Stress-diffusive regularizations of non-dissipative rate-type materials

Jan Burczak; Josef Málek; Piotr Minakowski

doi:10.3934/dcdss.2017067

Article Contents

2017, Volume 10, Issue 6: 1233-1256. Doi: 10.3934/dcdss.2017067

This issue Previous Article Iterative finite element solution of a constrained total variation regularized model problem Next Article Existence and linearization for the Souza-Auricchio model at finite strains

Stress-diffusive regularizations of non-dissipative rate-type materials

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
2.
OxPDE, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
3.
Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83,186 75 Prague 8, Czech Republic
4.
University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland
5.
Heidelberg University, Interdisciplinary Center for Scientific Computing, Im Neuenheimer Feld 205,69120 Heidelberg, Germany

^* Corresponding author: Piotr Minakowski: minak@mimuw.edu.pl
^* Corresponding author: Piotr Minakowski: minak@mimuw.edu.pl

Received: June 30, 2016

Revised: January 31, 2017

Published: December 2017

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Abstract

We consider non-dissipative (elastic) rate-type material models that are derived within the Gibbs-potential-based thermodynamic framework. Since the absence of any dissipative mechanism in the model prevents us from establishing even a local-in-time existence result in two spatial dimensions for a spatially periodic problem, we propose two regularisations. For such regularized problems we obtain well-posedness of the planar, spatially periodic problem. In contrast with existing results, we prove ours for a regularizing term present solely in the evolution equation for the stress.

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Mathematics Subject Classification: Primary: 35Q35, 74D10; Secondary: 35A01.

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