{
  "id": "2010.14079",
  "version": "v1",
  "published": "2020-10-27T05:44:26.000Z",
  "updated": "2020-10-27T05:44:26.000Z",
  "title": "Two Approximation Results for Divergence Free Vector Fields",
  "authors": [
    "Jesse Goodman",
    "Felipe Hernandez",
    "Daniel Spector"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "In this paper we prove two approximation results for divergence free vector fields. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \\in M_b(\\mathbb{R}^d;\\mathbb{R}^d)$ such that $\\operatorname*{div} F=0$ in the sense of distributions, there exist $C^1$ closed curves $\\{\\Gamma_{i,l}\\}_{\\{1,\\ldots,n_l\\}\\times \\mathbb{N}}$, with parameterization by arclength $\\gamma_{i,l} \\in C^1([0,L_{i,l}];\\mathbb{R}^d)$, $l \\leq L_{i,l} \\leq 2l$, for which \\[ F= \\lim_{l \\to \\infty} \\frac{\\|F\\|_{M_b(\\mathbb{R}^d;\\mathbb{R}^d)}}{n_l \\cdot l} \\sum_{i=1}^{n_l} \\dot{\\gamma}_{i,l} \\left.\\mathcal{H}^1\\right\\vert_{\\Gamma_{i,l}} \\] weakly-star as measures and \\begin{align*} \\lim_{l \\to \\infty} \\frac{1}{n_l \\cdot l} \\sum_{i=1}^{n_l} |\\Gamma_{i,l}| = 1. \\end{align*} The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \\[ \\left\\{ F \\in M_b(\\mathbb{R}^d;\\mathbb{R}^d): \\operatorname*{div}F=0 \\right\\} \\] with respect to the strict topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-27T05:44:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divergence free vector fields",
      "approximation results",
      "strict topology",
      "immediate consequence",
      "smooth compactly supported functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}