{
  "id": "2212.05368",
  "version": "v1",
  "published": "2022-12-10T21:52:05.000Z",
  "updated": "2022-12-10T21:52:05.000Z",
  "title": "Existence of asymmetric vortex patch for the generalized SQG equations",
  "authors": [
    "Edison Cuba",
    "Lucas C. F. Ferreira"
  ],
  "comment": "38 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $\\alpha\\in[1,2)$ in the whole plane, where $\\alpha=1$ corresponds to the surface quasi-geostrophic equations (SQG). More precisely, we construct non-trivial simply connected co-rotating and traveling patches with unequal vorticity magnitudes. The proof is carried out by means of a combination of a desingularization argument with the implicit function theorem on the linearization of contour dynamics equation. Our results extend recent ones in the range $\\alpha\\in[0,1)$ by Hassainia-Hmidi (DCDS-A, 2021) and Hassainia-Wheeler (SIAM J. Math. Anal., 2022) to more singular velocities, filling an open gap in the range of $\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-10T21:52:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76B03",
      "76B47"
    ],
    "keywords": [
      "asymmetric vortex patch",
      "generalized sqg equations",
      "non-trivial simply connected co-rotating",
      "surface quasi-geostrophic equations",
      "unequal vorticity magnitudes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}