On oscillations of an asymptotic equation of a nonlinear variational wave equation

The wave equation ∂_t²u - c∂_x(c∂_xu) = 0, where c = c(u) is a given function, arises in a number of different physical contexts and is the simplest example of an interesting class of nonlinear hyperbolic partial differential equations. For unidirectional weakly nonlinear waves, an asymptotic equation ∂_x(∂_tu + u∂_xu) = (1/2)(∂_xu)² has been derived. It has been shown through concrete examples that oscillations in v, v = ∂_xu, in the initial data persist into positive time for the asymptotic equation. In the first part of this paper, we show by applying Young measure theory that no oscillations are generated if there are no oscillations (around a nonnegative state v) in the initial data, which implies in particular the global existence of weak solutions to the asymptotic equation with nonnegative L^p(ℝ) initial data v with p > 2. In the second part, we obtain a regularity result for a large class of weak solutions to this equation by using its kinetic formulation. In particular this regularity result applies to both the conservative and dissipative weak solutions.