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

Abstract

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form ${\mathbf{u}}_{tt}=\mathrm{div}(\mathbb{T})+\mathbf{f}$ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $\mathbf{\epsilon }(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$, is assumed to be of the form $\mathbf{\epsilon }({\mathbf{u}}_t)+\alpha \mathbf{\epsilon }(\mathbf{u})=F(\mathbb{T})$, 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 show 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 $a$.

34 pages