Weak Solutions to a Nonlinear Variational Wave Equation

Abstract

We establish the global existence of weak solutions to the Cauchy problem for a nonlinear variational wave equation, where the wave speed is a given monotone function of the wave amplitude. The equation arises in modeling wave motions of nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use the Young-measure method in the setting of Lp spaces. We overcome the difficulty that oscillations get amplified by the growth terms of the equation.