We prove growth of Sobolev norms for the two-dimensional periodic gravity water waves with constant nonzero vorticity, in infinite depth and with periodic boundary conditions. Precisely we construct weakly turbulent solutions exhibiting arbitrary large growth of high Sobolev norms, while having lower-order norms of small size, yielding the first rigorous construction of weakly turbulent solutions for the water waves equations. 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 resonant transport operator that drives energy to high modes and causes Sobolev norm growth.

This is a joint work with B. Langella, F. Murgante and S. Terracina.