{
  "id": "0809.2664",
  "version": "v1",
  "published": "2008-09-16T09:28:49.000Z",
  "updated": "2008-09-16T09:28:49.000Z",
  "title": "Balance laws with integrable unbounded sources",
  "authors": [
    "Graziano Guerra",
    "Francesca Marcellini",
    "Veronika Schleper"
  ],
  "comment": "26 pages, 4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for a $n\\times n$ strictly hyperbolic system of balance laws $$ \\{{array}{c} u_t+f(u)_x=g(x,u), x \\in \\mathbb{R}, t>0 u(0,.)=u_o \\in L^1 \\cap BV(\\mathbb{R}; \\mathbb{R}^n), | \\lambda_i(u)| \\geq c > 0 {for all} i\\in \\{1,...,n\\}, \\|g(x,\\cdot)\\|_{\\mathbf{C}^2}\\leq \\tilde M(x) \\in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\\mathbf{L}^1$ norm of $\\|g(x,\\cdot)\\|_{\\mathbf{C}^1}$ and $\\|u_o\\|_{BV(\\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-16T09:28:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35L45",
      "35L60"
    ],
    "keywords": [
      "balance laws",
      "integrable unbounded sources",
      "discontinuous cross sectional area",
      "global entropy solutions",
      "strictly hyperbolic system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.2664G"
    }
  }
}