TY - JOUR

T1 - The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

AU - Di Francesco, Marco

AU - Stivaletta, Graziano

N1 - KAUST Repository Item: Exported on 2023-03-06
Acknowledgements: The authors are grateful to the anonymous referee for her/his comments to the first version of this paper, which led to an improvement of the results. We thank M. D. Rosini for his useful suggestions on the writing of this paper. 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2023/1/5

Y1 - 2023/1/5

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/668229

UR - https://www.worldscientific.com/doi/10.1142/S0219891622500205

UR - http://www.scopus.com/inward/record.url?scp=85146296842&partnerID=8YFLogxK

U2 - 10.1142/S0219891622500205

DO - 10.1142/S0219891622500205

M3 - Article

SN - 0219-8916

VL - 19

SP - 775

EP - 807

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

IS - 4

ER -