Critical mass in a
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
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.