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Global solutions of the equations
of elastodynamics for incompressible materials
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## Global solutions of the equations
of elastodynamics for incompressible materials

### David G. Ebin

**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.

*Copyright American Mathematical Society 1996*

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#### Article Info

- ERA Amer. Math. Soc.
**02** (1996), pp. 50-59
- Publisher Identifier: S 1079-6762(96)00006-5
- 1991
*Mathematics Subject Classification*. Primary
35L70, 35Q72, 73C50, 73D35
*Key words and phrases*. Non-linear hyperbolic,
elastodynamics, incompressible, global existence
- Received by the editors December 29, 1995
- Communicated by James Glimm
- Comments (When Available)

**David G. Ebin**

SUNY at Stony Brook, NY 11794-3651

*E-mail address:* `ebin@math.sunysb.edu`

Partially supported by NSF grant DMS 9304403

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