The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

Abstract

We consider the follow-the-leader particle approximation scheme for a [Formula: see text] scalar conservation law with non-negative compactly supported [Formula: see text] initial datum and with a [Formula: see text] concave flux, which is known to provide convergence towards the entropy solution [Formula: see text] to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density [Formula: see text] is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on [Formula: see text] (which imply concavity of [Formula: see text]) we prove that the continuum version [Formula: see text] of said condition allows to select a unique weak solution, despite [Formula: see text] is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition [Formula: see text]. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case [Formula: see text] the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.