Hyperbolic conservation laws on spacetimes

TL;DR

This paper extends Kruzkov's theory of hyperbolic conservation laws to manifolds, establishing existence, uniqueness, and stability of solutions on spacetimes with arbitrary topology, using geometric and differential form methods.

Contribution

It generalizes conservation law theory to (n+1)-manifolds with a novel geometric framework involving n-forms and hyperbolic spacetime conditions.

Findings

Proves existence and uniqueness of entropy solutions on spacetimes.

Derives a geometric L1 semi-group property for solutions.

Introduces an alternative flux framework with parametrized vector fields.

Abstract

We present a generalization of Kruzkov's theory to manifolds. Nonlinear hyperbolic conservation laws are posed on a differential (n+1)-manifold, called a spacetime, and the flux field is defined as a field of n-forms depending on a parameter. The entropy inequalities take a particularly simple form as the exterior derivative of a family of n-form fields. Under a global hyperbolicity condition on the spacetime, which allows arbitrary topology for the spacelike hypersurfaces of the foliation, we establish the existence and uniqueness of an entropy solution to the initial value problem, and we derive a geometric version of the standard L1 semi-group property. We also discuss an alternative framework in which the flux field consists of a parametrized family of vector fields.