{
  "id": "1303.1166",
  "version": "v2",
  "published": "2013-03-05T20:32:21.000Z",
  "updated": "2014-05-15T18:13:45.000Z",
  "title": "Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms",
  "authors": [
    "Wolfgang Arendt",
    "Dominik Dier",
    "Hafida Laasri",
    "El Maati Ouhabaz"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "\\begin{abstract}\\label{abstract} We consider a non-autonomous evolutionary problem \\[ \\dot{u} (t)+\\A(t)u(t)=f(t), \\quad u(0)=u_0 \\] where the operator $\\A(t):V\\to V^\\prime$ is associated with a form $\\fra(t,.,.):V\\times V \\to \\R$ and $u_0\\in V$. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\\in L^2(0,T;H)$ we are interested in solutions $u \\in H^1(0,T;H)\\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and symmetric. Moreover, we show that each solution is in $C([0,T];V)$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-15T18:13:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal regularity",
      "evolution equations",
      "non-autonomous forms",
      "non-autonomous evolutionary problem",
      "non-autonomous robin-boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.1166A"
    }
  }
}