{
  "id": "2002.05360",
  "version": "v1",
  "published": "2020-02-13T05:43:48.000Z",
  "updated": "2020-02-13T05:43:48.000Z",
  "title": "Existence and smoothness of the solution to the Navier-Stokes",
  "authors": [
    "Argyngazy Bazarbekov"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions. In this paper we shall study this problem. The Navier-Stokes equations are given by: $u_{it}(x,t)-\\rho\\triangle u_i(x,t)-u_j(x,t) u_{ix_j}(x,t)+p_{x_i}(x,t)=f_i(x,t)$ , $div\\textbf{u}(x,t)=0$ with initial conditions $\\textbf{u}|_{(t=0)\\bigcup\\partial\\Omega}=0$. We introduce the unknown vector-function: $\\big(w_i(x,t)\\big)_{i=1,2,3}: u_{it}(x,t)-\\rho\\triangle u_i(x,t)-\\frac{dp(x,t)}{dx_i}=w_i(x,t)$ with initial conditions: $u_i(x,0)=0,$ $u_i(x,t)\\mid_{\\partial\\Omega}=0$. The solution $u_i(x,t)$ of this problem is given by: $u_i(x,t)=\\int_0^t\\int_{\\Omega}G(x,t;\\xi,\\tau)~\\Big(w_i(\\xi,\\tau)+\\frac{dp(\\xi,\\tau)}{d\\xi_i}\\Big)d\\xi d\\tau$ where $G(x,t;\\xi,\\tau)$ is the Green function. We consider the following N-Stokes-2 problem: find a solution $\\textbf{w}(x,t)\\in \\textbf{L}_2(Q_t), p(x,t): p_{x_i}(x,t)\\in L_2(Q_t)$ of the system of equations: $w_i(x,t)-G\\Big(w_j(x,t)+\\frac{dp(x,t)}{dx_j}\\Big)\\cdot G_{x_j}\\Big(w_i(x,t)+\\frac{dp(x,t)}{dx_i}\\Big)=f_i(x,t)$ satisfying almost everywhere on $Q_t.$ Where the v-function $\\textbf{p}_{x_i}(x,t)$ is defined by the v-function $\\textbf{w}_i(x,t)$. Using the following estimates for the Green function: $|G(x,t;\\xi ,\\tau)| \\leq\\frac{c}{(t-\\tau)^{\\mu}\\cdot |x-\\xi|^{3-2\\mu}}; |G_{x}(x,t;\\xi,\\tau)|\\leq\\frac{c}{(t-\\tau)^{\\mu}\\cdot|x-\\xi|^{3-(2\\mu-1)}}(1/2<\\mu<1),$ from this system of equations we obtain: $w(t)<f(t)+b\\Big(\\int_0^{t}\\frac{w(\\tau)}{(t-\\tau)^{\\mu}} d\\tau\\Big)^2$; $w(t)=\\|\\textbf{w}(x,t)\\|_{L_2(\\Omega)}, f(t)=\\|\\textbf{f}(x,t)\\|_{L_2(\\Omega)}.$ Further, using the replacement of the unknown function by \\textbf{Riccati}, from this inequality we obtain the a priori estimate. By the Leray-Schauder's method and this a priori estimate the existence and uniqueness of the solution is proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-13T05:43:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "46E35"
    ],
    "keywords": [
      "smoothness",
      "priori estimate",
      "green function",
      "initial conditions",
      "navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}