Finite time blow-up for the nematic liquid crystal flow in dimension two

TL;DR

This paper develops a novel inner-outer gluing method to construct solutions for a simplified nematic liquid crystal flow in two dimensions that blow up at specified points in finite time, providing detailed blow-up descriptions.

Contribution

It introduces a new inner-outer gluing technique to explicitly construct finite-time blow-up solutions at predetermined points in 2D nematic liquid crystal flow.

Findings

Solutions blow up exactly at chosen points as time approaches T.

The method provides a precise characterization of the blow-up behavior.

Constructs solutions for any number of specified blow-up points.

Abstract

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $\Omega \subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.