Blow-up criterion of classical solutions for the incompressible nematic liquid crystal flows

TL;DR

This paper establishes a criterion for the potential breakdown of smooth solutions to the 3D incompressible nematic liquid crystal flow, linking solution regularity to boundedness of certain norms of velocity curl and director field gradient.

Contribution

It provides a new blow-up criterion for classical solutions of nematic liquid crystal flows based on BMO and L^4 norms, advancing understanding of solution regularity conditions.

Findings

Solution remains smooth if the integral of curl u in BMO and gradient d in L^4 is finite.

The criterion offers a condition to prevent finite-time singularity formation.

The results apply to short-time classical solutions in three dimensions.

Abstract

In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if $(u,d)$ is smooth up to time $T$ provided that $\int_0^T|\nabla\times u(t,\cdot)|_{BMO(\mathbb R^3)}+|\nabla d(t,\cdot)|^8_{L^4(\mathbb R^3)}dt<\infty $