Kinetic wave turbulence theory: rigorous results (in-person talk)

Kinetic wave turbulence theory: rigorous results (in-person talk)

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Zaher Hani, University of Michigan
Fine Hall 314

In-Person and Online Talk 

 Zoom Link: https://princeton.zoom.us/j/92147928280

Passcode required

The central claim of wave turbulence theory is that the generic longtime behavior of a nonlinear dispersive system is described by a corresponding kinetic equation called the "wave kinetic equation" (WKE). This happens at an appropriate timescale and in a thermodynamic limit in which the size of the system diverges to infinity and the strength of the nonlinearity vanishes (weak nonlinearity limit). This is the wave analog of how Boltzmann’s kinetic equation describes the generic behavior of a dilute gas of particles in a limit where the number of particles diverges to infinity and their radius vanishes. 

 

Mathematically, the main problem is to provide a proof of this conjecture, which amounts to giving a rigorous derivation of the wave kinetic equation starting from the nonlinear dispersive equation as a first principle. The particle analog of such a result is Lanford’s celebrated theorem (1975) which gives the rigorous derivation of the Boltzmann equation starting from Newtonian particle dynamics. In this talk, we shall describe our recent work, jointly with Yu Deng (USC), which gives the full derivation of the wave kinetic equation starting from the nonlinear Schrodinger equation as a microscopic system. We will also discuss some related questions pertaining to the nonequilbrium statistics of nonlinear waves, like propagation of chaos, behavior of higher moments, and the wave kinetic hierarchy. All this is joint work with Yu Deng (USC).