On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher

Details On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher

2011-09-07

arxiv.org/abs/1109.1543v1

Mathematics

Analysis of PDEs

26 pages

Scientific paper

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated.

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