Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations
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| Format: | Preprint |
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2024
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| _version_ | 1866909300249591808 |
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| author | Kinoshita, Shinya Sanwal, Akansha Schippa, Robert |
| author_facet | Kinoshita, Shinya Sanwal, Akansha Schippa, Robert |
| contents | We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$ exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On $\mathbb{R}^2$ we cover the full subcritical range, whereas on $\mathbb{R} \times \mathbb{T}$ the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb{R}^2$, the cylinder $\mathbb{R} \times \mathbb{T}$, and the torus $\mathbb{T}^2$. Finally, we point out that on tori $\mathbb{T}^2_γ$, KP-I equations are never semilinear. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations Kinoshita, Shinya Sanwal, Akansha Schippa, Robert Analysis of PDEs We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$ exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On $\mathbb{R}^2$ we cover the full subcritical range, whereas on $\mathbb{R} \times \mathbb{T}$ the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb{R}^2$, the cylinder $\mathbb{R} \times \mathbb{T}$, and the torus $\mathbb{T}^2$. Finally, we point out that on tori $\mathbb{T}^2_γ$, KP-I equations are never semilinear. |
| title | Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.16348 |