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

TL;DR

This paper proves local and global well-posedness of the Ericksen-Leslie liquid crystal model in compressible flow, extending previous incompressible results and utilizing techniques from compressible Navier-Stokes equations.

Contribution

It establishes the existence of local and global classical solutions for the compressible Ericksen-Leslie model under specific coefficient constraints and initial energy conditions.

Findings

Local-in-time existence of classical solutions with finite initial energy.

Global classical solutions under small initial energy and damping conditions.

Energy law is shown to be dissipative under certain Leslie coefficient constraints.

Abstract

We study the Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow. Inspired by our study for incompressible case \cite{Jiang-Luo-arXiv-2017} and some techniques from compressible Navier-Stokes equations, we prove the local-in-time existence of the classical solution to the system with finite initial energy, under some constraints on the Leslie coefficients which ensure 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 global classical solution can be established.