Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-01T19:46:35.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Global weak solutions for elastic equations with damping and different end states

Published online by Cambridge University Press:  14 November 2011

Tao Luo  and
Tong Yang
Tao Luo
Affiliation:
Institute of Mathematics, Academia Sinica, China; Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Tong Yang
Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong e-mail: matyang@cityu.edu.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study global weak solutions for elastic equations with damping using the compensated compactness method. When the two end states at ± ∞ are not equal, the selfsimilar solutions for the corresponding parabolic equation are used to get the entropic estimates for both the L∞ and L2 cases.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 797 - 807
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chueh, K. N., Conley, C. C. and Smoller, J. A.. Positively invariant regions for the systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 371–411.CrossRefGoogle Scholar
2Dafermos, C.. A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys. 46 Special Issue (1995), 294–307.Google Scholar
3Ding, X., Chen, G. and Luo, P.. Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. I. Acta Math. Sci. 4 (1985), 483–500.Google Scholar
4DiPerna, R. J.. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.CrossRefGoogle Scholar
5Hsiao, L. and Liu, T. P.. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys. 143 (1992), 599–605.CrossRefGoogle Scholar
6Lin, P.. Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics. Trans. Amer. Math. Soc. 329 (1992), 377–413.CrossRefGoogle Scholar
7Murat, F.. Compacite par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489–507.Google Scholar
8Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, IV, ed. Knops, R. J.. Pitman Research Notes in Mathematics, Vol. 4, 136–92 (Harlow: Longman, 1979).Google Scholar
9Yosida, K.. Functional Analysis (New York: Springer, 1968).CrossRefGoogle Scholar
10Zhu, J.. Convergence of the viscosity solutions for the system of nonlinear elasticity. J. Math. Anal. Appl. 209 (1997), 585–604.CrossRefGoogle Scholar