{
  "id": "1109.0816",
  "version": "v4",
  "published": "2011-09-05T07:24:16.000Z",
  "updated": "2012-01-02T08:42:35.000Z",
  "title": "$L^p$-maximal regularity of nonlocal parabolic equation and applications",
  "authors": [
    "Xicheng Zhang"
  ],
  "comment": "38 pages, Theorem 6.1 is improved",
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "By using Fourier's transform and Fefferman-Stein's theorem, we investigate the $L^p$-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric L\\'evy operators, and obtain the unique strong solvability of the corresponding nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. In particular, a characterization for the domain of pseudo-differential operators of L\\'evy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we show that a large class of non-symmetric L\\'evy operators generates an analytic semigroup in $L^p$-space. Moreover, as applications, we prove a Krylov's estimate for stochastic differential equation driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the well-posedness to a class of quasi-linear first order parabolic equation with critical diffusion. In particular, critical Hamilton-Jacobi equation and multidimensional critical Burger's equation are uniquely solvable and the smooth solutions are obtained.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-01-02T08:42:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G20",
      "45K05",
      "60H30"
    ],
    "keywords": [
      "nonlocal parabolic equation",
      "maximal regularity",
      "quasi-linear first order parabolic equation",
      "applications",
      "non-symmetric levy operators generates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.0816Z"
    }
  }
}