On well-posedness of Ericksen–Leslie’s parabolic–hyperbolic liquid crystal model in compressible flow

Abstract

We study the well-posedness of the Ericksen–Leslie’s parabolic–hyperbolic liquid crystal model in compressible flow. Inspired by our study for incompressible case [N. Jiang and Y.-L. Luo, On well-posedness of Ericsen–Leslie’s hyperbolic incompressible liquid crystal model, preprint (2017), arXiv:1709.06370v1 ] and some techniques from compressible Navier–Stokes equations, we first prove the local-in-time existence of the classical solution to the system with finite initial energy, under some natural constraints on the Leslie coefficients which ensure that the basic energy law is dissipative. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the existence of global solution can be established.