Finite time singularity of the nematic liquid crystal flow in dimension three

TL;DR

This paper demonstrates the occurrence of finite time singularities in three-dimensional nematic liquid crystal flows through two constructed examples, highlighting potential blow-up scenarios in such fluid models.

Contribution

It provides the first explicit examples of finite time singularities in 3D nematic liquid crystal flows, including axisymmetric and generic initial data cases.

Findings

Finite time singularities are possible in 3D nematic liquid crystal flows.

Explicit examples of singularities are constructed for specific initial conditions.

Singularities can occur even with small initial energy and nontrivial topology.

Abstract

In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data $(u_0,d_0)$ that has sufficiently small energy, and $d_0$ has a nontrivial topology