Skip to main content
SpringerLink
Log in

Shadow Waves: Entropies and Interactions for Delta and Singular Shocks

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

This paper deals with two key problems for delta (or singular) shock solutions to systems of conservation laws: that of their entropy admissibility conditions (which is connected to the notorious uniqueness problem) and that of their interaction. We choose to represent systems of conservation laws by nets of functions which are piecewise constant (or constant with respect only to the space variable), here called shadow waves. All the calculations can then be done on each element of such nets using only the usual Rankine–Hugoniot conditions. A 3 × 3 pressureless gas dynamics model is the main example in the paper.

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. Bouchut F., James F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. PDE 24, 2173–2190 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Bressan A.: Hyperbolic Systems of Conservation Laws. Oxford University Press, New York (2000)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen G.-Q., Liu H.: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Physica D 189, 141–165 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Chen G-Q., Wang D. (2002). The Cauchy problem for the Euler equations for compressible fluids. In: Friedlander S., Serre D. (eds)., Handbook of Mathematical Fluid Dynamics, Volume I, Elsevier, 421–543 (2002)

  7. Dafermos C.: Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Heidelberg (2000)

    MATH  Google Scholar 

  8. Danilov V.G., Shelkovich V.M.: Dynamics of propagation and interaction of shock waves in conservation law systems. J. Differ. Equations 211, 333–381 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Danilov V.G., Shelkovich V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 29, 401–427 (2005)

    MathSciNet  Google Scholar 

  10. E W., Rykov Y.G., Sinai Ya.G.: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys. 177, 349–380 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Huang F.: Weak solution to pressureless type system. Comm. Partial Differential Equations 30, 283–304 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. LeFloch, P.: An existence and uniqueness result for two nonstrictly hyperbolic systems. In: IMA Volumes in Math. and its Appl., Keyfitz, B.L., Shearer, M. (eds) Nonlinear evolution equations that change type, Springer Verlag, Vol 27, 126–138 (1990)

  14. Mitrović D., Nedeljkov M.: Delta shock waves as a limit of shock waves. J. Hyp. Diff. Equ. 4, 629– (2007)

    Article  MATH  Google Scholar 

  15. Nedeljkov M.: Unbounded solutions to some systems of conservation laws - split delta shocks waves. Mat. Ves., 54, 145–149 (2002)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Nedeljkov M.: Singular shock waves in interactions. Quart. Appl. Math. 66, 281–302 (2008)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Panov, E.Yu.: On a representation of the prolonged systems for a scalar conservation law and on higher-order entropies. Differential Equations 44, 1758–1763 (2008). Translated from Differentsialnye Uravneniya 44, 1694–1699 (2008)

    Google Scholar 

  20. Panov E.Yu., Shelkovich V.M.: δ′-Shock waves as a new type of solutions to systems of conservation laws. J. Differ. Equations 228, 49–86 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Serre D.: Systems of Conservation Laws I. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  22. Shelkovich V.M.: The Riemann problem admitting δ-, δ′-shocks, and vacuum states (the vanishing viscosity approach). J. Difer. Equations 231, 459–500 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yang H., Sun W.: The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws. Nonlinear Anal. 67, 3041–3049 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad Trg D. Obradovića 4, 21000, Novi Sad, Serbia

    Marko Nedeljkov

Authors
  1. Marko Nedeljkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marko Nedeljkov.

Additional information

Communicated by A. Bressan

Partially supported by Grant No. OI 144016, Ministry of Science, Republic of Serbia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nedeljkov, M. Shadow Waves: Entropies and Interactions for Delta and Singular Shocks. Arch Rational Mech Anal 197, 489–537 (2010). https://doi.org/10.1007/s00205-009-0281-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-009-0281-2

Keywords

Search

Navigation