On singularities of Ericksen-Leslie system in dimension three

TL;DR

This paper investigates singularity formation in the three-dimensional Ericksen-Leslie system modeling nematic liquid crystals, providing explicit examples of finite-time singularities under specific conditions.

Contribution

It constructs explicit finite-time singularity examples in 3D Ericksen-Leslie system, including axisymmetric and topologically nontrivial cases, and presents a counterexample to the maximum principle.

Findings

Finite-time singularities can occur in the Ericksen-Leslie system.

Explicit examples of singularities are constructed under special initial conditions.

A counterexample to the maximum principle is demonstrated.

Abstract

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for an initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one.