Quasi-resonant normal form and quadratic lifespan for 3D gravity-capillary water waves

Abstract

We study the long-time dynamics of small-amplitude solutions to the three-dimensional gravity-capillary water waves equations for an inviscid and irrotational fluid with periodic boundary conditions. We prove that, for almost all values of the surface tension parameter, solutions with initial size $\varepsilon$ exist and remain small over time intervals of order $\varepsilon^{-2}$. A major difficulty arises from the loss of derivatives caused by the quasilinear nature of the equations combined with severe quadratic and cubic small-divisor interactions in high space dimensions. Classical normal form methods applied to 3D water waves system typically fail to prevent derivative loss due to the accumulation of near-resonances. To overcome this obstruction, we develop a new analytical strategy that combines a sharp frequency partition with a quasi-resonant normal form transformation acting only on selected interaction scales. Our microlocal analysis reveals that the potentially dangerous interactions terms exhibit a block-diagonal structure, which stems from both the geometric properties of the quasi-resonant frequency sets and the Hamiltonian structure of the water waves system. As a consequence, these operators preserve Sobolev norms and do not produce energy growth. This structural insight, together with the quasi-resonant normal-form transformation, allows us to prevent derivative-loss mechanisms while avoiding the accumulation of harmful small denominators.