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

Adrien Blanchet | Zendy

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

Author(s) -

Adrien Blanchet

Publication year - 2014

Publication title -

séminaire laurent schwartz — edp et applications

Language(s) - English

Resource type - Journals

ISSN - 2266-0607

DOI - 10.5802/slsedp.6

Subject(s) - dictyostelium discoideum , diffusion , random walk , petri dish , dimension (graph theory) , realization (probability) , space (punctuation) , taxis , scale (ratio) , mathematics , biological system , physics , computer science , chemistry , pure mathematics , biology , thermodynamics , statistics , biochemistry , genetics , botany , quantum mechanics , gene , operating system

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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