Extending Lagrangian transformations to nonconvex scalar conservation laws

TL;DR

This paper develops a method to find Lagrangian transformations for scalar conservation laws with smooth flux by using weak diffeomorphisms, extending the applicability of such transformations to nonconvex cases.

Contribution

It introduces a novel approach to derive Lagrangian transformations for nonconvex scalar conservation laws via a modified Temple system.

Findings

Established existence of a strictly hyperbolic system derived from scalar laws.

Proved equivalence of entropy solutions between the system and the scalar law.

Provided a method to determine associated weak diffeomorphisms.

Abstract

The present paper studies a method of finding Lagrangian transformations, in the form of particle paths, for all scalar conservation laws having a smooth flux. These are found using the notion of weak diffeomorphisms. More precisely, from any given scalar conservation law, we derive a Temple system having one linearly degenerate and one genuinely nonlinear family. We modify the system to make it strictly hyperbolic and prove an existence result for it. Finally we establish that entropy admissible weak solutions to this system are equivalent to those of the scalar equation. This method also determines the associated weak diffeomorphism.