeljkov, M.: Unbounded solutions to some systems of conservation laws-split delta shock waves. Mat. Vesnik 54(3–4), 145–149 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Nedeljkov, M.: Delta and singular delta locus for one dimensional systems of conservation laws. Math. Methods Appl. Sci. 27(8), 931–955 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Nedeljkov, M., Oberguggenberger, M.: Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J. Math. Anal. Appl. 344(2), 1143–1157 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Oberguggenberger, M.: Multiplication of distribution and applications to partial differential equations. Pitman Res, vol. 259. Notes Math. Ser. Longman, Harlow (1992)

  43. Oelschläger, K.: On the connection between Hamiltonian many-particle systems and the hydrodynamical equation. Arch. Ration. Mech. Anal. 115(4), 297–310 (1991)

    MathSciNet  MATH  Google Scholar 

  44. Oelschläger, K.: An integro-differential equation modelling a Newtonian dynamics and its scaling limit. Arch. Ration. Mech. Anal. 137(2), 99–134 (1997)

    MathSciNet  MATH  Google Scholar 

  45. Paiva, A.: New \(\delta \)-shock waves in the \(p\)-system: a distributional product approach. Math. Mech. Solids 25(3), 619–629 (2020)

    MathSciNet  Google Scholar 

  46. Sarrico, C.O.R.: About a family of distributional products important in the applications. Port. Math. 45(3), 295–316 (1988)

    MathSciNet  MATH  Google Scholar 

  47. Sarrico, C.O.R.: Distributional products and global solutions for nonconservative inviscid Burgers equation. J. Math. Anal. Appl. 281(2), 641–656 (2003)

    MathSciNet  MATH  Google Scholar 

  48. Sarrico, C.O.R.: Collision of delta-waves in a turbulent model studied via a distributional product. Nonlinear Anal. Theory Methods Appl. 73(9), 2868–2875 (2010)

    MATH  Google Scholar 

  49. Sarrico, C.O.R.: The multiplication of distributions and the Tsodyks model of synapses dynamics. Int. J. Math. Anal. 6(21), 999–1014 (2012)

    MathSciNet  MATH  Google Scholar 

  50. Sarrico, C.O.R.: A distributional product approach to \(\delta \)-shock wave solutions for a generalized pressureless gas dynamics system. Int. J. Math. 25(1), 1450007 (2014)

    MathSciNet  MATH  Google Scholar 

  51. Sarrico, C.O.R., Paiva, A.: Products of distributions and collision of a \(\delta \)-wave with a \(\delta ^{\prime }\)-wave in a turbulent model. J. Nonlinear Math. Phys. 22(3), 381–394 (2015)

    MathSciNet  MATH  Google Scholar 

  52. Sarrico, C.O.R., Paiva, A.: The multiplication of distributions in the study of a Riemann problem in fluid dynamics. J. Nonlinear Math. Phys. 24(3), 328–345 (2017)

    MathSciNet  MATH  Google Scholar 

  53. Sarrico, C.O.R., Paiva, A.: New distributional travelling waves for the nonlinear Klein–Gordon equation. Differ. Integral Equ. 30(11–12), 853–878 (2017)

    MathSciNet  MATH  Google Scholar 

  54. Sarrico, C.O.R., Paiva, A.: Newton’s second law and the multiplication of distributions. J. Math. Phys. 59(1), 013505 (2018)

    MathSciNet  MATH  Google Scholar 

  55. Schandarin, S.F., Zel’dovich, Y.B.: The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys. 61(2), 185–220 (1989)

    MathSciNet  Google Scholar 

  56. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1965)

    Google Scholar 

  57. Sever, M.: Distribution Solutions of Nonlinear Systems of Conservation Laws. Mem. Amer. Math. Soc., vol. 190. AMS, Providence (2007)

    MATH  Google Scholar 

  58. Shelkovich, V.M.: \(\delta \)-and \(\delta ^{\prime }\)-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes. Russ. Math. Surv+, 63(3):73–149 (2008). Translated from Usp. Mat. Nauk 63(3), 473–546 (2008)

  59. Shen, C.: The Riemann problem for the Chaplygin gas equations with a source term. ZAMM-Z. Angew. Math. Mech. 96(6), 681–695 (2016)

    MathSciNet  Google Scholar 

  60. Shen, C.: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IMA J. Appl. Math. 81(1), 76–99 (2016)

    MathSciNet  MATH  Google Scholar 

  61. Shen, C., Sun, M.: Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J. Differ. Equ. 249(12), 3024–3051 (2010)

    MathSciNet  MATH  Google Scholar 

  62. Shen, C., Sun, M.: A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system. Int. J. Non-Linear Mech. 105, 105–112 (2018)

    Google Scholar 

  63. Sheng, W., Zhang, T.: The Riemann Problem for the Transportation Equations in Gas Dynamics. Mem. Amer. Math. Soc., vol. 137. AMS, Providence (1999)

    MATH  Google Scholar 

  64. Sun, M.: Delta shock waves for the chromatography equations as self-similar viscosity limits. Q. Appl. Math. 69(3), 425–443 (2011)

    MathSciNet  MATH  Google Scholar 

  65. Sun, M.: The multiplication of distributions in the study of delta shock wave for the nonlinear chromatography system. Appl. Math. Lett. 96, 61–68 (2019)

    MathSciNet  MATH  Google Scholar 

  66. Tan, D., Zhang, T.: Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws: I. four-j cases. J. Differ. Equ. 111(2), 203–254 (1994)

    MathSciNet  MATH  Google Scholar 

  67. Tan, D., Zhang, T., Zheng, Y.: Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differ. Equ. 112(1), 1–32 (1994)

    MathSciNet  MATH  Google Scholar 

  68. Thein, F., Hantke, M.: Singular and selfsimilar solutions for Euler equations with phase transitions. Bull. Braz. Math. Soc. New Ser. 47(2), 779–786 (2016)

    MathSciNet  MATH  Google Scholar 

  69. Tsien, H.: Two dimensional subsonic flow of compressible fluids. J. Aeronaut. Sci. 6(10), 399–407 (1939)

    MathSciNet  MATH  Google Scholar 

  70. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1973)

    MATH  Google Scholar 

  71. Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252(11), 5951–5993 (2012)

    MathSciNet  MATH  Google Scholar 

  72. Zel’dovich, Y.B.: Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)

    Google Scholar 

Download references

Acknowledgements

Supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2019. I cordially thank the referees for the helpful comments and valuable suggestions which significantly improved this paper.

Author information

Authors and Affiliations

  1. CMAFCIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016, Lisbon, Portugal

    Adelino Paiva

Authors
  1. Adelino Paiva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adelino Paiva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paiva, A. Formation of \(\delta \)-shock waves in isentropic fluids. Z. Angew. Math. Phys. 71, 110 (2020). https://doi.org/10.1007/s00033-020-01332-6

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01332-6

Keywords

Mathematics Subject Classification

Search

Navigation