TY  - JOUR
T1  - Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
A1  - Bulíček, Miroslav
A1  - Patel, Victoria
A1  - Şengül, Yasemin
A1  - Süli, Endre
JA  - Commun. Pure Appl. Anal.
Y1  - 2021
VL  - 20
IS  - 5
SP  - 1931
EP  - 1960
M2  - doi: 10.3934/cpaa.2021053
N2  - We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form
$\bu_{tt} = \mbox{div }\mathbb{T}   \boldf$ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $\beps(\bu)$ to the Cauchy stress tensor $\bbT$, is assumed to be of the form $\beps(\bu_t)   \alpha\beps(\bu) = F(\bbT)$, where we define \( F(\bbT) = ( 1   |\bbT|^a)^{-\frac{1}{a}}\bbT\), 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 $\bbT$ 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 $\bbT \in L^{1 \delta}(Q)^{d \times d}$ for a $\delta > 0$, the value of which depends only on $a$.
M1  - Preprint project = NCMM
M1  - Preprint year = 2020
M1  - Preprint number = 11
M1  - Preprint ID = NCMM/2020/11
ER  -