{
  "id": "1606.03965",
  "version": "v1",
  "published": "2016-06-13T14:15:48.000Z",
  "updated": "2016-06-13T14:15:48.000Z",
  "title": "Global strong solution for the Korteweg system with quantum pressure in dimension $N\\geq 2$",
  "authors": [
    "Boris Haspot"
  ],
  "comment": "To appear in Mathematische Annalen. arXiv admin note: text overlap with arXiv:1302.2617 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "This work is devoted to prove the existence of global strong solution in dimension $N\\geq 2$ for a isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \\cite{fDS}), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure which corresponds to consider the capillary coefficient $\\kappa(\\rho)=\\frac{\\kappa_1}{\\rho}$ with $\\kappa_1>0$. In a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span $T^*$. This one depends on the norm of the initial data $(\\rho_0,v_0)$. The second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient ( where $\\kappa_1=\\mu^2$) and on the viscosity tensor which corresponds to the viscous shallow water case $-2\\mu{\\rm div}(\\rho Du)$. To do this we derivate different energy estimate on the density and the effective velocity $v$ which ensures that the strong solution can be extended beyond $T^*$. The main difficulty consists in controlling the vacuum or in other words to estimate the $L^\\infty$ norm of $\\frac{1}{\\rho}$. The proof relies mostly on a method introduced by De Giorgi \\cite{DG} (see also Ladyzhenskaya et al in \\cite{La} for the parabolic case) to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-13T14:15:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global strong solution",
      "quantum pressure",
      "korteweg system",
      "capillary coefficient",
      "main difficulty consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}