These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higher dimensions, obtained throught a gradient flow approach in the Wassertein metric. These models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc 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.
chemo-taxis; Keller-Segel model; degenerate diffusion; minimising scheme; Monge-Kantorovich distance;
Adrien Blanchet, « A gradient flow approach to the Keller-Segel systems », RIMS Kokyuroku's lecture note, vol. 1837, juin 2013, p. 52–73.