{
  "id": "1803.04613",
  "version": "v1",
  "published": "2018-03-13T04:31:00.000Z",
  "updated": "2018-03-13T04:31:00.000Z",
  "title": "Apllication BMO type space to parabolic equations of Navier-Stokes type with the Neumann boundary condition",
  "authors": [
    "Yang Minghua",
    "Zhang Chao"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\L$ be a Neumann operator of the form $\\L=-\\Delta_{N}$ acting on $L^2(\\mathbb R^n)$. Let ${BMO}_{\\Delta_{N}}(\\mathbb R^n)$ denote the BMO space on $\\mathbb R^n$ associated to the Neumann operator $\\L$. In this article we will show that a function $f\\in { BMO}_{\\Delta_{N}}(\\mathbb R^n)$ is the trace of the solution of $${\\mathbb L}u=u_{t}+\\L u=0, u(x,0)= f(x),$$ where $u$ satisfies a Carleson-type condition \\begin{eqnarray*} \\sup_{x_B, r_B} r_B^{-n}\\int_0^{r_B^2}\\int_{B(x_B, r_B)} |\\nabla u(x,t)|^2 {dx dt } \\leq C <\\infty, \\end{eqnarray*} for some constant $C>0$. Conversely, this Carleson condition characterizes all the ${\\mathbb L}$-carolic functions whose traces belong to the space ${BMO}_{\\Delta_{N}}(\\mathbb R^n)$. This result extends the analogous characterization founded by E. Fabes and U. Neri in (\\textit{Duke Math. J.} \\textbf{42} (1975), 725-734) for the classical BMO space of John and Nirenberg. Furthermore, based on the characterization of ${BMO}_{\\Delta_{N}}(\\mathbb R^n)$ space mentioned above, we prove global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on intial data $u_{0}\\in {{ BMO}_{\\Delta_{N}}^{-1}(\\mathbb R^n)}$, which is motivated by the work of P. Auscher and D. Frey (\\textit{J. Inst. Math. Jussieu} \\textbf{16(5)} (2017), 947-985).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-13T04:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "42B37",
      "35J10",
      "47F05"
    ],
    "keywords": [
      "apllication bmo type space",
      "neumann boundary condition",
      "navier-stokes type",
      "parabolic equations",
      "neumann operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}