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Eric Carlen

2011-01-12
09:00:00 - 09:50:00

R102 , Astronomy and Mathematics Building

We investigate the long time behavior of two non-linear evolution equationsinvolving entropy and the the 2D Newtonian potential. Both may be effectively investigated in a Wasserstein metric setting, pursuing a path opened up by Felix Otto, and both have interesting connections.

The first is 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 the interplay between the two Lyapunov functionals, we determine basins of attraction for the steady state solutions. The proof relies heavily on mass transportation techniques, and has a number of novel features. For instance, the second Lyapunov functional, has a dissipation functional that is a difference of two terms, neither of which need to be small when the dissipation is small, and the level sets of this dissipation functional are not compact. We introduce a strategy of controlled concentration to deal with these 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 towards the steady states. This is joint work with A. Blanchet and J.a. Carrillo.

The second example is a modified version of the 2D Navier-Stokes equation introduced by Cagliotti, Pulvirenti and Rousset. The modifications are such that for certain initial data, the long time behavior of the modified equations is the intermediate time behavior of the original equation. This time, instead of dealing with multiple Lyapunov functionals, we deal with multiple conservations laws, and contrained gradient, as studied in previous work with Gangbo.

For material related to this talk, click here.