Skip to main content
SpringerLink
Log in

Long time asymptotics of periodic generalized entropy solutions of scalar conservation laws

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We prove that the periodic generalized entropy solution of a one-dimensional conservation law converges in time to a traveling wave. In this case, the flow function is linear on the minimal interval containing the essential image of the traveling wave profile and the wave velocity coincides with the angular coefficient of the flow function bounded on this interval.

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. S. N. Kruzhkov, “First order quasilinear equations with several independent variables,” Mat. Sb. 81 (123) (2), 228–255 (1970).

    MathSciNet  MATH  Google Scholar 

  2. E. Yu. Panov, “Measure-Valued solutions of the Cauchy problem for first-order quasilinearwith an unbounded domain of dependence on the initial data,” Dinamika Sploshnoi Sredy 88, 102–108 (1988).

    MathSciNet  MATH  Google Scholar 

  3. S. N. Kruzhkov and E. Yu. Panov, “First-order conservative quasilinear laws with an infinite domain of dependence on the initial data,” Dokl. Akad. Nauk SSSR 314 (1), 79–84 (1990) [Soviet Math. Dokl. 42 (2), 316–321 (1991)].

    MathSciNet  MATH  Google Scholar 

  4. S. N. Kruzhkov and E. Yu. Panov, “Osgood’s type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order,” Ann. Univ. Ferrara Sez. VII (N. S. ) 40, 31–54 (1994).

    MathSciNet  MATH  Google Scholar 

  5. E. Yu. Panov, “Uniqueness of the solution of the Cauchy problem for a first-order quasilinear equation with an admissible strictly convex entropy,” Mat. Zametki 55 (5), 116–129 (1994) [Math. Notes 55 (5–6), 517–525 (1994)].

    MathSciNet  MATH  Google Scholar 

  6. E. Yu. Panov, “On the theory of generalized entropy sub-and supersolutions of the Cauchy problem for a first-order quasilinear equation,” Differ. Uravn. 37 (2), 252–259 (2001) [Differ. Equations 37 (2), 272–280 (2001)].

    MathSciNet  Google Scholar 

  7. E. Yu. Panov, “On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws,” Netw. Heterog. Media 11 (2), 349–367 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Tartar, “Compensated compactness and applications to partial differential equations,” in Res. Notes in Math., Vol. 39: Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, Vol. IV (Pitman, Boston, MA, 1979), pp. 136–212.

    Google Scholar 

  9. G. Q. Chen and Y. G. Lu, “The study on application way of the compensated compactness theory,” Chinese Sci. Bull. 34 (1), 15–19 (1989).

    MathSciNet  MATH  Google Scholar 

  10. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 2010), Vol. 325.

    Book  Google Scholar 

  11. E. Panov, “On weak completeness of the set of entropy solutions to a scalar conservation law,” SIAM J. Math. Anal. 41 (1), 26–36 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. V. Lysukho and E. Yu. Panov, “Renormalized entropy solutions of the Cauchy problem for first-order quasilinear conservation laws in the class of periodic functions,” Probl. Matem. Anal. 59, 25–42 (2011).

    MathSciNet  MATH  Google Scholar 

  13. G.-Q. Chen and H. Frid, “Decay of entropy solutions of nonlinear conservation laws,” Arch. Ration. Mech. Anal. 146 (2), 95–127 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  14. C. M. Dafermos, “Long time behavior of periodic solutions to scalar conservation laws in several space dimensions,” SIAM J. Math. Anal. 45 (4), 2064–2070 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. R. J. DiPerna, “Measure-valued solutions to conservation laws,” Arch. Ration. Mech. Anal. 88 (3), 223–270 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Panov, “On weak completeness of the set of entropy solutions to a degenerate nonlinear parabolic equation,” SIAM J. Math. Anal. 44 (1), 513–535 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Murat, “L’injection du cône positif de H-1 dans W-1, q est compacte pour tout q < 2,” J. Math. Pures Appl. (9) 60 (3), 309–322 (1981).

    MathSciNet  MATH  Google Scholar 

  18. F. Murat, “Compacité par compensation,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (3), 489–507 (1978).

    MathSciNet  MATH  Google Scholar 

  19. E. Yu. Panov, “Existence of strong traces for generalized solutions of multidimensional scalar conservation laws,” J. Hyperbolic Differ. Equ. 2 (4), 885–908 (2005).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yaroslav Mudryi Novgorod State University, Novgorod, Russia

    E. Yu. Panov

  2. St. Petersburg State University, St. Petersburg, Russia

    E. Yu. Panov

Authors
  1. E. Yu. Panov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Yu. Panov.

Additional information

Original Russian Text © E. Yu. Panov, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 1, pp. 133–143.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panov, E.Y. Long time asymptotics of periodic generalized entropy solutions of scalar conservation laws. Math Notes 100, 113–122 (2016). https://doi.org/10.1134/S0001434616070105

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434616070105

Keywords

Search

Navigation