Wave equations on non-smooth space-times

TL;DR

This paper extends the theory of wave equations to non-smooth Lorentzian manifolds, establishing existence and uniqueness of solutions in both smooth and weakly singular metric cases, advancing mathematical understanding of wave behavior in irregular spacetimes.

Contribution

It introduces a novel framework for solving wave equations on non-smooth space-times using Colombeau's algebra, covering both local and global solutions for weakly singular metrics.

Findings

Global unique solvability for smooth globally hyperbolic space-times.

Existence and uniqueness results for weakly singular, locally bounded metrics.

Extension of classical solution theory to low-regularity Lorentzian manifolds.

Abstract

We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth globally hyperbolic space-times. Then we turn to the case where the metric is non-smooth and present a local as well as a global existence and uniqueness result for a large class of Lorentzian manifolds with a weakly singular, locally bounded metric in Colombeau's algebra of generalized functions.