{
  "id": "2408.12204",
  "version": "v1",
  "published": "2024-08-22T08:27:56.000Z",
  "updated": "2024-08-22T08:27:56.000Z",
  "title": "Stochastic Homogenization of Parabolic Equations with Lower-order Terms",
  "authors": [
    "Man Yang"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients: \\begin{equation*} \\partial_{t}p^\\epsilon-\\nabla\\cdot\\left(\\mathbf{a}\\left( \\dfrac{x}{\\epsilon},\\dfrac{t}{\\epsilon^2}\\right)\\nabla p^\\epsilon\\right)-\\mathbf{b}\\left( \\dfrac{x}{\\epsilon},\\dfrac{t}{\\epsilon^2}\\right)\\nabla p^\\epsilon -\\mathbf{d}\\left( \\dfrac{x}{\\epsilon},\\dfrac{t}{\\epsilon^2}\\right) p^\\epsilon=0. \\end{equation*} Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of $p^\\epsilon$ in $H^1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-22T08:27:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower-order terms",
      "stochastic homogenization",
      "general parabolic equation",
      "caccioppoli inequality",
      "studying homogenization results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}