On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof

Javier Gómez-Serrano, Rafael Granero-Belinchón

Research output: Contribution to journalArticle

20 Scopus citations

Abstract

We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

Original languageEnglish (US)
Pages (from-to)1471-1498
Number of pages28
JournalNonlinearity
Volume27
Issue number6
DOIs
StatePublished - Jun 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Keywords

  • Darcys law
  • blow-up
  • computerassisted
  • inhomogeneous Muskat problem
  • singularity
  • turning
  • water waves

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