dc.contributor.author Francesco, Marco Di dc.contributor.author Stivaletta, Graziano dc.date.accessioned 2021-03-24T07:40:11Z dc.date.available 2021-03-24T07:40:11Z dc.date.issued 2021-02-26 dc.identifier.uri http://hdl.handle.net/10754/668229 dc.description.abstract We consider the follow-the-leader particle approximation scheme for a $1d$ scalar conservation law with nonnegative $L^\infty_c$ initial datum and with a $C^1$ concave flux, which is known to provide convergence towards the entropy solution $\rho$ 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 $\rho^n$ 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 $\left(f(\rho)/\rho\right)_x\leq 1/t$ of said condition allows to select a unique weak solution, despite $\left(f(\rho)/\rho\right)_x\leq 1/t$ is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition $f'(\rho)_x\leq 1/t$. 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 $f(\rho)=\rho(A-\rho^\gamma)$ 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. dc.description.sponsorship We thank M. D. Rosini for his useful suggestions on the writing of this manuscript. Part of this work was carried out during the visit of MDF to King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. MDF is deeply grateful for the warm hospitality by people at KAUST, for the excellent scientific environment, and for the support in the development of this work. MDF acknowledges support from the “InterMaths” project of the DISIM Department at the University of L’Aquila. dc.publisher arXiv dc.relation.url https://arxiv.org/pdf/2103.00061.pdf dc.rights Archived with thanks to arXiv dc.title The one-sided Lipschitz condition in the follow-the-leader approximation of scalar conservation laws dc.type Preprint dc.eprint.version Pre-print dc.contributor.institution DISIM - Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio 1 (Coppito) 67100 L’Aquila (AQ) - Italy dc.identifier.arxivid 2103.00061
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