# American Institute of Mathematical Sciences

December  2017, 10(6): 1233-1256. doi: 10.3934/dcdss.2017067

## 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

To Tomáš Roubíček on the occasion of his 60th birthday.

Received  July 2016 Revised  February 2017 Published  June 2017

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.

Citation: Jan Burczak, Josef Málek, Piotr Minakowski. Stress-diffusive regularizations of non-dissipative rate-type materials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1233-1256. doi: 10.3934/dcdss.2017067
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