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

Article type: Research Article

Authors: Zhang, Ping | Zheng, Yuxi

Affiliations: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, P.R. China | Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Abstract: The wave equation \curpartial _t^2u - c\curpartial _x(c\curpartial _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 \curpartial _x(\curpartial _tu + u\curpartial _xu) =(1/2)(\curpartial _xu)^2 has been derived. It has been shown through concrete examples that oscillations in v, v=\curpartial _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(\mathbb{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.

Keywords: Existence, kinetic formulation, regularity, Young measure

Journal: Asymptotic Analysis, vol. 18, no. 3-4, pp. 307-327, 1998