## 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 language | English (US) |
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Pages (from-to) | 1471-1498 |

Number of pages | 28 |

Journal | Nonlinearity |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - 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