Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux I

TL;DR

This paper explores various notions of continuous solutions to a balance law with smooth flux, establishing their equivalence and uniqueness, and analyzing the ODE reduction along characteristics under specific conditions.

Contribution

It extends previous work to general smooth fluxes, proves the equivalence of distributional and Lagrangian solutions, and characterizes continuous solutions as Kruzkov iso-entropy solutions.

Findings

Continuous solutions are Kruzkov iso-entropy solutions.

Equivalence between distributional and Lagrangian solutions is established.

ODE reduction on characteristics holds under negligible inflection points.

Abstract

We discuss different notions of continuous solutions to the balance law \[u_t + (f(u ))_x =g \] with $g$ bounded, $f\in C^{2}$, extending previous works relative to the flux $f(u)=u^{2}$. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also establish the ODE reduction on any characteristics under the sharp assumption that the set of inflection points of the flux $f$ is negligible. The correspondence of the source terms in the two settings is matter of a companion work, where we also provide counterexamples when the negligibility on inflection points fails.