Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system
An approximation to the measure valued, global in time solutions to the (Patlak-)Keller-Segel 2D model basing on systems of stochastic interacting particles is given. The proposed approach, by the presented numerical method, reproduces the well-known dichotomy (with the crucial mass \(8\pi\)) in the qualitative behavior of the system and gives the solution even after the blow-up events. Chemotaxis is a biological phenomenon in which living organisms direct their movements according to certain chemical in the environment. The (Patlak-)Keller-Segel model describes the collective motion of cells which are attracted by a self-emitted chemical substance. This model shows that there is a competition between the tendency of cells to spread by diffusion and the tendency to aggregate. The balance between these two mechanisms happens precisely at the critical mass \(M=8\pi\). In the literature there can be find many papers dedicated to the analysis of the (Patlak-)Keller-Segel system, but the authors found only six that deals with its numerical treatment, thus they proposed a discrete approximation of the equations presented in Section~1 (namely (8)--(10)) with a system of interacting particles. The details can be found in Section~2 where one can read about the formal derivation of the stochastic approximation and formal calculation of the critical mass with description of system behavior when the system consists of light particles only and the system has a supercritical total mass \(M>8\pi\). After the discretization in time presented in Section~3 where the collisions between particles are considered as coarsening of the discrete grid, a some kind of operator-splitting strategy is presented with divagations about the choice of the time step length \(\Delta t\) to ensure the stability of the system of particles and on the other hand to guarantee the acceptable costs of computations. In the presented approach the interactions of a given particle with near particles are evaluated one-to-one, while the clusters of far particles are approximated by their centers of gravity. To illustrate the functionality of the proposed approach two examples of numerical results are presented. In both cases the number \(N=2000\) particles and the time step \(\Delta t=10^{-3}\) are taken. In the last section the authors gave a convergence proof for the regularized model - the proof is based on a BBGKY-like approach. It is worth to mention that the paper is based on a previous work of the authors [Stochastic particle approximation to the global measure valued solutions of the Keller-Segel model in 2D. in: Proceedings of the Conference EQUADIFF (2007)], but here we can find the detailed description, numerical results and discussion about the considered problem.
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