@article{10902/29577,
year = {2014},
month = {6},
url = {https://hdl.handle.net/10902/29577},
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.},
organization = {Acknowledgments: The authors are supported by the Grant MTM2011-26696 from Ministerio de Ciencia e Innovación (MICINN) and MINECO: ICMAT Severo Ochoa project SEV-2011- 0087. Javier Gómez-Serrano is supported by StG-203138CDSIF of the ERC. Rafael Granero-Belinchón is grateful to Luigi Berselli and Rafael Orive for productive comments in an early version of these results. Javier Gómez-Serrano thanks Rafael de la Llave for fruitful discussions. We thank Diego Córdoba for his guidance and useful suggestions. We wish to thank the Instituto de Ciencias Matemáticas (Madrid) for computing facilities.},
publisher = {Institute of Physics},
publisher = {Nonlinearity, 2014, 27, 1471},
title = {On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof},
author = {Gomez Serrano, Javier and Granero Belinchón, Rafael},
}