A hybrid variational principle for the Keller–Segel system in ℝ²

¹ TSE (GREMAQ, Université Toulouse 1 Capitole), 21 Allée de Brienne, 31015 Toulouse cedex 6, France.
adrien.blanchet@ut-capitole.fr
² Department of Mathematics, Imperial College London, London SW7 2AZ, UK.
carrillo@imperial.ac.uk
³ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
davidk@andrew.cmu.edu
⁴ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.
kowalczy@dim.uchile.cl
⁵ Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, 31062 Toulouse cedex 9, France.
Philippe.Laurencot@math.univ-toulouse.fr
⁶ Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia, Italy.
stefano.lisini@unipv.it

Received: 25 March 2015

Abstract

We construct weak global in time solutions to the classical Keller–Segel system describing cell movement by chemotaxis in two dimensions when the total mass is below the established 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 minimizing implicit scheme for Wasserstein distances introduced by [R. Jordan, D. Kinderlehrer and F. Otto, SIAM J. Math. Anal. 29 (1998) 1–17].