Vol. 10, No. 2, 2017

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A note on stability shifting for the Muskat problem, II: From stable to unstable and back to stable

Diego Córdoba, Javier Gómez-Serrano and Andrej Zlatoš

Vol. 10 (2017), No. 2, 367–378
Abstract

In this note, we show that there exist solutions of the Muskat problem which shift stability regimes in the following sense: they start stable, then become unstable, and finally return back to the stable regime. This proves the existence of double stability shifting in the direction opposite to (as well as more difficult and more surprising than) the one shown by Cordoba et al. (Philos. Trans. Roy. Soc. A 373:2050 (2015), art. id. 20140278).

Keywords
Muskat problem, interface, incompressible fluid, porous media, Rayleigh–Taylor, computer-assisted
Mathematical Subject Classification 2010
Primary: 35Q35
Secondary: 35R35, 65G30, 76B03
Supplementary material

Auxiliary tables and code for the proof of Lemmas 2.2.1 and 2.2.2

Milestones
Received: 27 January 2016
Revised: 26 October 2016
Accepted: 26 December 2016
Published: 23 February 2017
Authors
Diego Córdoba
Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas
c/ Nicolas Cabrera
13-15 Campus Cantoblanco UAM
28049 Madrid
Spain
Javier Gómez-Serrano
Department of Mathematics
Princeton University
610 Fine Hall, Washington Rd
Princeton, NJ 08544
United States
Andrej Zlatoš
Department of Mathematics
University California San Diego
9500 Gilman Dr. #0112
La Jolla, CA 92093
United States