Global regularity for 2D Muskat equations with finite slope

Peter Constantin, Francisco Gancedo, Roman Shvydkoy, Vlad Vicol

Research output: Contribution to journalArticlepeer-review


We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W2,p, 1<p≤∞. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

Original languageEnglish (US)
Pages (from-to)1041-1074
Number of pages34
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Issue number4
StatePublished - Jul 2017


  • Darcy's law
  • Maximum principle
  • Muskat problem
  • Porous medium

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics


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