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 1d scalar conservation law with non-negative compactly supported L∞ initial datum and with a C1 concave flux, which is known to provide convergence towards the entropy solution ρ 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 ρn is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f(ρ)/ρ)x ≤ 1/t of said condition allows to select a unique weak solution, despite (f(ρ)/ρ)x ≤ 1/t is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition f'(ρ)x ≤ 1/t. 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 f(ρ) = ρ(A-ργ) 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.