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 Free Access
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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.
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- F. John, Formation of singularities in elastic waves, Lect. Notes in Physics, vol. 195, Springer-Verlag, 1984, pp. 194–210.
- ---, Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29–51.
- ---, 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.
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- ---, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), 321–332.
- ---, 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.
- J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983.
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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
