Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves

Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a widely open problem). In this paper we prove the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. This is a very hard small divisors problem for Partial Differential Equations due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension with a very complicated geometry of the resonances. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equations on compact domains.