{
  "id": "1904.09468",
  "version": "v1",
  "published": "2019-04-20T16:09:20.000Z",
  "updated": "2019-04-20T16:09:20.000Z",
  "title": "Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions",
  "authors": [
    "Sam G. Krupa",
    "Alexis F. Vasseur"
  ],
  "comment": "46 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\\frac{u^2}{2})_x=\\mathbf{H}[u]$ where $\\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-20T16:09:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35L60",
      "35L03",
      "35B65",
      "76B15",
      "35B35",
      "35D30",
      "35L67"
    ],
    "keywords": [
      "piecewise smooth solutions",
      "large class",
      "uniqueness",
      "burgers-hilbert equation",
      "strong trace condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}