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

Adrien Blanchet, José A. Carrillo+4 more|arXiv (Cornell University)|Jul 21, 2014

Mathematical Biology Tumor Growth22 references4 citations

TL;DR

This paper introduces a hybrid variational principle to construct global weak solutions for the 2D Keller-Segel system with total mass below the critical threshold $8\pi$. By leveraging a gradient flow structure in a Wasserstein product space, the authors extend the Jordan-Kinderlehrer-Otto minimising scheme to handle the coupled parabolic-parabolic system, ensuring mass preservation and long-time existence without blow-up.

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).

Motivation & Objective

• To establish the existence of global weak solutions for the parabolic-parabolic Keller-Segel system in $\mathbb{R}^2$ when the total mass is below the critical value $8\pi$.
• To develop a variational framework that captures the gradient flow structure of the system in a product space of probability measures and $H^1$ functions.
• To generalize the minimising implicit scheme of Jordan, Kinderlehrer, and Otto to a hybrid setting suitable for chemotaxis models with coupled dynamics.
• To ensure the constructed solutions preserve mass and avoid finite-time blow-up, even in the presence of nonlinear drift and diffusion.
• To provide a rigorous variational construction that yields free energy solutions with improved regularity and compactness properties.

Proposed method

• Formulate the Keller-Segel system as a gradient flow in a product space of probability measures and $H^1$ functions, using the free energy functional $\mathcal{E}$.
• Apply a hybrid variational scheme combining time-discretisation and Wasserstein distance minimisation to define incremental minimisers at each time step.
• Use the Matthes-McCann-Savaré flow interchange technique to derive regularity estimates for the minimisers in the discrete scheme.
• Establish uniform bounds on the discrete energy and dissipation using a modified Biler-Hebisch-Nadzieja inequality and Carleman-type estimates.
• Prove compactness of the interpolated solutions via narrow convergence and weak convergence in $L^2$ using Proposition C.1 on vector field compactness.
• Pass to the limit in the time-discretised scheme to obtain a global weak solution satisfying the continuity equation and transport equation in a distributional sense.

Experimental results

Research questions

• RQ1 Can a global weak solution be constructed for the 2D parabolic-parabolic Keller-Segel system when the total mass is below the critical threshold $8\pi$?
• RQ2 How can the Jordan-Kinderlehrer-Otto minimising scheme be generalised to handle systems with coupled parabolic equations and nonlocal interactions?
• RQ3 What variational structure underlies the Keller-Segel system that allows for a stable time-discretisation and energy dissipation control?
• RQ4 What compactness and regularity properties emerge from the variational scheme that ensure convergence to a weak solution?
• RQ5 How does the hybrid variational principle preserve mass and prevent finite-time blow-up in the critical mass regime?

Key findings

• Global weak solutions exist for the 2D parabolic-parabolic Keller-Segel system when the chemotactic sensitivity $\chi$ is below $8\pi$, the critical mass threshold.
• The constructed solutions are free energy solutions that preserve the total mass and remain bounded in time, avoiding finite-time blow-up.
• The hybrid variational principle ensures discrete energy dissipation and uniform bounds on the $L^2$-norm of the density and its logarithmic derivative.
• A modified Biler-Hebisch-Nadzieja inequality and Carleman-type estimate are used to control the entropy and logarithmic moments, enabling uniform estimates.
• Compactness of the interpolants is established via narrow convergence of measures and weak convergence of vector fields, allowing passage to the continuous limit.
• The method yields a solution that is a probability measure for all time, with $\rho(t,\cdot) \in \mathcal{P}(\mathbb{R}^2)$, ensuring physical consistency.