Variational properties of the kinetic solutions of scalar conservation laws

TL;DR

This paper explores the variational properties of kinetic solutions to scalar conservation laws, demonstrating that these solutions can be viewed as curves in a Hilbert space with unique tangent minimizers of an interaction functional.

Contribution

It introduces a novel variational framework for kinetic solutions, showing their geometric interpretation as curves with unique tangents in a Hilbert space.

Findings

Kinetic solutions form curves in a Hilbert space.

Tangents to these curves are unique minimizers of an interaction functional.

Provides a new perspective on the structure of solutions to scalar conservation laws.

Abstract

We discuss properties of kinetic solutions of scalar conservation laws in the variational approach developed by Eu. Panov and Y. Brenier. Our main result shows that such solutions can be considered as curves in a suitable Hilbert space with tangents that are unique minimizers of an interaction functional.