{
  "id": "1308.4285",
  "version": "v1",
  "published": "2013-08-20T12:23:30.000Z",
  "updated": "2013-08-20T12:23:30.000Z",
  "title": "Almost critical local well-posedness for the space-time Monopole equation in Lorenz gauge",
  "authors": [
    "Achenef Tesfahun"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in $H^s $ with $s>\\frac14$. The equation is $L^2$-critical, and hence a $\\frac14$ derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier-Lebesgue space $\\hat{H_p^s}$ for $1<p\\le 2$ which coincides with $H^s$ when $p=2$ but scales like lower regularity Sobolev spaces for $1<p< 2$. In particular, we will see that as $p\\rightarrow 1^+$, the critical exponent $s^c_p\\rightarrow 1^-$, in which case $\\hat{\\dot H_{1+}^{1-}}$ is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to $p$ close to 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-20T12:23:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35L70"
    ],
    "keywords": [
      "space-time monopole equation",
      "lorenz gauge",
      "critical local well-posedness",
      "initial data",
      "lower regularity sobolev spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1250067,
      "adsabs": "2013arXiv1308.4285T"
    }
  }
}