Nonlinear Analysis and Optimization Seminar[ Edit ]
Moderator: Simeon Reich
The parabolic-elliptic Patlak-Keller-Segel system for $n$-populations describes the collective motion of cells interacting via a self-produced chemical agent (called chemoattractant). In two space dimensions and for a single population ($n=1$) it is known that the $L^1$-norm of the initial datum is a salient parameter which separates the dichotomy between global in time existence and finite time blow up (or, chemotactic collapse). In other words, there exists an (explicitly known) critical value $\beta_c$ such that a solution exists globally if and only if the initial $L^1$ norm is smaller than or equal to $\beta_c$. In this talk I will discuss the global existence of solutions for $n$-populations. Exploring the gradient flow structure of the system in Wasserstein space, we show that whenever the $L^1$ norm of the initial datum satisfies an appropriate notion of 'sub-criticality', the system admits a global solution.
This is joint work with Gershon Wolansky (arXiv: 1902.10736)