Transfer of energy for pure-gravity water waves with constant vorticity

TL;DR

This paper proves the existence of smooth, small-amplitude water wave solutions with energy transfer to high frequencies, demonstrating a form of weak turbulence in a quasilinear hydrodynamic system with constant vorticity.

Contribution

It introduces a novel mechanism for energy cascades in quasilinear dispersive PDEs, providing the first rigorous construction of weakly turbulent solutions for such water waves.

Findings

High Sobolev norms grow arbitrarily large while lower norms stay small.

Energy transfer occurs simultaneously in the free surface and velocity field.

The mechanism relies on quasi-resonances from 2-wave interactions.

Abstract

We consider two-dimensional periodic gravity water waves with constant nonzero vorticity $\gamma$, in infinite depth and with periodic boundary conditions. We prove that, if the characteristic wave number $\frac{\gamma^2}{g}$ is rational, the system admits smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while lower-order norms remain arbitrarily small, thereby exhibiting a genuine transfer of energy toward high frequencies. This yields the first rigorous construction of weakly turbulent solutions for a quasilinear hydrodynamic wave system, in a regime where the flow remains smooth. Moreover, the growth occurs simultaneously in the free surface and in the vertical component of the velocity at the interface, showing that the instability involves the full hydrodynamic evolution. The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure. A central ingredient is to exploit quasi-resonances from 2-wave interactions to produce a transport operator that drives energy to high modes and causes Sobolev norm growth. A virial-type argument then shows that the resulting instability affects both the free surface elevation and the velocity field.