Agent-Based Simulation of the Two-Dimensional Patlak-Keller-Segel Model

The two-dimensional Keller-Segel model after blow-up

Grow-Up Rate and Refined Asymptotics for a Two-Dimensional Patlak-Keller-Segel Model in a Disk

We consider a special case of the Patlak-Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the...

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak-Keller-Segel Model

Abstract. Variational steepest descent approximation schemes for the modified Patlak-KellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recov...

A finite volume scheme for the Patlak-Keller-Segel chemotaxis model

A finite volume method is presented to discretize the Patlak-Keller-Segel (PKS) model for chemosensitive movements. On the one hand, we prove existence and uniqueness of a numerical solution to the proposed scheme. On the other hand, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation convergences to the solution to the PK...

The Patlak-Keller-Segel Model and Its Variations: Properties of Solutions via Maximum Principle

In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concen...