Modul:   MAT673  Seminar PDE and Mathematical Physics

[Video Seminar] Invariant measures and long time behavior for the Benjamin-Ono equation

Talk by Prof. Dr. Nikolay Tzvetkov

Date: 25.09.14  Time: 18.15 - 19.15  Room: Y27H35/36

The KdV and the Benjamin-Ono equations are basic models, derived from the water waves equations for the propagation of long, small amplitude one dimensional waves. The solutions of the KdV equations, posed on the torus are known to be almost periodic in time. The long time behavior of the Benjamin-Ono equation, posed on the torus is much less understood. In this talk, we will present some progress on this problem. Namely, we shall construct an infinite sequence of weighted gaussian measures which are invariant by the flow of the Benjamin-Ono equation. These measures are supported by Sobolev spaces of increasing regularities. The "probabilistic view point" is essential in our analysis. In particular our arguments are less dependent on the particular behavior of each trajectory, compared to previous works on the subject. The talk is based on a series of works by Yu Deng (Princeton University), Nicola Visciglia (University of Pisa) and the speaker.