{
  "id": "1609.00122",
  "version": "v1",
  "published": "2016-09-01T06:36:37.000Z",
  "updated": "2016-09-01T06:36:37.000Z",
  "title": "Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains",
  "authors": [
    "Qiang Xu"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in \\cite{SGZWS,G} to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and the basic idea comes from \\cite{QX2}, originally motivated by \\cite{SZW2,SZW12,TS}. We obtain an almost-sharp $O(\\varepsilon\\ln(r_0/\\varepsilon))$ convergence rate in $L^2$ space, and a sharp $O(\\varepsilon)$ error estimate in $L^{\\frac{2d}{d-1}}$ space by a little stronger assumption. Under the dimensional condition $d=2$, we also establish the optimal $O(\\varepsilon)$ convergence rate on pressure terms in $L^2$ space. Then utilizing the convergence rates we can derive the $W^{1,p}$ estimates uniformly down to microscopic scale $\\varepsilon$ without any smoothness assumption on the coefficients, where $|\\frac{1}{p}-\\frac{1}{2}|<\\frac{1}{2d}+\\epsilon$ and $\\epsilon$ is a positive constant independent of $\\varepsilon$. Combining the local estimates, based upon $\\text{VMO}$ coefficients, consequently leads to the uniform $W^{1,p}$ estimates. Here the proofs do not rely on the well known compactness methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-01T06:36:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "convergence rate",
      "stokes systems",
      "lipschitz domains",
      "homogenization theory",
      "basic idea comes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}