Archive for Rational Mechanics and Analysis

, Volume 94, Issue 1, pp 15–38

The initial-value problem for elastodynamics of incompressible bodies

  • David G. Ebin
  • Ralph A. Saxton
Article

DOI: 10.1007/BF00278241

Cite this article as:
Ebin, D.G. & Saxton, R.A. Arch. Rational Mech. Anal. (1986) 94: 15. doi:10.1007/BF00278241
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Abstract

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials.

Our method consists in:
  1. a)

    Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.

     
  2. b)

    Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

     
  1. 1)

    Using energy estimates to bound the growth of various Sobolev norms of solutions.

     
  2. 2)

    Finding the solution as the limit of a sequence of solutions of linearized problems.

     

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.

Copyright information

© Springer-Verlag GmbH & Co 1986

Authors and Affiliations

  • David G. Ebin
    • 1
    • 2
  • Ralph A. Saxton
    • 1
    • 2
  1. 1.Department of MathematicsState University of New York at Stony BrookStony Brook
  2. 2.Department of MathematicsUniversity of New Orleans LakefrontNew Orleans