论文标题
存在大型数据全球弱解决方案，用于应变粘弹性体的模型
Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
论文作者
论文摘要
我们证明了一个独特的大数据全球薄弱的解决方案，以$ \ mathbf {u} _ {tt} = \ m atrm {div}（\ mathbb {t}） + \ mathbf {f} $ claincoealasties clainistion clainistion等方程，在构成方程的情况下， $ \boldsymbolε（\ MathBf {u}）$ to Cauchy storce Tensor $ \ Mathbb {t} $，假定为$ \boldsymbolε（\ MathBf {u} _t _t） $ f（\ mathbb {t}）=（1 + | \ mathbb {t} |^a）^{ - \ frac {1} {a}} \ mathbb {t} $，对于常数参数$α\ in（in（0，\ infty）$ in（0，\ infty）$ in（0，\ norder）， 状况。 cauchy stress $ \ mathbb {t} $显示为$ l^1（q）^{d \ times d} $上的时空域$ q $。特别是，在三个空间维度中，如果$ a \ in（0，\ frac {2} {7}）$，则实际上$ \ mathbb {t} \ in l^{1+δ}（q）（q）^{q）
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 $\boldsymbolε(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$, is assumed to be of the form $\boldsymbolε(\mathbf{u}_t) +α\boldsymbolε(\mathbf{u})= F(\mathbb{T})$, where we define $F(\mathbb{T}) = (1 + |\mathbb{T}|^a)^{-\frac{1}{a}}\mathbb{T}$, for constant parameters $α\in (0, \infty)$ 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 $\mathbb{T}\in L^{1+δ}(Q)^{d\times d}$ for a $δ>0$, the value of which depends only on $a$.
