{
  "id": "math/0612063",
  "version": "v2",
  "published": "2006-12-02T22:28:46.000Z",
  "updated": "2006-12-20T18:54:06.000Z",
  "title": "Borel summability of Navier-Stokes equation in $\\mathbb{R}^3$ and small time existence",
  "authors": [
    "O. Costin",
    "S. Tanveer"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the Navier-Stokes initial value problem, $$v_t - \\nabla v = -\\mathcal{P} [ v \\cdot \\nabla v \\right ] + f, v(x, 0) = v_0 (x), x \\in \\mathbb{R}^3 $$ where $\\mathcal{P}$ is the Hodge-Projection to divergence free vector fields in the assumption that $ | f |_{\\mu, \\beta} < \\infty $ and $| v_0 |_{\\mu+2, \\beta} < \\infty$ for $\\beta \\ge 0, \\mu > 3$, where $$ | {\\hat f} (k) | = \\sup_{k \\in \\mathbb{R}^3} e^{\\beta |k|} (1+|k|)^\\mu | {\\hat f} (k) |$$ and ${\\hat{f}} (k) = \\mathcal{F} [f (\\cdot)] (k) $ is the Fourier transform in $x$. By Borel summation methods we show that there exists a classical solution in the form $$ v(x, t) = v_0 + \\int_0^\\infty e^{-p/t} U(x, p) dp $$ $t\\in\\CC$, $ \\Re \\frac{1}{t} > \\alpha$, and we estimate $\\alpha$ in terms of $| {\\hat v}_0 |_{\\mu+2, \\beta}$ and $ | {\\hat f} |_{\\mu, \\beta}$. We show that $| {\\hat v} (\\cdot; t) |_{\\mu+2, \\beta} < \\infty $. Existence and $t$-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of $v$ beyond $t=\\alpha^{-1}$ becomes a growth rate question of $U(\\cdot, p)$ as $ p \\to \\infty$, $U$ being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of $v$ for $v_0$ and $f$ analytic. In particular, we obtain Gevrey-1 asymptotics results: $ v \\sim v_0 + \\sum_{m=1}^\\infty v_m t^m $, where $ |v_m | \\le m! A_0 B_0^m$, with $A_0$ and $B_0$ are given in terms of to $v_0$ and $f$ and for small $t$, with $m(t)=\\lfloor B_0^{-1}t^{-1}\\rfloor$, $$ | v(x, t) - v_0 (x) - \\sum_{m=1}^{m(t)} v_m (x) t^m | \\le A_0 m(t)^{1/2} e^{-m(t)} $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-12-20T18:54:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D05",
      "76D03"
    ],
    "keywords": [
      "small time existence",
      "borel summability",
      "navier-stokes equation",
      "navier-stokes initial value problem",
      "divergence free vector fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12063C"
    }
  }
}