Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - S. Lisini - E. Mainini

Uniqueness for Keller-Segel-type chemotaxis models

created by mainini on 27 Dec 2012
modified by lisini on 05 Sep 2014


Published Paper

Inserted: 27 dec 2012
Last Updated: 5 sep 2014

Journal: Discrete Contin. Dyn. Syst.-Series A
Volume: 34
Number: 4
Pages: 1319–1338
Year: 2014
Doi: 10.3934/dcds.2014.34.1319


We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

Keywords: displacement convexity, Wasserstein distance, Gradient flows, Chemotaxis, Keller-Segel model