Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
复制标题
应变限制粘弹性体模型大数据全局弱解的存在性
DOI：
10.3934/cpaa.2021053
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发表时间：
2020
影响因子：
1
通讯作者：
E. Suli
中科院分区：
文献类型：
--
作者：
Miroslav Bul'ivcek;Victoria Patel;Yasemin cSengul;E. Suli
对于具有应变限制行为的粘弹体，我们证明了一类形式为$\mathbf{u}_{tt}=\mathm{div}(\mathbb{T})+\mathbf{f}$的模型存在唯一的大数据整体时间弱解，其中本构方程将线性化的应变张量与柯西应力张量$\mathbb{T}$联系起来。假设形式为$\boldSymbol{\epsilon}(\mathbf{u}_t)+\α\boldSymbol{\epsilon}(\mathbf{u})=F(\mathbb{T})$，其中我们定义$F(\mathbb{T})=(1+|\mathbb{T}|^a)^{-\frac{1}{a}}\mathbb{T}$，表示(0，\inty)$和$a\in(0，\inty)$中的常量参数$\α\具有周期边界条件。柯西应力在时空域$Q$上属于$L^1(Q)^{d\次d}$。特别地，在三个空间维度中，如果$a\in(0，FRAC{2}{7})$，则实际上$\mathbb{T}\在L^{1+\Delta}(Q)^{d\x d}$中为$\Delta>；0$，其值仅取决于$a$。
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 $\boldsymbol{\epsilon}(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$, is assumed to be of the form $\boldsymbol{\epsilon}(\mathbf{u}_t) +\alpha \boldsymbol{\epsilon}(\mathbf{u})= F(\mathbb{T})$, 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 $a$.