ABSTRACT

The gravity water wave kinetic equation, written down by Hasselmann in 1962, underlies modern ocean wave forecasting and climate modelling and was central to Hasselmann's 2021 Nobel Prize. Yet the basic mathematical question — does it admit solutions? — has remained open for 64 years, owing to a collision kernel that is widely considered the most singular in physical kinetic theory. In this talk I will present joint work with Yulin Pan establishing the first local-in-time existence of strong $L^1$ solutions in dimensions $d \ge 2$, for nonnegative initial data in a weighted $L^2 \cap L^\infty$ space. The proof rests on (i) a new kernel estimate that rigorously confirms the Zakharov–Geogjaev asymptotics — at most quadratic growth in the large wavenumber — and (ii) a decomposition of the linearised collision operator into a dissipative part plus a bounded perturbation, preserved under polynomial weights. Quadratic kernel growth turns out to be exactly the threshold at which this decomposition closes. The result also supplies an analytic ingredient needed for the full derivation programme connecting the free-surface Euler equations to the wave kinetic description.