BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Charles Collot (Courant institute of mathematical Sciences) DTSTART:20200423T140000Z DTEND:20200423T145000Z DTSTAMP:20260404T143401Z UID:IMS/5 DESCRIPTION:Title: On the derivation of the homogeneous kinetic wave equation\nby Char les Collot (Courant institute of mathematical Sciences) as part of PDE sem inar via Zoom\n\n\nAbstract\nThe kinetic wave equation arises in many phys ical situations: the description of small random surface waves\, or out of equilibria dynamics for large quantum systems for example. In this talk w e are interested in its derivation as an effective equation from the nonli near Schrodinger equation (NLS) for the microscopic description of a syste m. More precisely\, we will consider (NLS) in a weakly nonlinear regime on the torus in any dimension greater than two\, and for highly oscillatory random Gaussian fields as initial data. A conjecture in statistical physic s is that there exists a kinetic time scale on which\, statistically\, the Fourier modes evolve according to the kinetic wave equation. We prove thi s conjecture up to an arbitrarily small polynomial loss in a particular re gime\, and obtain a more restricted time scale in other regimes. The main difficulty\, that I will comment on\, is that one needs to identify the le ading order statistically observable nonlinear effects. This means underst anding correlation between Fourier modes\, and relating randomness with st ability and local well-posedness. The key idea of the analysis is the use of Feynman interaction diagrams to understand the solution as colliding li near waves. We use this framework to construct an approximate solution as a truncated series expansion\, and use in addition random matrices tools t o obtain its nonlinear stability in Bourgain spaces. This is joint work wi th P. Germain.\n LOCATION:https://stable.researchseminars.org/talk/IMS/5/ END:VEVENT END:VCALENDAR