Publication 14-CNA-023

A Hybrid Variational Principle for the Keller-Segel System In $ \mathbb { R^2 }$

Adrien Blanchet
TSE (GREMAQ, Universite Toulouse 1 Capitole)
Toulouse, France
Adrien.Blanchet@ut-capitole.fr

Jose Antonio Carrillo
Department of Mathematics
Imperial College London
London
carrillo@imperial.ac.uk

David Kinderlehrer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
davidk@andrew.cmu.edu

Michal Kowalczyk
Departamento de Ingeniera Matematica and
Centro de Modelamiento Matematico (UMI 2807 CNRS)
Universidad de Chile, Casilla
Santiago, Chile
kowalczy@dim.uchile.cl

Philippe Laurençot
Institut de Mathematiques de Toulouse
Toulouse, France
Philippe.Laurencot@math.univ-toulouse.fr

Stefano Lisini
Dipartimento di Matematica "F. Casorati"
Universita degli Studi di Pavia
Pavia, Italy
stefano.lisini@unipv.it

Abstract: We construct weak global in time solutions to the classical Keller-Segel system cell movement by chemotaxis in two dimensions when the total mass is below the well-known critical value. Our construction takes advantage of the fact that the Keller-Segel system can be realized as a gradient flow in a suitable functional product space. This allows us to employ a hybrid variational principle which is a generalisation of the minimising implicit scheme for Wasserstein distances introduced by Jordan, Kinderlehrer and Otto (1998).

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