A New Structure for the 2D water wave equation: Energy stability and Global well-posedness
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909555923877888 |
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| author | Su, Qingtang Wang, Siwei |
| author_facet | Su, Qingtang Wang, Siwei |
| contents | We study the two-dimensional gravity water waves with a one-dimensional interface with small initial data. Our main contributions include the development of two novel localization lemmas and a Transition-of-Derivatives method, which enable us to reformulate the water wave system into the following simplified structure: $$(D_t^2-iA\partial_α)θ=i\frac{t}α|D_t^2ζ|^2D_tθ+R$$ where $R$ behaves well in the energy estimate. As a key consequence, we derive the uniform bound $$
\sup_{t\geq 0}\Big(\norm{D_tζ(\cdot,t)}_{H^{s+1/2}}+\norm{ζ_α(\cdot,t)-1}_{H^s}\Big)\leq Cε, $$ which enhances existing global uniform energy estimates for 2D water waves by imposing less restrictive constraints on the low-frequency components of the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Structure for the 2D water wave equation: Energy stability and Global well-posedness Su, Qingtang Wang, Siwei Analysis of PDEs We study the two-dimensional gravity water waves with a one-dimensional interface with small initial data. Our main contributions include the development of two novel localization lemmas and a Transition-of-Derivatives method, which enable us to reformulate the water wave system into the following simplified structure: $$(D_t^2-iA\partial_α)θ=i\frac{t}α|D_t^2ζ|^2D_tθ+R$$ where $R$ behaves well in the energy estimate. As a key consequence, we derive the uniform bound $$ \sup_{t\geq 0}\Big(\norm{D_tζ(\cdot,t)}_{H^{s+1/2}}+\norm{ζ_α(\cdot,t)-1}_{H^s}\Big)\leq Cε, $$ which enhances existing global uniform energy estimates for 2D water waves by imposing less restrictive constraints on the low-frequency components of the initial data. |
| title | A New Structure for the 2D water wave equation: Energy stability and Global well-posedness |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.22142 |