Advances in Differential Equations

Almost global existence of weak solutions for the nonlinear elastodynamics system for a class of strain energies

Sébastien Court and Karl Kunisch

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Abstract

The aim of this paper is to prove the existence of almost global weak solutions for the unsteady nonlinear elastodynamics system in dimension $d=2$ or $3$, for a range of strain energy density functions satisfying some given assumptions. These assumptions are satisfied by the main strain energies generally considered. The domain is assumed to be bounded, and mixed boundary conditions are considered. Our approach is based on a nonlinear parabolic regularization technique, involving the $p$-Laplace operator. First we prove the existence of a local-in-time solution for the regularized system, by a fixed point technique. Next, using an energy estimate, we show that if the data are small enough, bounded by $\varepsilon >0$, then the maximal time of existence does not depend on the parabolic regularization parameter, and the behavior of the lifespan $T$ is $\gtrsim \log (1/\varepsilon)$, defining what we call here almost global existence. The solution is thus obtained by passing this parameter to zero. The key point of our proof is due to recent nonlinear Korn's inequalities proven by Ciarlet and Mardare in $W^{1,p}$ spaces, for $p>2$.

Article information

Source
Adv. Differential Equations Volume 23, Number 1/2 (2018), 135-160.

Dates
First available in Project Euclid: 26 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1508983364

Subjects
Primary: 74B20: Nonlinear elasticity 35L70: Nonlinear second-order hyperbolic equations 35L53: Initial-boundary value problems for second-order hyperbolic systems 74H20: Existence of solutions 35A01: Existence problems: global existence, local existence, non-existence 35D30: Weak solutions 35K92: Quasilinear parabolic equations with p-Laplacian

Citation

Court, Sébastien; Kunisch, Karl. Almost global existence of weak solutions for the nonlinear elastodynamics system for a class of strain energies. Adv. Differential Equations 23 (2018), no. 1/2, 135--160. https://projecteuclid.org/euclid.ade/1508983364


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