Abstract
The Keller-Segel equation describes the movement of cells by chemotaxis. Cells diffuse in the plane, and emit a chemical. This product, which also diffuses, attracts the cells. This leads to a singular interaction between the cells (via the product). This interaction is critical in the sense that, depending on the values of the constants, there may be global existence of a solution, or formation of a cluster of cells in fini time. A new proof of non-explosion in the subcritical case will be described, which allows this equation to be approximated by stochastic particle systems, in the elliptic setting where the product diffuse instantaneously. We'll also discuss the approximation of the solution by a particle system in the parabolic frame (a very subcritical case), where the product diffuse at a finie speed. Finally, the explosion of the particle system in the supercritical elliptic frame will be precisely described.
From work with B. Jourdain, Y. Tardy and M. Tomasevic.
