Weak Diffeomorphisms and Solutions to Conservation Laws

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. © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2021.