Global strong solution of the 3D inhomogeneous 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 inhomogeneous liquid crystal flows with density-dependent viscosity, even without small initial velocity, under certain conditions on initial data.

Contribution

It establishes the first global strong solution result for 3D inhomogeneous liquid crystal flows without requiring small initial velocity.

Findings

Global strong solution exists under specified initial conditions.

Unique solution is guaranteed for density-dependent viscosity with power > 1.

No smallness condition on initial velocity is needed.

Abstract

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient $\mu(\rho)$ is a power function of the density with the power larger than $1$, that is $\mu(\rho)=\mu\rho^\alpha$ with $\alpha>1$, 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 inhomogeneous liquid crystal flows without smallness of velocity.