Uniform Regularity and Vanishing Viscosity Limit for the Compressible Nematic Liquid Crystal Flows in Three Dimensional Bounded Domain

TL;DR

This paper establishes uniform regularity and convergence results for compressible nematic liquid crystal flows in three dimensions, demonstrating the existence of solutions independent of viscosity and quantifying the vanishing viscosity limit.

Contribution

It proves uniform bounds and convergence rates for solutions of compressible nematic liquid crystal flows in a bounded domain, advancing understanding of their vanishing viscosity behavior.

Findings

Existence of unique strong solutions independent of viscosity

Uniform bounds for density, velocity, and director field

Quantitative convergence rate to inviscid solutions

Abstract

In this paper, we study the uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the compressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity coefficient. The solutions are uniform bounded in a conormal Sobolev space. Furthermore, we prove that the density and velocity are uniform bounded in $W^{1, \infty}$, and the director field is uniform bounded in $W^{3,\infty}$ respectively. Based on these uniform estimates, one also obtains the convergence rate of the viscous solutions to the inviscid ones with a rate of convergence.