Functional Inequalities, Mass Transportation, and Asymptotic for the Critical Mass Keller-Segel Model
abstract: We investigate the long time behavior of the critical mass Keller-Segel equation, which has a one parameter family of steady-state solutions whose second moment is not bounded. The equation is gradient flow in the $2$-Wasserstein metric for a non-displacement convex functional. However, it has a second Lyapunov function that is displacement convex. Using this, we show that the steady state solutions. The proof relies heavily on mass transportation techniques, and has a number of novel features. For instance, although the second Lyapunov functional, which is an entropy for the fast diffusion equation, also has a strictly positive dissipation for the Keller-Segal evolution, this dissipation turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality, and then an interplay between the two Lyapunov functionals to prove the existence of basins of attraction for each stationary state composed by certain initial data converging mtowards the steady states. This is joint work with A. Blanchet and J.A. Carrillo.
timetable:
Tue 12 Oct, 9:00 - 9:50, Aula Dini
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