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The initial-value problem for elastodynamics of incompressible bodies

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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.

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Bibliography

  1. Adams, R. A., Sobolev Spaces, Academic Press, 1975.

  2. Arnold, V., Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.

  3. Ball, J. M., Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Arch. Rational Mech. Anal. 63, 1977, 337–403.

    Google Scholar 

  4. J. P. scBourguignon & H. scBrezis, Remarks on the Euler Equation. Journal of Functional Analysis 15, 1974, pp. 341–363.

    Google Scholar 

  5. Dafermos, C. M., & W. J. scHrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics. Arch. Rational Mech. Anal. 87, 1985, 267–292.

    Google Scholar 

  6. Ebin, D. G., The manifold of Riemannian Metrics, Proc. Symposia Pure Math., Global Analysis 15 (1970), 11–40.

    Google Scholar 

  7. Ebin, David G., & Jerrold scMarsden, Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Annals of Mathematics 92, 1970, 102–163.

    Google Scholar 

  8. Hughes, T. J. R., T. Kato, & J. E. Marsden, Well-posed Quasi-linear Secondorder Hyperbolic Systems with Applications to Nonlinear Elastodynamics and General Relativity. Arch. Rational Mech. Anal. 63, 1977, 273–294.

    Google Scholar 

  9. Hughes, T. J. R., & J. E. Marsden, Mathematical Foundations of Elasticity. Prentice-Hall, 1983.

  10. Kato, T., The Cauchy Problem for Quasi-Linear Symmetric Hyperbolic Systems. Arch. Rational Mech. Anal. 58, 1975, 181–205.

    Google Scholar 

  11. Kato, T., & C. V. scLai, Nonlinear Evolution Equations and the Euler Flow. Journal of Functional Analysis 56, 1984, 15–28.

    Google Scholar 

  12. Lions, J. L., et E. Magenes, Problèmes aux Limites non Homogènes et Applications. Vols. 1 et 2, Dunod, 1968.

  13. Moser, J., On the Volume Elements on a Manifold. Trans. Amer. Math. Soc. 120, 1965, 286–294.

    Google Scholar 

  14. Ogden, R. W., On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids. Proc. Roy. Soc. Lond. A 326, 1972, 565–584.

    Google Scholar 

  15. Spanier, E. H., Algebraic Topology. McGraw-Hill, 1966.

  16. Truesdell, C., & W. Noll, The Non-Linear Field Theories of Mechanics, Handbuch der Physik III/3, ed: S. Flügge, Springer-Verlag, 1965.

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Authors and Affiliations

  1. Department of Mathematics, State University of New York at Stony Brook, Stony Brook

    David G. Ebin & Ralph A. Saxton

  2. Department of Mathematics, University of New Orleans Lakefront, New Orleans

    David G. Ebin & Ralph A. Saxton

Authors
  1. David G. Ebin
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  2. Ralph A. Saxton
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Communicated by S. Antman

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Ebin, D.G., Saxton, R.A. The initial-value problem for elastodynamics of incompressible bodies. Arch. Rational Mech. Anal. 94, 15–38 (1986). https://doi.org/10.1007/BF00278241

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  • DOI: https://doi.org/10.1007/BF00278241

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