On Oscillations of an Asymptotic Equation of a Nonlinear
Variational Wave Equation
Ping Zhang & Yuxi Zheng
Abstract
The wave equation $u_{tt} - c(cu_x)_x=0$, 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 $(u_t + uu_x)_x =1/2(u_x)^2$ has been derived.
It has been shown through concrete examples that oscillations in $v$,
$v=u_x$, 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 $L^p(\Bbb R)$ 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.