A hybrid variational principle for the Keller-Segel system in $\mathbb R^2$

Adrien Blanchet (1) , José A. Carrillo (2) , David Kinderlehrer (3) , Michal Kowalczyk (4) , Philippe Laurençot (5) , Stefano Lisini (6)

1 GREMAQ - Groupe de recherche en économie mathématique et quantitative
2 Department of Mathematics [Imperial College London]
3 Department of Mathematical Sciences [Pittsburgh]
4 DIM - Depto Ingeniería Matemática
5 IMT - Institut de Mathématiques de Toulouse UMR5219
6 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]

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).

Mots clés: Wasserstein distance; chemotaxis; Keller-Segel model; minimising scheme; Kantorovich-Rubinstein-Wasserstein distance

Domaines: Equations aux dérivées partielles [math.AP]

Adrien Blanchet, José A. Carrillo, David Kinderlehrer, Michal Kowalczyk, Philippe Laurençot, et al.. A hybrid variational principle for the Keller-Segel system in $\mathbb R^2$. ESAIM: Mathematical Modelling and Numerical Analysis, 2015, 49 (6), pp.1553 - 1576. ⟨10.1051/m2an/2015021⟩. ⟨hal-01026281⟩