{
  "id": "0706.1489",
  "version": "v1",
  "published": "2007-06-11T15:03:13.000Z",
  "updated": "2007-06-11T15:03:13.000Z",
  "title": "New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System",
  "authors": [
    "Lorenzo Brandolese",
    "Francois Vigneron"
  ],
  "comment": "26 pages, article to appear in Journal de Math\\'ematiques Pures et Appliqu\\'ees",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\\R^d$ ($d\\geq2$) starting from mild decaying data $a$ behave as $|x|\\to\\infty$ as a potential field: u(x,t) = e^{t\\Delta}a(x) + \\gamma_d\\nabla_x(\\sum_{h,k} \\frac{\\delta_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\\mathfrak{o}(\\frac{1}{|x|^{d+1}}) where $\\gamma_d$ is a constant and $K_{h,k}=\\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \\le |u(x,t)|\\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-11T15:03:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D05",
      "35Q30"
    ],
    "keywords": [
      "asymptotic profiles",
      "nonstationnary solutions",
      "lower bound holds",
      "non-stationnary incompressible navier-stokes system",
      "arbitrary small measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.1489B"
    }
  }
}