Global Solution to a nonlinear wave equation of liquid crystal in the constant electric field

TL;DR

This paper develops a global weak solution for a nonlinear wave equation modeling nematic liquid crystals under a constant electric field, advancing mathematical understanding of such complex physical systems.

Contribution

It constructs the first global conservative weak solution for a nonlinear variational wave equation derived from liquid crystal models with electric fields.

Findings

Established existence of global weak solutions

Demonstrated conservation of energy in solutions

Applicable to a broad class of smooth wave speed functions

Abstract

We construct a global conservative weak solution to the Cauchy problem for the non-linear variational wave equation $v_{tt} - c(v)(c(v)v_x)_x + \frac{1}{2}(v+v^3)= 0$ where $c(\cdot)$ is any smooth function with uniformly positive bounded value. This wave equation is derived from a wave system modelling nematic liquid crystals in a constant electric field.