发表机构
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究三维不可压缩粘性形态弹性系统，通过ε阶扩散正则化构造弱解，证明标量变量强收敛但Jaumann项产生缺陷，并给出缺陷为零的充分条件。
AI 中文摘要
我们研究三维不可压缩粘性形态弹性系统，其中欧拉有效应变通过Zaremba-Jaumann率输运，并通过Kelvin-Voigt应力与Navier-Stokes动量平衡耦合。应变保持对称，并分解为关于其迹的标量输运-松弛方程和包含非线性拉伸项与Jaumann项的偏量方程。我们通过ε阶扩散对两个方程进行正则化，这在力学上可解释为具有特征长度尺度的弱非局部重构。对于固定的ε>0，我们通过截断Galerkin格式构造Leray-Hopf型弱解，并利用标量变量的最大值原理事后移除截断。当ε→0时，通过有界Lipschitz域上的DiPerna-Lions交换子论证，标量变量在L^p((0,T)×Ω)中对每个1≤p<∞强收敛，使得拉伸项能在极限中被识别。相反，Jaumann交换子是弱收敛序列的乘积，其极限无法从现有估计中识别。因此，我们在极限偏量方程中获得缺陷D∈L^2(0,T;(H^2(Ω;S_0))^*)。我们给出D=0的两个充分条件。相关共旋转粘弹性模型中使用的能量级消去在此不闭合，因为拉伸项的系数本身是未知的。
英文摘要
We study an incompressible visco-morphoelastic system in three dimensions in which the Eulerian effective strain is transported by the Zaremba-Jaumann rate and coupled to a Navier-Stokes momentum balance through a Kelvin-Voigt stress. The strain remains symmetric and splits into a scalar transport-relaxation equation for its trace and a deviatoric equation containing the nonlinear stretching and Jaumann terms. We regularize both equations by diffusion of order $\varepsilon$, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed $\varepsilon>0$ we construct Leray-Hopf type weak solutions by a truncated Galerkin scheme and remove the truncation a posteriori using a maximum principle for the scalar variable. As $\varepsilon\to0$, the scalar variable converges strongly in $L^p((0,T)\timesΩ)$ for every $1\le p<\infty$, by a DiPerna-Lions commutator argument on bounded Lipschitz domains, allowing the stretching term to be identified in the limit. In contrast, the Jaumann commutator is a product of two weakly convergent sequences, and its limit cannot be identified from the available estimates. We therefore obtain a defect \[\boldsymbol{\mathcal{D}}\in L^2(0,T;(H^2(Ω;\mathbb{S}_0))^\ast)\] in the limiting deviatoric equation. We give two sufficient conditions for $\boldsymbol{\mathcal{D}}=0$. The energy-level cancellation used in related corotational viscoelastic models does not close here because the coefficient of the stretching term is itself an unknown.
Comments30 pages
