{
  "id": "2104.10388",
  "version": "v1",
  "published": "2021-04-21T07:22:12.000Z",
  "updated": "2021-04-21T07:22:12.000Z",
  "title": "A regularized gradient flow for the $p$-elastic energy",
  "authors": [
    "Simon Blatt",
    "Christopher Hopper",
    "Nicole Vorderobermeier"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove long-time existence for the negative $L^2$-gradient flow of the $p$-elastic energy, $p\\geq 2$, with an additive positive multiple of the length of the curve. To achieve this result we regularize the energy by adding a small multiple of a higher order energy, namely the square of the $L^2$-norm of the normal gradient of the curvature $\\kappa$. Long-time existence is proved for the gradient flow of these new energies together with the smooth sub-convergence of the evolution equation's solutions to critical points of the regularized energy in $W^{2,p}$. We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the $p$-elastic energies. These latter weak solutions also sub-converge to critical points of the $p$-elastic energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-21T07:22:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44"
    ],
    "keywords": [
      "elastic energy",
      "regularized gradient flow",
      "long-time existence",
      "weak solution",
      "higher order energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}