Uniqueness for Keller-Segel-type chemotaxis models

TL;DR

This paper establishes the uniqueness of solutions for Keller-Segel chemotaxis models, including fully parabolic and classical cases, using displacement convexity and energy methods, thereby advancing the mathematical understanding of these systems.

Contribution

It proves uniqueness of solutions for a broad class of Keller-Segel models, including nonlinear diffusions, using novel quasi-Lipschitz and displacement convexity techniques.

Findings

Uniqueness holds for fully parabolic and classical Keller-Segel models.

Displacement convexity of the free energy is derived from its evolution.

The methods extend to nonlinear diffusions satisfying McCann's convexity condition.

Abstract

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.