The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$
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| Format: | Preprint |
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2024
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| _version_ | 1866912808473460736 |
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| author | Herr, Sebastian Schippa, Robert Tzvetkov, Nikolay |
| author_facet | Herr, Sebastian Schippa, Robert Tzvetkov, Nikolay |
| contents | We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_12222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$ Herr, Sebastian Schippa, Robert Tzvetkov, Nikolay Analysis of PDEs We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate. |
| title | The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.12222 |