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

Abstract

This article is focused on a multidimensional nonlinear variational wave equation that is the Euler-Lagrange equation of a variational principle arising from 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 break down in finite time, even for an arbitrarily small initial energy. The key point is the energy equation derived from the variational wave equation, which is also used to establish the existence of energy-conservative weak solutions in a finite period.