Hyperbolic conservation laws and spacetimes with limited regularity

TL;DR

This paper explores the mathematical foundations of hyperbolic conservation laws on manifolds, addressing existence and behavior of weak solutions in contexts like geophysical flows and general relativity.

Contribution

It develops a framework for weak solutions of hyperbolic conservation laws on Riemannian and Lorentzian manifolds, including cases where the metric is an unknown.

Findings

Established existence results for weak solutions.

Analyzed qualitative behavior of solutions.

Applied framework to Einstein-Euler equations.

Abstract

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either be fixed (shallow water equations on the sphere, for instance) or be one of the unknowns of the theory (Einstein-Euler equations of general relativity).