A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation

TL;DR

This paper introduces a Finsler type Lipschitz optimal transport metric for a quasilinear wave equation, proving Lipschitz continuity of solutions and establishing generic regularity results for large data solutions.

Contribution

It constructs a novel Finsler type metric for the wave equation and demonstrates Lipschitz continuity of the solution flow under this metric, with generic regularity results.

Findings

Solution flow is Lipschitz under the new metric

Piecewise smooth transportation paths exist among solutions

Results apply to large data solutions without size restrictions

Abstract

We consider the global well-posedness of weak energy conservative solution to a general quasilinear wave equation through variational principle, where the solution may form finite time cusp singularity, when energy concentrates. As a main result in this paper, we construct a Finsler type optimal transport metric, then prove that the solution flow is Lipschitz under this metric. We also prove a generic regularity result by applying Thom's transversality theorem, then find piecewise smooth transportation paths among a dense set of solutions. The results in this paper are for large data solutions, without restriction on the size of solutions.