On a Hamiltonian regularization of scalar conservation laws

Abstract

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $\ell >0$ and conserves an $H^1$ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $\ell$ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization. This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.

References in corpus (3)

• Global weak solutions of a Hamiltonian regularised Burgers equation
• Global weak solutions of the Serre-Green-Naghdi equations with surface tension
• On the blow-up scenario for some modified Serre-Green-Naghdi equations

Cited by in corpus (2)

• Hamiltonian regularisation of the unidimensional barotropic Euler equations
• Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom