On the global existence for the Muskat problem

Peter Constantin, Diego Córdoba, Francisco Gancedo, Robert M. Strain

Research output: Contribution to journalArticle

58 Scopus citations

Abstract

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L2(R) maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ||f||1 ≤ 1/5. Previous results of this sort used a small constant ε ≪ 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy ||f 0||L∞ < ∞ and ||δxf 0||L∞ < 1. We take advantage of the fact that the bound ||δxf0||L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

Original languageEnglish (US)
Pages (from-to)201-227
Number of pages27
JournalJournal of the European Mathematical Society
Volume15
Issue number1
DOIs
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Fluid interface
  • Global existence
  • Incompressible flows
  • Porous media

Fingerprint Dive into the research topics of 'On the global existence for the Muskat problem'. Together they form a unique fingerprint.

  • Cite this