{
  "id": "1610.02221",
  "version": "v1",
  "published": "2016-10-07T10:53:17.000Z",
  "updated": "2016-10-07T10:53:17.000Z",
  "title": "On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type",
  "authors": [
    "Félix del Teso",
    "Jørgen Endal",
    "Espen R. Jakobsen"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\\partial_tu-A\\varphi(u)=0.$$ These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity $\\varphi$ and the largest class of linear symmetric nonlocal diffusion operators $A$ considered so far. The operators are defined from a bilinear energy form $\\mathcal{E}$ and may be degenerate and have some $x$-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump L\\'evy processes are included. The main results are (i) an Ole\\u{\\i}nik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new well-posedness results for both notions of solutions. Finally, we obtain quantitative energy and related $L^p$-estimates for distributional solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-07T10:53:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A02",
      "35B30",
      "35D30",
      "35K55",
      "35K65",
      "35R09",
      "35R11"
    ],
    "keywords": [
      "finite energy",
      "nonlocal equations",
      "symmetric nonlocal diffusion operators",
      "degenerate nonlinear diffusion equations",
      "porous medium type equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}