# Preprint 2015-018

# Lipschitz metrics for a class of nonlinear wave equations

## Alberto Bressan and Geng Chen

**Abstract:**
The nonlinear wave equation
$u_{tt}-c(u)\bigl(c(u)u_x\bigr)_x=0$
determines a flow of conservative solutions
taking values in the space $H^1(\mathbb{R})$.
However, this flow is not continuous w.r.t. the natural $H^1$ distance.
Aim of this paper is to construct a new metric which renders the
flow uniformly Lipschitz continuous on bounded subsets of $H^1(\mathbb{R})$.
For this purpose, $H^1$ is given the structure of a Finsler manifold,
where the norm of tangent vectors is defined
in terms of an optimal transportation problem.
For paths of piecewise smooth solutions,
one can carefully estimate how the weighted length grows in time.
By a recent result on generic regularity,
these piecewise regular paths are dense
and can be used to construct a
geodesic distance with the desired Lipschitz property.