Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients

TL;DR

This paper compares different solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients, focusing on generalized characteristics and energy estimates, and illustrates their independence through examples.

Contribution

It analyzes and contrasts various solution frameworks for hyperbolic equations with irregular coefficients, highlighting their differences and independence.

Findings

Different solution concepts are logically independent.

Examples illustrate the applicability of each solution notion.

Theories extend classical methods to less regular coefficients.

Abstract

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts.