Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 3D compressible nematic liquid crystal flows with large oscillations and vacuum, under small initial energy, extending previous local and Cauchy problem results.

Contribution

It extends prior work by establishing global strong solutions with large oscillations and vacuum in bounded domains, using refined estimates and analysis.

Findings

Global existence and uniqueness of strong solutions under small initial energy.

Blow-up mechanisms are prevented with sufficiently small initial energy.

The results extend previous local and Cauchy problem solutions to bounded domains.

Abstract

We investigate compressible nematic liquid crystal flows in three-dimensional (3D) bounded domains with slip boundary condition for velocity and Neumann boundary condition for orientation field. By applying piecewise-estimate method and delicate analysis based on the effective viscous flux and vorticity, we derive the global existence and uniqueness of strong solutions provided that the initial total energy is suitably small. Our result is an extension of the works of Huang-Wang-Wen (J. Differential Equations 252: 2222-2265, 2012) and Li-Xu-Zhang (J. Math. Fluid Mech. 20: 2105-2145, 2018), where the local strong solutions in three dimensions and the global strong solutions for 3D Cauchy problem were established, respectively. Moreover, it also shows that blow up mechanism for local strong solutions obtained by Huang-Wang-Wen (Arch. Ration. Mech. Anal. 204: 285-311, 2012) cannot occur if the initial total energy is sufficiently small.