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

François  James; Nicolas Vauchelet

doi:10.3934/dcds.2016.36.1355

Article Contents

2016, Volume 36, Issue 3: 1355-1382. Doi: 10.3934/dcds.2016.36.1355

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

François James^1, and
Nicolas Vauchelet^2,

1.
Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans & CNRS UMR 7349, Fédération Denis Poisson, Université d'Orléans & CNRS FR 2964, 45067 Orléans Cedex 2
2.
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, INRIA Paris-Rocquencourt, EPI MAMBA, F-75005, Paris

Received: December 31, 2013

Revised: May 31, 2015

Published: May 2017

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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 self-consistent interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)) [18], a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developed. 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 developed in the spirit of James and Vauchelet (NoDEA (2013)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.

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Mathematics Subject Classification: Primary: 35B40, 35D30, 35L60, 35Q92, 49K20.

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