{
  "id": "1704.05602",
  "version": "v1",
  "published": "2017-04-19T03:43:43.000Z",
  "updated": "2017-04-19T03:43:43.000Z",
  "title": "Partial regularity for type two doubly nonlinear parabolic systems",
  "authors": [
    "Ryan Hynd"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study weak solutions ${\\bf v}:U\\times (0,T)\\rightarrow \\mathbb{R}^m$ of the nonlinear parabolic system $$ D\\psi({\\bf v}_t)=\\text{div}DF(D{\\bf v}), $$ where $\\psi$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in mathematical models used to study various structural properties of materials. Under the assumption that the second derivatives of $F$ are H\\\"older continuous, we show that $D^2{\\bf v}$ and ${\\bf v}_t$ are locally H\\\"older continuous except for possibly on a lower dimensional subset of $U\\times (0,T)$. Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for $D^2{\\bf v}$ and ${\\bf v}_t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-19T03:43:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "doubly nonlinear parabolic systems",
      "partial regularity",
      "fractional time derivative estimates",
      "general doubly nonlinear evolutions",
      "lower dimensional subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}