A Beale--Kato--Majda criterion for the 3-D Compressible Nematic Liquid Crystal Flows with Vacuum

TL;DR

This paper establishes a blow-up criterion for 3-D compressible nematic liquid crystal flows, showing that the solution remains global if the velocity gradient's $L^{1}(0,T;L^{ty})$-norm stays bounded, improving previous results.

Contribution

It introduces a new Beale--Kato--Majda type criterion based solely on the velocity gradient for these flows, enhancing understanding of solution blow-up conditions.

Findings

Global existence criterion based on velocity gradient norm

Improved blow-up condition over previous results

Applicable to flows with vacuum states

Abstract

In this paper, we prove a Beale--Kato--Majda blow-up criterion in terms of the gradient of the velocity only for the strong solution to the 3-D compressible nematic liquid crystal flows with nonnegative initial densities. More precisely, the strong solution exists globally if the $L^{1}(0,T;L^{\infty})$-norm of the gradient of the velocity $u$ is bounded. Our criterion improves the recent result of X. Liu and L. Liu (\cite{LL}, A blow-up criterion for the compressible liquid crystals system, arXiv:1011.4399v2 [math-ph] 23 Nov. 2010).