Global semigroup of conservative solutions of the nonlinear variational wave equation

TL;DR

This paper establishes a global semigroup framework for conservative solutions to a nonlinear wave equation, accommodating measure-valued initial data and energy concentration phenomena, with new variables and numerical methods.

Contribution

It introduces a novel approach using characteristic-related variables to handle singularities and proves energy concentration only occurs on measure-zero sets.

Findings

Existence of a global semigroup for solutions with measure-valued initial data.

Energy concentration occurs only on measure-zero sets or where $c'(u)=0$.

A new numerical method for constructing conservative solutions is demonstrated.

Abstract

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$. We allow for initial data $u|_{t=0}$ and $u_t|_{t=0}$ that contain measures. We assume that $0<\kappa^{-1}\le c(u) \le \kappa$. Solutions of this equation may experience concentration of the energy density $(u_t^2+c(u)^2u_x^2)dx$ into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may only focus on a set of times of zero measure or at points where $c'(u)$ vanishes. A new numerical method to construct conservative solutions is provided and illustrated on examples.