ON THE HYDRODYNAMICS OF HYPERBOLIC LIQUID CRYSTAL ELASTOMERS

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 from 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 an energetic variational approach inspired by Liu [An introduction of elastic complex fluids: An energetic variational approach, in Multi-scale Phenomena in Complex Fluids, Contemp. Appl. Math. CAM 12, World Scientific, Singapore, 2009, pp. 286-337]. (ii) Then we study the hyperbolic liquid crystal elastomers from a mathematical perspective, which is a fully quasilinear hyperbolic system. The local well-posedness for large data is proved by a 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.