Global regularity for 2D Muskat equations with finite slope

Peter Constantin, Francisco Gancedo, Roman Shvydkoy, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

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
Volume34
Issue number4
DOIs
StatePublished - Jul 1 2017

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Global Regularity
Slope
Fluids
Porous materials
Fluid
Darcy's Law
Regular Solution
Uniformly continuous
Local Existence
Unique Solution
Porous Media
Deduce
Curvature
Lower bound
Series
Operator

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics

Cite this

Global regularity for 2D Muskat equations with finite slope. / Constantin, Peter; Gancedo, Francisco; Shvydkoy, Roman; Vicol, Vlad.

In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, Vol. 34, No. 4, 01.07.2017, p. 1041-1074.

Research output: Contribution to journalArticle

Constantin, Peter ; Gancedo, Francisco ; Shvydkoy, Roman ; Vicol, Vlad. / Global regularity for 2D Muskat equations with finite slope. In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire. 2017 ; Vol. 34, No. 4. pp. 1041-1074.
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