Critical mass in a Keller-Segel model
Nader Masmoudi (CIMS)

The Patlak-Keller-Segel (PKS) model describes the collective motion of cells which are attracted by a self-emitted chemical substance. The long time behavior depends on the total mass which is assumed here to be conserved. It was conjectured by S. Childress and K. Percus that in two space dimensions there is a threshold number above which there is a chemotactic collapse.

One can prove that if the initial mass is below a critical mass 8 pi then the solution is global and spreads when t goes to infinity. If the initial mass is above the critical mass  8 pi then there is blow up in finite time.  For the critical mass 8 pi, there is infinite time aggregation. One of the main tools in proving these results is the use of the free-energy of the system combined with a Logarithmic Hardy-Littlewood-Sobolev inequality with a sharp constant.

We will also discuss the derivation of the model from kinetic models and show some numerics.