{
  "id": "2409.03955",
  "version": "v1",
  "published": "2024-09-06T00:39:31.000Z",
  "updated": "2024-09-06T00:39:31.000Z",
  "title": "The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG",
  "authors": [
    "Tsukasa Iwabuchi"
  ],
  "comment": "20pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space $\\mathbb R^2$, we demonstrate that the uniqueness of the mild solution holds in $L^2$. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivativesfor the nonlinear term involving fractional derivatives of the Dirichlet Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-06T00:39:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35Q86"
    ],
    "keywords": [
      "quadratic nonlinearity",
      "derivative structure",
      "uniqueness",
      "two-dimensional surface quasi-geostrophic equation",
      "three-dimensional incompressible navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}