{
  "id": "1606.01332",
  "version": "v1",
  "published": "2016-06-04T06:08:56.000Z",
  "updated": "2016-06-04T06:08:56.000Z",
  "title": "Measure-valued evolution equations with solution-dependent velocities on bounded domains: Mild solutions are weak solutions",
  "authors": [
    "Joep H. M. Evers"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1507.05730",
  "categories": [
    "math.AP"
  ],
  "abstract": "In [J. Differential Equations, 259 (2015), pp. 1068-1097], we proved the well-posedness of a measure-valued evolution equation posed on the bounded domain $[0,1]$. This result was obtained in the framework of (measure-valued) mild solutions; that is, solutions to the variation of constants formula. We were restricted there to a prescribed velocity field $v:[0,1]\\to\\mathbb{R}$ that is only a function of the spatial variable. We generalized this result in [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953] to velocity fields that depend on the solution itself: $v=v[\\mu](x)$. To that aim, we first constructed so-called Euler approximations. These are approximate solutions, per time interval $(t_j,t_{j+1}]$ driven by an appropriate $v_j=v_j(x)$. In [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953], mild solutions were defined as the limit of a sequence of Euler approximations as the mesh size went to zero. Such mild solutions were showed to exist and be unique. In the current paper we define weak solutions (for $v=v(x)$ and $v=v[\\mu]$, respectively) and show that, in either case, the aforementioned mild solutions are weak solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-04T06:08:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A33",
      "34A12",
      "45D05",
      "35F16"
    ],
    "keywords": [
      "mild solutions",
      "measure-valued evolution equation",
      "bounded domain",
      "solution-dependent velocities",
      "euler approximations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}