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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.23908 |
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| _version_ | 1866912981543026688 |
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| author | Ifrim, Mihaela Wilkening, Jon Zhao, Xinyu |
| author_facet | Ifrim, Mihaela Wilkening, Jon Zhao, Xinyu |
| contents | We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23908 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions Ifrim, Mihaela Wilkening, Jon Zhao, Xinyu Analysis of PDEs 35Q35, 42B37, 76B15 We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data. |
| title | Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions |
| topic | Analysis of PDEs 35Q35, 42B37, 76B15 |
| url | https://arxiv.org/abs/2603.23908 |