Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762

 
 

 

Global solutions of the equations of elastodynamics for incompressible materials


Author: David G. Ebin
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 50-59
MSC (1991): Primary 35L70, 35Q72, 73C50, 73D35
DOI: https://doi.org/10.1090/S1079-6762-96-00006-6
MathSciNet review: 1405969
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equations of the dynamics of an elastic material are a non-linear hyperbolic system whose unknowns are functions of space and time. If the material is incompressible, the system has an additional pseudo-differential term. We prove that such a system has global (classical) solutions if the initial data are small. This contrasts with the case of compressible materials for which F. John has shown that such solutions may not exist even for arbitrarily small data.


References [Enhancements On Off] (What's this?)

  • [ChKl] D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton, NJ 1993. MR 95k:83006
  • [Eb] D. G. Ebin, Global solutions of the equations of elastodynamics of incompressible neo-Hookean materials, Proc. Nat. Acad. Sci. 90 (1993), 3802--3805. MR 94b:35279
  • [EbSa] D. G. Ebin and R. Saxton, The initial value problem for elastodynamics of incompressible bodies, Arch. Rational Mech. Anal. 94 (1986), 15--38. MR 87h:58028
  • [EbSi1] D. G. Ebin and S. R. Simanca, Small deformations of incompressible bodies with free boundary, Comm. Partial Diff. Eq. 15 (1990), 1589--1616. MR 91j:35288
  • [EbSi2] ------, Deformations of incompressible bodies with free boundary, Arch. Rat. Mech. Anal. 120 (1992), 61--97. MR 94a:73017
  • [Hö] L. Hörmander, Non-linear hyperbolic differential equations, Lecture Notes, University of Lund, Sweden, 1988.
  • [HrRe] W. J. Hrusa and M. Renardy, An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials, Arch. Rat. Mech. Anal. 102 (1988), 95--118. MR 89i:35096
  • [Jo1] F. John, Formation of singularities in elastic waves, Lect. Notes in Physics, vol. 195, Springer-Verlag, 1984, pp. 194--210. MR 85h:73018
  • [Jo2] ------, Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29--51. MR 83d:35096
  • [Jo3] ------, Nonlinear wave equations, formation of singularities (Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989), University Lecture Series, vol. 2, Amer. Math. Soc., Providence, RI, 1990. MR 91g:35001
  • [Kla1] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), 43--101. MR 81b:35050
  • [Kla2] ------, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), 321--332. MR 86i:35091
  • [Kla3] ------, The null condition and global existence to nonlinear wave equations, Lectures in Applied Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293--326. MR 87h:35217
  • [MaHu] J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983. MR 95h:73022

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 35L70, 35Q72, 73C50, 73D35

Retrieve articles in all journals with MSC (1991): 35L70, 35Q72, 73C50, 73D35


Additional Information

DOI: https://doi.org/10.1090/S1079-6762-96-00006-6
Keywords: Non-linear hyperbolic, elastodynamics, incompressible, global existence
Received by editor(s): December 29, 1995
Additional Notes: Partially supported by NSF grant DMS 9304403
Communicated by: James Glimm
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society