Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations - Archive ouverte HAL Access content directly
Journal Articles Discrete and Continuous Dynamical Systems - Series A Year : 2016

Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

Abstract

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a selfconsistant interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)), a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developped. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developped in the spirit of James and Vauchelet (NoDEA (2013)). However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
Fichier principal
Vignette du fichier
GF_dual_corr1.pdf (396.71 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-00803709 , version 1 (22-03-2013)
hal-00803709 , version 2 (17-01-2014)
hal-00803709 , version 3 (04-09-2014)

Identifiers

Cite

Francois James, Nicolas Vauchelet. Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations. Discrete and Continuous Dynamical Systems - Series A, 2016, 36 (3), pp.1355-1382. ⟨10.3934/dcds.2016.36.1355⟩. ⟨hal-00803709v3⟩
679 View
528 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More