{
  "id": "1710.11399",
  "version": "v1",
  "published": "2017-10-31T10:23:06.000Z",
  "updated": "2017-10-31T10:23:06.000Z",
  "title": "Unconditional well-posedness below energy norm for the Maxwell-Klein-Gordon system",
  "authors": [
    "Hartmut Pecher"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The Maxwell-Klein-Gordon equation $ \\partial^{\\alpha} F_{\\alpha \\beta} = -Im(\\Phi \\overline{D_{\\beta} \\Phi}) $ , $ D^{\\mu}D_{\\mu} \\Phi = m^2 \\Phi $ , where $F_{\\alpha \\beta} = \\partial_{\\alpha} A_{\\beta} - \\partial_{\\beta} A_{\\alpha}$, $D_{\\mu} = \\partial_{\\mu} - iA_{\\mu} $, in the (3+1)-dimensional case is known to be unconditionally well-posed in energy space, i.e. well-posed in the natural solution space. This was proven by Klainerman-Machedon and Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge. The main purpose of the present paper is to establish that for both gauges this also holds true for data $\\Phi(0)$ in Sobolev spaces $H^s$ with less regularity, i.e. $s < 1$, but $s$ sufficently close to $1$. This improves the (conditional) well-posedness results in both cases, i.e. uniqueness in smaller solution spaces of Bourgain-Klainerman-Machedon type, which were essentially known by Cuccagna, Selberg and the author for $s > \\frac{3}{4}$ , and which in Coulomb gauge is also contained in the present paper. In fact, the proof consists in demonstrating that any solution in the natural solution space for some $s > s_0$ belongs to a Bourgain-Klainerman-Machedon space where uniqueness is known. Here $s_0 \\approx 0.914$ in Coulomb gauge and $s_0 \\approx 0.907$ in Lorenz gauge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-31T10:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35L70"
    ],
    "keywords": [
      "unconditional well-posedness",
      "energy norm",
      "maxwell-klein-gordon system",
      "coulomb gauge",
      "natural solution space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}