Weak diffeomorphisms and solutions to conservation laws

TL;DR

This paper extends the classical equivalence of Eulerian and Lagrangian frameworks to weak solutions of hyperbolic conservation laws, introducing a new approach using weak diffeomorphisms and Riemann invariants for systems without explicit velocity fields.

Contribution

It develops a novel framework that generalizes particle paths to scalar and one-dimensional systems lacking explicit velocity fields, using weak diffeomorphisms and Riemann invariants.

Findings

Established equivalence for weak solutions of conservation laws.

Introduced a new framework using weak diffeomorphisms.

Extended particle path concepts to broader classes of problems.

Abstract

Evolution equations which describe the changes in a velocity field over time have been classically studied within the Eulerian or Lagrangian frame of reference. Classically, these frameworks are equivalent descriptions of the same problem, and the equivalence can be demonstrated by constructing particle paths. For hyperbolic conservation laws, we extend the equivalence between these frameworks to weak solutions for a broad class of problems. Our main contribution in this paper is that we develop a new framework to extend the idea of a particle path to scalar equations and to systems in one dimension which do not explicitly include velocity fields. For systems, we use Riemann invariants as the tool to develop an analog to particle paths.