Global strong solution of the 3D compressible liquid crystal flows with density-dependent viscosity and large velocity

TL;DR

This paper proves the existence of a unique global strong solution for 3D compressible liquid crystal flows with density-dependent viscosity, even with large initial velocities, under certain initial conditions.

Contribution

It is the first to establish global strong solutions for 3D compressible liquid crystal flows without requiring small initial velocities.

Findings

Global strong solution exists under specified initial conditions.

Unique solution is guaranteed for large initial velocities.

First such result for 3D flows with density-dependent viscosity.

Abstract

This paper concerns the Cauchy problem of three-dimensional compressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficients $\mu_1(\rho),\mu_2(\rho)$ are power functions of the density with the power larger than $1$, it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and $L^3$-norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional compressible liquid crystal flows without smallness of velocity.