关于双曲型液晶弹性体的流体动力学
On the hydrodynamics of hyperbolic liquid crystal elastomers
AI总结：
本文利用能量变分方法推导含惯性效应的液晶弹性体流体动力学，并证明完全拟线性双曲系统的大数据局部适定性及平衡态附近小初值三维全局正则性。
摘要：
液晶弹性体是一类特殊的交联聚合物材料，它将弹性体的大弹性变形能力与液晶的取向序相结合。由于机械弹性变形与取向向量之间存在强耦合，该模型表现出与液晶模型显著不同的现象。我们的结果包含三个方面。（i）首先，我们采用受 Chun Liu [33] 启发的能量变分方法，推导了具有惯性效应的液晶弹性体的流体动力学。（ii）随后，我们从数学角度研究双曲型液晶弹性体，它是一个完全拟线性双曲系统。通过零黏性极限方法，我们证明了大数据的局部适定性。（iii）最后，我们在三维情形中证明了常数平衡态附近的小而光滑初始数据的全局正则性；这一结果是通过仔细研究二阶物质导数的结构以及系统中的抵消关系实现的。这是关于双曲型液晶弹性体的首个全局结果。
英文摘要：
Liquid crystal elastomers are special cross-linked polymer materials combining the large elastic deformability of elastomers with the orientational orders of liquid crystals. This model exhibits markedly different phenomena than the liquid crystal model due to the strong coupling between mechanical elastic deformation and orientation vector. Our results are threefold. (i) First we derive the hydrodynamics of liquid crystal elastomers with inertial effect by energetic variational approach inspired by Chun Liu [33]. (ii) Then we study the hyperbolic liquid crystal elastomers from mathematical point, which is a fully quasilinear hyperbolic system. The local well-posedness for large data is proved by zero-viscosity limit method. (iii) Finally, we show the global regularity for small and smooth initial data near the constant equilibrium in three dimensions, which is achieved by carefully investigating the structure of second-order material derivatives and the cancellations in the system. This is the first global result on hyperbolic liquid crystal elastomers.
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