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

Abstract

<p style='text-indent:20px;'>We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol{u}_{tt} = \mbox{div }\mathbb{T} + \boldsymbol{f} $\end{document}</tex-math></inline-formula> for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor <inline-formula><tex-math id="M2">\begin{document}$ \boldsymbol{\varepsilon}( \boldsymbol{u}) $\end{document}</tex-math></inline-formula> to the Cauchy stress tensor <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{T} $\end{document}</tex-math></inline-formula>, is assumed to be of the form <inline-formula><tex-math id="M4">\begin{document}$ \boldsymbol{\varepsilon}( \boldsymbol{u}_t) + \alpha \boldsymbol{\varepsilon}( \boldsymbol{u}) = F( \mathbb{T}) $\end{document}</tex-math></inline-formula>, where we define \(F(\mathbb{T}) = (1 + | \mathbb{T}|^a)^{-\frac{1}{a}} \mathbb{T}\), for constant parameters <inline-formula><tex-math id="M5">\begin{document}$ \alpha \in (0,\infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ a \in (0,\infty) $\end{document}</tex-math></inline-formula>, in any number <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> of space dimensions, with periodic boundary conditions. The Cauchy stress <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{T} $\end{document}</tex-math></inline-formula> is shown to belong to <inline-formula><tex-math id="M9">\begin{document}$ L^{1}(Q)^{d \times d} $\end{document}</tex-math></inline-formula> over the space-time domain <inline-formula><tex-math id="M10">\begin{document}$ Q $\end{document}</tex-math></inline-formula>. In particular, in three space dimensions, if <inline-formula><tex-math id="M11">\begin{document}$ a \in (0,\frac{2}{7}) $\end{document}</tex-math></inline-formula>, then in fact <inline-formula><tex-math id="M12">\begin{document}$ \mathbb{T} \in L^{1+\delta}(Q)^{d \times d} $\end{document}</tex-math></inline-formula> for a <inline-formula><tex-math id="M13">\begin{document}$ \delta &gt; 0 $\end{document}</tex-math></inline-formula>, the value of which depends only on <inline-formula><tex-math id="M14">\begin{document}$ a $\end{document}</tex-math></inline-formula>.