{
  "id": "2012.14239",
  "version": "v1",
  "published": "2020-12-28T14:13:11.000Z",
  "updated": "2020-12-28T14:13:11.000Z",
  "title": "Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D",
  "authors": [
    "Hartmut Pecher"
  ],
  "comment": "24 pages. arXiv admin note: text overlap with arXiv:2010.06170",
  "categories": [
    "math.AP"
  ],
  "abstract": "The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $\\phi$ and the potential $A$, respectively. The minimal regularity assumptions are $s > \\frac{1}{2}$ and $l > \\frac{1}{4}$ , which leaves a gap of $\\frac{1}{2}$ and $\\frac{1}{4}$ to the critical regularity with respect to scaling $s_c = l_c =0$ . This gap can be reduced for data in Fourier-Lebesgue spaces $\\widehat{H}^{s,r}$ and $\\widehat{H}^{l,r}$ to $s> \\frac{21}{16}$ and $l > \\frac{9}{8}$ for $r$ close to $1$ , whereas the critical exponents with respect to scaling fulfill $s_c \\to 1$ , $ l_c \\to 1 $ as $r \\to 1$ . Here $\\|f\\|_{\\widehat{H}^{s,r}} := \\| \\langle \\xi \\rangle^s \\tilde{f}\\|_{L^{r'}_{\\tau \\xi}} \\, , \\, 1 < r \\le 2 \\, , \\, \\frac{1}{r}+\\frac{1}{r'} = 1 \\, . $ Thus the gap is reduced for $\\phi$ as well as $A$ in both gauges.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-28T14:13:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35L70"
    ],
    "keywords": [
      "maxwell-klein-gordon system",
      "well-posedness results",
      "minimal regularity assumptions",
      "local well-posedness problem",
      "lorenz gauge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}