Blow up criterion for compressible nematic liquid crystal flows in dimension three

TL;DR

This paper establishes a criterion for the finite-time breakdown of strong solutions in three-dimensional compressible nematic liquid crystal flows, linking solution blow-up to specific norms of velocity deformation and director field gradients.

Contribution

It introduces a novel blow-up criterion for strong solutions in 3D compressible nematic liquid crystal flows based on deformation and director field gradients.

Findings

Blow-up occurs when the integral of the deformation tensor norm exceeds a threshold.

Finite-time breakdown is characterized by the growth of the director field gradient.

The criterion provides a mathematical condition for solution continuation or breakdown.

Abstract

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.