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.