# Preprint 2014-016

# Unique conservative solutions to a variational wave equation

## Alberto Bressan, Geng Chen, and Qingtian Zhang

**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
Hölder 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
$u(0,·) ∈ H^1(\mathbb{R})$, $u_t(0,·) ∈ L^2(\mathbb{R})$.