Hyperbolic conservation laws on manifolds with limited regularity

TL;DR

This paper develops a framework for solving nonlinear hyperbolic conservation laws on manifolds with limited regularity, including Lorentzian manifolds, establishing existence and uniqueness of weak solutions with entropy and boundary conditions.

Contribution

It introduces a novel formulation for hyperbolic conservation laws on manifolds with minimal regularity assumptions, covering cases like Lorentzian manifolds.

Findings

Existence of an L1 semi-group of weak solutions

Uniqueness of solutions under entropy and boundary conditions

Applicable to manifolds with limited geometric regularity

Abstract

We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions.