{
  "id": "1403.2461",
  "version": "v6",
  "published": "2014-03-11T02:14:26.000Z",
  "updated": "2021-08-23T03:48:16.000Z",
  "title": "Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\\dot B^{-1}_{\\infty,q}$",
  "authors": [
    "Baoxiang Wang"
  ],
  "comment": "25 Pages, in this new version, we add a reference of Iwabuchi T. and Nakamura M",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Cauchy problem for the incompressible Navier-Stokes equation \\begin{align} u_t -\\Delta u+u\\cdot \\nabla u +\\nabla p=0, \\ \\ {\\rm div} u=0, \\ \\ u(0,x)= \\delta u_0. \\label{NS} \\end{align} For arbitrarily small $\\delta>0$, we show that the solution map $\\delta u_0 \\to u$ in critical Besov spaces $\\dot B^{-1}_{\\infty,q}$ ($\\forall \\ q\\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$. Taking notice of the embedding $\\dot B^{-1}_{\\infty,q} \\subset BMO^{-1}$ ($q\\le 2$), we see that for sufficiently small $\\delta>0$, $u_0\\in \\dot B^{-1}_{\\infty,q} $ ($q\\le 2$) can guarantee that the Navier-Stokes equation has a unique global solution in $BMO^{-1}$, however, this solution is instable in $ \\dot B^{-1}_{\\infty,q} $ and the solution can have an inflation in $\\dot B^{-1}_{\\infty,q}$ for certain initial data. So, our result indicates that two different topological structures in the same space may determine the well and ill posedness, respectively.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-05-22T14:13:45.000Z",
      "comment": "25 Pages, in this new version, we correct some misprints and a mistake, the real valued case is considered",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2021-08-23T03:48:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "35K55"
    ],
    "keywords": [
      "critical besov spaces",
      "ill-posedness",
      "unique global solution",
      "incompressible navier-stokes equation",
      "small data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.2461W"
    }
  }
}