Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

Miroslav Bulíček; Victoria Patel; Yasemin Şengül; Endre Süli

doi:10.3934/cpaa.2021053

Article Contents

2021, Volume 20, Issue 5: 1931-1960. Doi: 10.3934/cpaa.2021053

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Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

1.
Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83,186 75 Prague 8, Czech Republic
2.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK
3.
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

^* Corresponding author: V. Patel
^* Corresponding author: V. Patel

Received: September 30, 2020

Early access: March 2021

Published: May 2021

M. Bulíček's work is supported by the project 20-11027X financed by GAČR. M. Bulíček is a member of the Nečas Center for Mathematical Modeling. V. Patel is supported by the UK Engineering and Physical Sciences Research Council [EP/L015811/1]. Y. Şengül is partially supported by the Scientific and Technological Research Council of Turkey (TÜBITAK) under the grant 116F093

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Abstract

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form $ \boldsymbol{u}_{tt} = \mbox{div }\mathbb{T} + \boldsymbol{f} $ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $ \boldsymbol{\varepsilon}( \boldsymbol{u}) $ to the Cauchy stress tensor $ \mathbb{T} $, is assumed to be of the form $ \boldsymbol{\varepsilon}( \boldsymbol{u}_t) + \alpha \boldsymbol{\varepsilon}( \boldsymbol{u}) = F( \mathbb{T}) $, where we define $F(\mathbb{T}) = (1 + | \mathbb{T}|^a)^{-\frac{1}{a}} \mathbb{T}$, for constant parameters $ \alpha \in (0,\infty) $ and $ a \in (0,\infty) $, in any number $ d $ of space dimensions, with periodic boundary conditions. The Cauchy stress $ \mathbb{T} $ is shown to belong to $ L^{1}(Q)^{d \times d} $ over the space-time domain $ Q $. In particular, in three space dimensions, if $ a \in (0,\frac{2}{7}) $, then in fact $ \mathbb{T} \in L^{1+\delta}(Q)^{d \times d} $ for a $ \delta > 0 $, the value of which depends only on $ a $.

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Mathematics Subject Classification: 35M13, 35K99, 74D10, 74H20.

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