Quasi-resonant normal form and quadratic lifespan for 3D gravity-capillary water waves
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| Format: | Preprint |
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2026
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| author | Feola, Roberto Montalto, Riccardo Murgante, Federico |
| author_facet | Feola, Roberto Montalto, Riccardo Murgante, Federico |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quasi-resonant normal form and quadratic lifespan for 3D gravity-capillary water waves Feola, Roberto Montalto, Riccardo Murgante, Federico Analysis of PDEs Mathematical Physics 35Q35, 35B35, 35B40, 35S50, 76B45 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. |
| title | Quasi-resonant normal form and quadratic lifespan for 3D gravity-capillary water waves |
| topic | Analysis of PDEs Mathematical Physics 35Q35, 35B35, 35B40, 35S50, 76B45 |
| url | https://arxiv.org/abs/2604.20968 |