A remark on duality solutions for some weakly nonlinear scalar conservation laws

TL;DR

This paper studies duality solutions for weakly nonlinear scalar conservation laws with nonlocal interactions, proposing a flux-based selection criterion to ensure uniqueness of solutions.

Contribution

It introduces a flux-based approach to select unique duality solutions for scalar conservation laws with nonlocal interactions, extending previous notions.

Findings

A notion of duality solution without uniqueness is identified.

A flux definition is proposed to ensure solution uniqueness.

The approach generalizes existing frameworks for nonlinear conservation laws.

Abstract

We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following the work of Bouchut and James (Comm. Partial Diff. Eq., 24, 1999), a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. Then we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.