A local existence result for a system of viscoelasticity with physical viscosity

Marta Lewicka; Piotr B. Mucha

doi:10.3934/eect.2013.2.337

Article Contents

2013, Volume 2, Issue 2: 337-353. Doi: 10.3934/eect.2013.2.337

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A local existence result for a system of viscoelasticity with physical viscosity

Marta Lewicka^1, and
Piotr B. Mucha^2,

1.
University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260
2.
Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02097 Warszawa

Received: September 2012

Revised: February 2013

Published online: March 2013

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Abstract

We prove the local in time existence of regular solutions to the system of equations of isothermal viscoelasticity with clamped boundary conditions. We deal with a general form of viscous stress tensor $\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition is compatible with the balance of angular momentum, frame invariance and the Claussius-Duhem inequality. We give examples of linear and nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these required conditions.

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Mathematics Subject Classification: Primary: 74B20; Secondary: 35Q74.

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