A user’s guide to PDE models for chemotaxis

By Thomas Hillen, Kevin J. Painter. Journal of Mathematical Biology, 2009

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Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “For literature on PKS we refer the reader to the review articles [33, 32]; see also [31, 26, 19] for parabolic-parabolic Keller-Segel systems.”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “This system is generally considered the fundamental mathematical model for the study of aggregation by chemotaxis of certain microorganisms [59, 49, 45, 44].”
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model (2010, methodology): “We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards ̺λ.”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, neutral): “Also note that the mass is conserved∫ R2 ρ(t, x) dx = ∫ R2 ρ0(x) dx =: M .”
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion (2013, neutral): “[8]), { nt = ∇ · ( D(n)∇n) − ∇ · (S(n)∇c), (1.”