Analysis Seminar

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Long-time random solutions for the water waves equations

Speaker: Fabio Pusateri, University of Toronto

Location: Warren Weaver Hall 1314

Date: Tuesday, April 22, 2025, 2:30 p.m.

Synopsis:

We will present some recent results about the long-time regularity of solutions of the full gravity water waves system for initial data with random i.i.d. Gaussian phases.

The main new mechanism we employ is a combination of sharp deterministic energy estimates for quasilinear equations, and probabilistic and combinatorial arguments based on Feynman trees expansions.

The motivation for the construction of such random solutions comes from the theory of Wave Turbulence put forth by Hasselmann in the 60s. This theory aims to describe free surface hydrodynamics, which is mathematically modelled by the water waves equations, via effective/macroscopic kinetic-type equations.
In the context of the cubic NLS equation, the derivation of the wave kinetic equation has been rigorously proved in a series of recent works by Deng-Hani.
Our present work initiates the rigorous analysis of Wave Turbulence in the context of the full fluids equations.

This is joint work with Yu Deng (U Chicago) and Alex Ionescu (Princeton).