Skip to main content
SpringerLink
Log in

Entropy Solutions in L ∞ for the Euler Equations in Nonlinear Elastodynamics and Related Equations

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

Abstract.

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L ∞ for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.

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. Chen, G.-Q., Dafermos, C.M.: Global solutions in L ∞ for a system of conservation laws of viscoelastic materials with memory. J. Partial Diff. Eqs. 10, 369–383 (1997)

    Google Scholar 

  2. Chen, G.-Q., Dafermos, C.M.: The vanishing viscosity method in one-dimensional thermoelasticity. Trans. Amer. Math. Soc. 347, 531–541 (1995)

    Google Scholar 

  3. Chen, G.-Q., Frid, H.: Decay of entropy solutions of nonlinear conservation laws. Arch. Ration. Mech. Anal. 146, 95–127 (1999)

    Google Scholar 

  4. Chen, G.-Q., Frid, H.: On the compressible Euler equations in thermoelasticity. Mat. Contemp. 15, 67–86 (1998)

    Google Scholar 

  5. Chen, G.-Q., LeFloch, Ph.: Compressible Euler equations with general pressure law. Arch. Ration. Mech. Anal. 153, 221–259 (2000)

    Google Scholar 

  6. Coleman, B.D., Gurtin, M.E.: Waves in materials with memory II: On the growth and decay of one dimensional acceleration waves. Arch. Rational Mech. Anal. 19, 239–265 (1965)

    Google Scholar 

  7. Coleman, B.D., Gurtin, M.E., Herrera, I.R.: Waves in materials with memory I. Arch. Ration. Mech. Anal. 19, 1–19 (1965)

    Google Scholar 

  8. Dafermos, C.M.: Dissipation in materials with memory. In: Viscoelasticity and Rheology A.S. Lodge, J.A. Nohel, M. Renardy, (eds.), New York: Academic Press, 1985, pp. 221–234

  9. Dafermos, C.M.: Estimates for conservation laws with little viscosity. SIAM J. Math. Anal. 18, 409–421 (1987)

    Google Scholar 

  10. Dafermos, C.M.: Solutions in L ∞ for a conservation law with memory. In: Analyse Mathématique et Applications, Gauthier-Villars, Paris, 1988, pp. 117–128

  11. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I). Acta Math. Sci. 5, 415–432 (1985); 7, 467–480 (1987) in Chinese

    MathSciNet  MATH  Google Scholar 

  12. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (II). Acta Math. Sci. 5, 433–472 (1985); 8, 61–94 (1988) in Chinese

    Google Scholar 

  13. DiPerna, R.J.: Convergence of approximate solutions to conservation law. Arch. Ration. Mech. Anal. 82, 27–70 (1983)

    Google Scholar 

  14. Gripenberg, G.: Compensated compactness and one-dimensional elastodynamics. Ann. Scuola Norm. Sup. Pisa Cl Sci. (4) 22, 227–240 (1995)

  15. Lin, P.: Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics. Trans. Am. Math. Soc. 329, 377–413 (1992)

    Google Scholar 

  16. MacCamy, R.C.: A model for one-dimensional nonlinear viscoelasticity. Quart. Appl. Math. 35, 22–33 (1977)

    Google Scholar 

  17. Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 5, 489–507 (1978)

    Google Scholar 

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

    Google Scholar 

  19. Nohel, J.A., Rogers, R.C., Tzavaras, A.E.: Weak solutions for a nonlinear system in viscoelasticity. Commun. Partial Diff. Eqs. 13, 97–127 (1988)

    Google Scholar 

  20. Serre, D.: La compacité par compensation pour les systèms hyperboliques non linéaires de deux équations à une dimension despace. J. Math. Pures Appl. 65, 433–468 (1986)

    Google Scholar 

  21. Shearer, J.W.: Global existence and compactness in L p for the quasi-linear wave equation. Comm. Partial Diff. Eqs. 19, 1829–1877 (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA

    Gui-Qiang Chen

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P. R. China

    Bang-He Li

  3. Department of Mathematics, Standford University, Stanford, CA, 94305-2125

    Tian-Hong Li

Authors
  1. Gui-Qiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bang-He Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tian-Hong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Qiang Chen.

Additional information

Communicated by C.M. Dafermos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, GQ., Li, BH. & Li, TH. Entropy Solutions in L ∞ for the Euler Equations in Nonlinear Elastodynamics and Related Equations. Arch. Rational Mech. Anal. 170, 331–357 (2003). https://doi.org/10.1007/s00205-003-0284-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-003-0284-3

Keywords

Search

Navigation