2021/02/26 by Marco Di Francesco, Di Francesco, Marco, Graziano Stivaletta +1
Engineering · Mathematics · Physics and Astronomy · #35A24 #35A35 #35Q70. Secondary: 35F55 #65N75 #90B20 #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Primary: 35L65
paper · pdf · doi:10.48550/arxiv.2103.00061
openalex publication_date 2021/02/26 · openalex created_date 2023/01/09 · openalex updated_date 2026/07/28
We consider the follow-the-leader particle approximation scheme for a 1d scalar conservation law with nonnegative L^∞c 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 approximating 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 proof. 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.