{
  "id": "2408.16348",
  "version": "v1",
  "published": "2024-08-29T08:35:42.000Z",
  "updated": "2024-08-29T08:35:42.000Z",
  "title": "Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations",
  "authors": [
    "Shinya Kinoshita",
    "Akansha Sanwal",
    "Robert Schippa"
  ],
  "comment": "72 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "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 $\\mathbb{T}^2_\\gamma$, KP-I equations are never semilinear.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-29T08:35:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp local well-posedness",
      "semilinear kp-i equations",
      "well-posedness results",
      "sharp well-posedness",
      "nonlinear loomis-whitney inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}