Unique Conservative Solutions to a Variational Wave Equation

TL;DR

This paper proves the uniqueness of conservative solutions to a variational wave equation with H"older continuous wave speed by analyzing characteristics and transforming the problem into a semilinear system with smooth coefficients.

Contribution

It introduces a novel method to establish uniqueness of solutions even when the wave speed is only H"older continuous, using characteristic analysis and a change of variables.

Findings

Uniqueness of conservative solutions is established for the variational wave equation.

A new set of variables constant along characteristics is introduced.

Solutions satisfy a semilinear system with smooth coefficients, ensuring uniqueness.

Abstract

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous in the $t$-$x$ plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables $X,Y$, constant along characteristics, we prove that $t,x,u$, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $u(0,\cdot)\in H^1(\mathbb{R})$, $u_t(0,\cdot)\in L^2(\mathbb{R})$.