{
  "id": "1101.3504",
  "version": "v4",
  "published": "2011-01-18T17:15:20.000Z",
  "updated": "2012-02-17T20:23:43.000Z",
  "title": "Maximal $L^p$-regularity for stochastic evolution equations",
  "authors": [
    "Jan van Neerven",
    "Mark Veraar",
    "Lutz Weis"
  ],
  "comment": "Accepted for publication in SIAM Journal on Mathematical Analysis",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We prove maximal $L^p$-regularity for the stochastic evolution equation \\[\\{{aligned} dU(t) + A U(t)\\, dt& = F(t,U(t))\\,dt + B(t,U(t))\\,dW_H(t), \\qquad t\\in [0,T], U(0) & = u_0, {aligned}.\\] under the assumption that $A$ is a sectorial operator with a bounded $H^\\infty$-calculus of angle less than $\\frac12\\pi$ on a space $L^q(\\mathcal{O},\\mu)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\\in (2,\\infty)$ and $q\\in [2,\\infty)$ and initial conditions $u_0$ in the real interpolation space $\\XAp $ we prove existence of unique strong solution with trajectories in \\[L^p(0,T;\\Dom(A))\\cap C([0,T];\\XAp),\\] provided the non-linearities $F:[0,T]\\times \\Dom(A)\\to L^q(\\mathcal{O},\\mu)$ and $B:[0,T]\\times \\Dom(A) \\to \\g(H,\\Dom(A^{\\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain $\\OO\\subseteq \\R^d$ with $d\\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\\OO))^d$ is shown.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-02-17T20:23:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35D10",
      "35R60",
      "46B09",
      "47D06",
      "47A60"
    ],
    "keywords": [
      "stochastic evolution equation",
      "regularity",
      "unique strong local solution",
      "stochastic partial differential equations",
      "abstract hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.3504V"
    }
  }
}