{
  "id": "2411.18284",
  "version": "v1",
  "published": "2024-11-27T12:16:08.000Z",
  "updated": "2024-11-27T12:16:08.000Z",
  "title": "Existence of curvature flow with forcing in a critical Sobolev space",
  "authors": [
    "Yuning Liu",
    "Yoshihiro Tonegawa"
  ],
  "comment": "24 pages, comment welcome!",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Suppose that a closed $1$-rectifiable set $\\Gamma_0\\subset \\mathbb R^2$ of finite $1$-dimensional Hausdorff measure and a vector field $u$ in a dimensionally critical Sobolev space are given. It is proved that, starting from $\\Gamma_0$, there exists a non-trivial flow of curves with the velocity given by the sum of the curvature and the given vector field $u$. The motion law is satisfied in the sense of Brakke and the flow exists through singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-27T12:16:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53E10",
      "49Q15"
    ],
    "keywords": [
      "curvature flow",
      "vector field",
      "dimensional hausdorff measure",
      "motion law",
      "dimensionally critical sobolev space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}