{
  "id": "1912.10121",
  "version": "v1",
  "published": "2019-12-20T22:13:50.000Z",
  "updated": "2019-12-20T22:13:50.000Z",
  "title": "On the global wellposedness for free boundary problem for the Navier-Stokes systems with surface tension",
  "authors": [
    "Hirokazu Saito",
    "Yoshihiro Shibata"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The aim of this paper is to show the global existence of solutions to the Navier-Stokes equations, including surface tension and gravity, with a free surface on an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time $t$ tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem on a time-dependent domain $\\Omega_t$, $t>0$, to a problem on the lower half-space $\\mathbf{R}_-^3=\\{(x',x_3) \\mid x'=(x_1,x_2)\\in\\mathbf{R}^2,x_3<0\\}$. We then establish some time-weighted estimate of solutions, in an $L_p$-in-time and $L_q$-in-space setting, for the linearized problem around the trivial steady state with the help of $L_r\\text{-}L_s$ decay estimates of semigroup associated with the linearized problem. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem on $\\mathbf{R}_-^3$ admits in the $L_p\\text{-}L_q$ setting a unique solution global in time and that the solution decays polynomially as time $t$ tends to infinity under the assumption that the exponents $p$, $q$ satisfy the conditions: $2<p<\\infty$, $3<q<16/5$, and $(2/p)+(3/q)<1$. Finally, we apply the inverse transformation of Hanzawa's one to the solution on $\\mathbf{R}_-^3$ to prove our main results mentioned above for the original problem on $\\Omega_t$. Here, we want to emphasize that it is not allowed to take $p=q$ in the above assumption about $p$, $q$, which means that the different exponents $p$, $q$ of $L_p\\text{-}L_q$ setting play an essential role in our approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-20T22:13:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free boundary problem",
      "surface tension",
      "global wellposedness",
      "navier-stokes systems",
      "solution decays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}