Skip to main content
SpringerLink
Log in
Book cover

Amorphous Polymers and Non-Newtonian Fluids pp 33–55Cite as

Solutions with Shocks for Conservation Laws with Memory

  • Chapter

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 6))

Abstract

The equations of motion of a one-dimensional body with unit reference density and zero body force, in Lagrangian coordinates, read

$$\left\{ {\begin{array}{*{20}{c}} {{{\partial }_{t}}u(x,t) - {{\partial }_{x}}v(x,t) = 0} \\ {{{\partial }_{t}}v(x,t) - {{\partial }_{x}}\sigma (x,t) = 0} \\ \end{array} } \right.$$
(0.1)

where u is deformation gradient, v is velocity, and denotes σ stress.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Coleman, B. D., and Gurtin, M. E., Waves in materials with memory, II. On the growth and decay of one-dimensional acceleration waves. Arch. Rational Mech. Anal. 19 (1965), 266–298.

    Article  Google Scholar 

  2. Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana U. Math. J. 26 (1977), 1097–1119.

    Article  Google Scholar 

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

    Google Scholar 

  4. Dafermos, C. M., Development of singularities in the motion of materials with fading memory. Arch. Rational Mech. Anal. 91 (1986), 193–205.

    Article  Google Scholar 

  5. Dafermos, C. M., Solutions in L∞ for a conservation law with memory. (To appear.)

    Google Scholar 

  6. Dafermos, C. M., and Nohel, J. A., A nonlinear hyperbolic Volterra equation in viscoelasticity. Am. J. Math. Suppl. dedicated to P. Hartman (1981), 87–116.

    Google Scholar 

  7. DiPerna, R. J., Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.

    Article  Google Scholar 

  8. Filippov, A. F., Differential equations with discontinuous right-hand side. Mat. Sbornik (N. S.) 51 (93) (1960), 99–128.

    Google Scholar 

  9. Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697–715.

    Article  Google Scholar 

  10. Glimm, J., and Lax, P. D., Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs A. M. S. 101 (1970).

    Google Scholar 

  11. Greenberg, J. M., The existence and qualitative properties of solutions of = 0. J. Math. Anal. Appl. 42 (1973), 205–220.

    Article  Google Scholar 

  12. Hrusa, J. W., and Nohel, J. A., The Cauchy problem in one-dimensional nonlinear viscoelasticity. J. Diff. Eqs. 59 (1985), 388–412.

    Article  Google Scholar 

  13. Klainerman, S., and Majda, A., Formation of singularities for wave equations including the nonlinear vibrating string. Comm. Pure Appl. Math. 33 (1980), 241–263.

    Article  Google Scholar 

  14. Lax, P. D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Physics 5 (1964), 611–613.

    Article  Google Scholar 

  15. MacCamy, R. C., A model for one-dimensional, nonlinear viscoelasticity. Quart. Appl. Math. 35 (1977), 21–33.

    Google Scholar 

  16. Malek-Madani, R., and Nohel, J. A., Formation of singularities for a conservation law with memory. SIAM J. Math. Anal. 16 (1985), 530–540.

    Article  Google Scholar 

  17. Nohel, J. A., A nonlinear conservation law with memory. Volterra and Functional Differential Equations, pp. 91–123 (K. B. Hannsgen, T. L. Herdman and R. L. Wheeler, eds.) Marcel Dekker 1982.

    Google Scholar 

  18. Nohel, J. A., and Renardy, M., Development of singularities in nonlinear viscoelasticity. This volume.

    Google Scholar 

  19. Oleinik, O. A., Discontinuous solutions of non-linear differential equations. Usp. Mat. Nauk (N. S.) 12(3) (1957), 3–73.

    Google Scholar 

  20. Rascle, M., Un resultat de “compacite par compensation a coefficients variables.” Application a l’elasticite nonlineaire. Compt. Rend. Acad. Sci. Paris, Serie I, 302 (1986), 311–314.

    Google Scholar 

  21. Renardy, M., Recent developments and open problems in the mathematical theory of viscoelasticity. Viscoelasticity and Rheology, pp. 345–360 (A. Lodge, J. A. Nohel, and M. Renardy, eds.) Academic Press 1985.

    Google Scholar 

  22. Truesdell, C. A., and Noll, W., The Nonlinear Field Theories of Mechanics. Handbuch der Physik III/3. Springer-Verlag, Berlin 1965.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lefschetz Center for Dynamical Systems Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912, USA

    C. M. Dafermos

Authors
  1. C. M. Dafermos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The Institute for Mathematics and Its Applications, 55455, Minneapolis, Minnesota, USA

    Constantine Dafermos , J. L. Ericksen  & David Kinderlehrer ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer Science+Business Media New York

About this chapter

Cite this chapter

Dafermos, C.M. (1987). Solutions with Shocks for Conservation Laws with Memory. In: Dafermos, C., Ericksen, J.L., Kinderlehrer, D. (eds) Amorphous Polymers and Non-Newtonian Fluids. The IMA Volumes in Mathematics and Its Applications, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1064-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1064-1_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7000-3

  • Online ISBN: 978-1-4612-1064-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation