{
  "id": "2203.05799",
  "version": "v1",
  "published": "2022-03-11T08:45:36.000Z",
  "updated": "2022-03-11T08:45:36.000Z",
  "title": "Almost global existence for some nonlinear Schr{ö}dinger equations on $\\mathbb{T}^d$ in low regularity",
  "authors": [
    "Joackim Bernier",
    "Benoît Grébert"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are interested in the long time behavior of solutions of the nonlinear Schr{\\\"o}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\\mathbb T^d)$ with $s_0>d/2$. We prove that, even in this context of low regularity, the $H^s$-norms, $s\\geq 0$, remain under control during times, $T_\\varepsilon= \\exp \\big(-\\frac{|\\log\\varepsilon|^2}{4\\log|\\log\\varepsilon|} \\big)$, exponential with respect to the initial size of the initial datum in $H^{s_0}$, $\\|u(0)\\|_{H^{s_0}}=\\varepsilon$. For this, we add to the linear part of the equation a random Fourier multiplier in $\\ell^\\infty(\\mathbb Z^d)$ and show our stability result for almost any realization of this multiplier. In particular, with such Fourier multipliers, we obtain the almost global well posedness of the nonlinear Schr{\\\"o}dinger equation on $H^{s_0}(\\mathbb T^d)$ for any $s_0>d/2$ and any $d\\geq1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-11T08:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "low regularity",
      "global existence",
      "dinger equations",
      "nonlinear schr",
      "initial datum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}