Weak solutions to a nonlinear variational wave equation with general data

Abstract

We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation utt−c(u)(c(u)ux)x=0 with general initial data (u(0),ut(0))=(u0,u1)∈W1,2×L2 under the assumptions that the wave speed c(u) satisfies c′(⋅)⩾0 and c′(u0(⋅))>0. Moreover, we obtain high regularity for the spatial derivative ∂xu of the wave amplitude u away from where c′(u)=0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of Lp spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for ∂xu to handle possible concentrations. This result improves our earlier existence result for initial data in the space W1,∞×L∞ to the natural space W1,2×L2.