{
  "id": "1111.6246",
  "version": "v1",
  "published": "2011-11-27T11:06:22.000Z",
  "updated": "2011-11-27T11:06:22.000Z",
  "title": "SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension",
  "authors": [
    "Stefano Bianchini",
    "Laura Caravenna"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that if $t \\mapsto u(t) \\in \\mathrm {BV}(\\R)$ is the entropy solution to a $N \\times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \\[ u_t + f(u)_x = 0, \\] then up to a countable set of times $\\{t_n\\}_{n \\in \\mathbb N}$ the function $u(t)$ is in $\\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \\[ u(t) = \\sum_{i=1}^N v_i(t) \\tilde r_i(t), \\quad v_i(t) \\in \\mathcal M(\\R), \\ \\tilde r_i(t) \\in \\mathrm R^N, \\] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $\\mu_{i,\\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\\mu_{i,\\mathrm{jump}}$ is positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-27T11:06:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35B05",
      "35D10"
    ],
    "keywords": [
      "strictly hyperbolic system",
      "conservation laws",
      "sbv regularity",
      "space dimension",
      "cantorian part"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-012-1480-5",
      "journal": "Communications in Mathematical Physics",
      "year": 2012,
      "month": "Jul",
      "volume": 313,
      "number": 1,
      "pages": 1
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012CMaPh.313....1B"
    }
  }
}