{
  "id": "2405.12703",
  "version": "v1",
  "published": "2024-05-21T11:50:35.000Z",
  "updated": "2024-05-21T11:50:35.000Z",
  "title": "Constructions of bounded solutions of $div\\, {\\mathbf u}=f$ in critical spaces",
  "authors": [
    "Albert Cohen",
    "Ronald DeVore",
    "Eitan Tadmor"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct uniformly bounded solutions of the equation $div\\, {\\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(\\Omega)$, where $\\Omega$ is a domain of ${\\mathbb R}^d$. This question was addressed by Bourgain & Brezis, [On the equation ${\\rm div}\\, Y=f$ and application to control of phases, JAMS 16(2) (2003) 393-426], who proved that although the problem has a uniformly bounded solution, it is critical in the sense that there exists no linear solution operator for general $L^d$-data. We first discuss the validity of this existence result under weaker conditions than $f\\in L^d(\\Omega)$, and then focus our work on constructive processes for such uniformly bounded solutions. In the $d=2$ case, we present a direct one-step explicit construction, which generalizes for $d>2$ to a $(d-1)$-step construction based on induction. An explicit construction is proposed for compactly supported data in $L^{2,\\infty}(\\Omega)$ in the $d=2$ case. We also present constructive approaches based on optimization of a certain loss functional adapted to the problem. This approach provides a two-step construction in the $d=2$ case. This optimization is used as the building block of a hierarchical multistep process introduced in [E. Tadmor, Hierarchical construction of bounded solutions in critical regularity spaces, CPAM 69(6) (2016) 1087-1109] that converges to a solution in more general situations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-21T11:50:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical spaces",
      "direct one-step explicit construction",
      "linear solution operator",
      "existence result",
      "weaker conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}