Nematic liquid crystal flow with partially free boundary

TL;DR

This paper investigates the dynamics of nematic liquid crystal flow with natural boundary conditions, constructing smooth solutions that blow up at specified points, advancing understanding of singularity formation in such systems.

Contribution

It introduces a coupled Navier-Stokes and harmonic map system with physically natural boundary conditions and demonstrates finite-time blow-up solutions at arbitrary points.

Findings

Constructed smooth solutions with finite-time blow-up

Blow-up can occur at boundary or interior points

Advances understanding of singularities in liquid crystal flows

Abstract

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.