On the hydrodynamics of hyperbolic liquid crystal elastomers

TL;DR

This paper develops a comprehensive hydrodynamic model for hyperbolic liquid crystal elastomers, proving local well-posedness for large data and establishing the first global regularity result for small initial data in three dimensions.

Contribution

It introduces a new hyperbolic model for liquid crystal elastomers, proves local well-posedness for large data, and establishes the first global regularity result in three dimensions.

Findings

Derived hydrodynamics with inertial effects using energetic variational approach.

Proved local well-posedness for large initial data.

Established global regularity for small initial data in 3D.

Abstract

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.