The Filippov characteristic flow for the aggregation equation with mildly singular potentials

José Antonio Carrillo 1 Francois James 2 Frédéric Lagoutière 3 Nicolas Vauchelet 4, 5
1 Department of Mathematics [Imperial College London]
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay
4 LJLL - Laboratoire Jacques-Louis Lions
5 MAMBA - Modelling and Analysis for Medical and Biological Applications
LJLL - Laboratoire Jacques-Louis Lions, Inria de Paris
Abstract : Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.
Keywords : finite volume schemes. Filippov's flow aggregation equation finite volume schemes measure-valued solutions nonlocal conservation equations gradient flow
Type de document :
Article dans une revue
Journal of Differential Equations, Elsevier, 2016, 260 (1), pp.304-338
Liste complète des métadonnées

Littérature citée [45 références]  Voir  Masquer  Télécharger
https://hal.archives-ouvertes.fr/hal-01061991
Contributeur : Nicolas Vauchelet <>
Soumis le : lundi 8 septembre 2014 - 23:12:48
Dernière modification le : jeudi 7 février 2019 - 16:19:11
Document(s) archivé(s) le : mardi 9 décembre 2014 - 12:50:38

Fichiers

aggregation_v6.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01061991, version 1
  • ARXIV : 1409.2811

Collections

UNIV-PARIS7 | INRIA | UNIV-ORLEANS | INSMI | UPMC | LJLL | LM-ORSAY | MSL | MSL-THESE | TDS-MACS | USPC | UNIV-PARIS-SACLAY | IDP | INRIA-ROCQ | SORBONNE-UNIVERSITE

Citation

José Antonio Carrillo, Francois James, Frédéric Lagoutière, Nicolas Vauchelet. The Filippov characteristic flow for the aggregation equation with mildly singular potentials. Journal of Differential Equations, Elsevier, 2016, 260 (1), pp.304-338. 〈hal-01061991〉

Exporter

BibTeX TEI DC DCterms EndNote

Partager

Métriques

Consultations de la notice

1117

Téléchargements de fichiers

346

Contact

support.ccsd.cnrs.fr
hal.support@ccsd.cnrs.fr
Logo CCSD