Lipschitz Metrics for a Class of Nonlinear Wave Equations

TL;DR

This paper introduces a new Lipschitz metric for a class of nonlinear wave equations, ensuring the flow is uniformly Lipschitz continuous in the $H^1$ space by using a Finsler manifold structure and optimal transportation techniques.

Contribution

It constructs a novel metric on $H^1$ that makes the flow of solutions Lipschitz continuous, addressing the discontinuity of the natural $H^1$ distance for these equations.

Findings

Constructed a Finsler manifold structure on $H^1$.

Defined a new metric using optimal transportation.

Proved the metric makes the flow Lipschitz continuous.

Abstract

The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ 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 the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.