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:20260423T021120Z UID:IMS/5 DESCRIPTION:Title: On t he derivation of the homogeneous kinetic wave equation\nby Charles Col lot (Courant institute of mathematical Sciences) as part of PDE seminar vi a Zoom\n\n\nAbstract\nThe kinetic wave equation arises in many physical si tuations: the description of small random surface waves\, or out of equili bria dynamics for large quantum systems for example. In this talk we are i nterested in its derivation as an effective equation from the nonlinear Sc hrodinger equation (NLS) for the microscopic description of a system. More precisely\, we will consider (NLS) in a weakly nonlinear regime on the to rus in any dimension greater than two\, and for highly oscillatory random Gaussian fields as initial data. A conjecture in statistical physics is th at there exists a kinetic time scale on which\, statistically\, the Fourie r modes evolve according to the kinetic wave equation. We prove this conje cture up to an arbitrarily small polynomial loss in a particular regime\, and obtain a more restricted time scale in other regimes. The main difficu lty\, that I will comment on\, is that one needs to identify the leading o rder statistically observable nonlinear effects. This means understanding correlation between Fourier modes\, and relating randomness with stability and local well-posedness. The key idea of the analysis is the use of Feyn man interaction diagrams to understand the solution as colliding linear wa ves. We use this framework to construct an approximate solution as a trunc ated series expansion\, and use in addition random matrices tools to obtai n its nonlinear stability in Bourgain spaces. This is joint work with P. G ermain.\n LOCATION:https://researchseminars.org/talk/IMS/5/ END:VEVENT END:VCALENDAR