Singularity for a multidimensional variational wave equation arising from nematic liquid crystals

TL;DR

This paper investigates a multidimensional nonlinear wave equation from nematic liquid crystal theory, showing that smooth solutions can break down in finite time despite small initial energy, using the method of characteristics.

Contribution

It demonstrates finite-time breakdown of smooth solutions for a spherically-symmetric variational wave equation in nematic liquid crystals.

Findings

Smooth solutions break down in finite time

Breakdown occurs even with small initial energy

Method of characteristics used for proof

Abstract

This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using the method of characteristics, we show that the smooth solutions for the spherically-symmetric variational wave equation breakdown in finite time, even for the arbitrarily small initial energy.