{
  "id": "1503.05892",
  "version": "v1",
  "published": "2015-03-19T19:10:33.000Z",
  "updated": "2015-03-19T19:10:33.000Z",
  "title": "Homogenization of initial boundary value problems for parabolic systems with periodic coefficients",
  "authors": [
    "Yu. M. Meshkova",
    "T. A. Suslina"
  ],
  "comment": "68 pages. arXiv admin note: text overlap with arXiv:1406.7530",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\mathcal{O} \\subset \\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\\mathcal{O};\\mathbb{C}^n)$, we consider matrix elliptic second order differential operators $\\mathcal{A}_{D,\\varepsilon}$ and $\\mathcal{A}_{N,\\varepsilon}$ with the Dirichlet or Neumann boundary condition on $\\partial \\mathcal{O}$, respectively. Here $\\varepsilon>0$ is the small parameter. The coefficients of the operators are periodic and depend on $\\mathbf{x}/\\varepsilon$. The behavior of the operator $e^{-\\mathcal{A}_{\\dag ,\\varepsilon}t}$, $\\dag =D,N$, for small $\\varepsilon$ is studied. It is shown that, for fixed $t>0$, the operator $e^{-\\mathcal{A}_{\\dag ,\\varepsilon}t}$ converges in the $L_2$-operator norm to $e^{-\\mathcal{A}_{\\dag}^0 t}$, as $\\varepsilon \\to 0$. Here $\\mathcal{A}_{\\dag}^0$ is the effective operator with constant coefficients. For the norm of the difference of the operators $e^{-\\mathcal{A}_{\\dag ,\\varepsilon}t}$ and $e^{-\\mathcal{A}_{\\dag}^0 t}$ a sharp order estimate (of order $O(\\varepsilon)$) is obtained. Also, we find approximation for the exponential $e^{-\\mathcal{A}_{\\dag ,\\varepsilon}t}$ in the $(L_2\\rightarrow H^1)$-norm with error estimate of order $O(\\varepsilon ^{1/2})$; in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-19T19:10:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "initial boundary value problems",
      "parabolic systems",
      "periodic coefficients",
      "second order differential operators",
      "homogenization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 68,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}