Germs for scalar conservation laws: the Hamilton-Jacobi equation point of view

TL;DR

This paper establishes a connection between entropy solutions of scalar conservation laws with discontinuous flux and derivatives of Hamilton-Jacobi solutions with discontinuous Hamiltonians, using numerical schemes and viscosity theory.

Contribution

It characterizes maximal and complete germs and relaxation operators in the context of scalar conservation laws and Hamilton-Jacobi equations with discontinuities, extending existing theories.

Findings

Proved equivalence of entropy solutions and Hamilton-Jacobi derivatives.

Characterized maximal and complete germs in the AKR theory.

Validated convergence of numerical schemes for discontinuous flux and Hamiltonians.

Abstract

We prove that the entropy solution to a scalar conservation law posed on the real line with a flux that is discontinuous at one point (in the space variable) coincides with the derivative of the solution to a Hamilton-Jacobi (HJ) equation whose Hamiltonian is discontinuous. Flux functions (Hamiltonians) are not assumed to be convex in the state (gradient) variable. The proof consists in proving the convergence of two numerical schemes. We rely on the theory developed by B.~Andreianov, K.~H.~Karlsen and N.~H.~Risebro (\textit{Arch. Ration. Mech. Anal.}, 2011) for such scalar conservation laws and on the viscosity solution theory developed by the authors (\textit{arxiv}, 2023) for the corresponding HJ equation. This study allows us to characterise certain germs introduced in the AKR theory (namely maximal and complete ones) and relaxation operators introduced in the viscosity solution framework.