{
  "id": "1103.3140",
  "version": "v2",
  "published": "2011-03-16T10:25:55.000Z",
  "updated": "2011-11-29T10:17:10.000Z",
  "title": "On Singularity formation for the L^2-critical Boson star equation",
  "authors": [
    "Enno Lenzmann",
    "Mathieu Lewin"
  ],
  "comment": "24 pages. Accepted in Nonlinearity",
  "journal": "Nonlinearity 24 (2011) 3515-3540",
  "doi": "10.1088/0951-7715/24/12/009",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\\partial_t u = \\sqrt{-\\Delta+m^2} \\, u - (|x|^{-1} \\ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a \"finite speed of propagation\" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\\\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-29T10:17:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularity formation",
      "critical boson star equation",
      "unique weak limit",
      "large data blowup conjecture",
      "radial finite-time blowup solution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Nonlinearity",
      "year": 2011,
      "month": "Dec",
      "volume": 24,
      "number": 12,
      "pages": 3515
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011Nonli..24.3515L"
    }
  }
}