On a Hamiltonian regularization of scalar conservation laws

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $\mathrm{\ell }>0$ and conserves an $H^1$ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $\mathrm{\ell }$ 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.