Compactness and large-scale regularity for Darcy's law

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3 Scopus citations


This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior C1,α and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform Wk,p estimates for 1<p<∞.

Original languageEnglish
Pages (from-to)673-701
Number of pages29
JournalJournal des Mathematiques Pures et Appliquees
StatePublished - Jul 2022

Bibliographical note

Funding Information:
Supported in part by NSF grant DMS-1856235.

Publisher Copyright:
© 2022 Elsevier Masson SAS


  • Darcy law
  • Large-scale regularity
  • Perforated domain
  • Stokes equations

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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