{
  "id": "2204.01363",
  "version": "v1",
  "published": "2022-04-04T10:17:44.000Z",
  "updated": "2022-04-04T10:17:44.000Z",
  "title": "On the failure of the chain rule for the divergence of Sobolev vector fields",
  "authors": [
    "Miriam Buck",
    "Stefano Modena"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \\geq 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = {\\rm div}\\, h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \\in W^{1, \\tilde p}$ and a density $\\rho \\in L^p$ for which ${\\rm div}\\, (\\rho u) =0$ but ${\\rm div}\\, (\\beta(\\rho) u) = T$, provided $1/p + 1/\\tilde p \\geq 1 + 1/(d-1)$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T10:17:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divergence",
      "chain rule problem",
      "incompressible sobolev vector field",
      "incompressible vector fields",
      "renormalization map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}