{
  "id": "2204.08295",
  "version": "v1",
  "published": "2022-04-18T12:51:49.000Z",
  "updated": "2022-04-18T12:51:49.000Z",
  "title": "Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces",
  "authors": [
    "Jinlu Li",
    "Yanghai Yu",
    "Weipeng Zhu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in $\\R^d$ is well-posed from $\\dot{B}_{p, q}^{-2}$ to $\\mathbb{P} \\dot{B}_{p, q}^{0}$ with $p=d$ and $1 \\leq q < 2$. In this paper, we demonstrate that for the case $1\\leq q<2$ the 4D stationary Navier-Stokes equations is ill-posed from $\\dot{B}_{4, q}^{-2}(\\R^4)$ to $\\mathbb{P} \\dot{B}_{4, q}^{0}(\\R^4)$ by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of $q$, there exists a sequence of external forces which converges to zero in $\\dot{B}_{4, q}^{-2}$ and yields a sequence of solutions which does not converge to zero in $\\dot{B}_{4, q}^{0}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-18T12:51:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical besov spaces",
      "4d stationary navier-stokes equations",
      "external forces",
      "ill-posedness",
      "solution map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}