Vol. 15, No. 2, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 6, 1375–1616
Issue 5, 1131–1373
Issue 4, 891–1130
Issue 3, 567–890
Issue 2, 273–566
Issue 1, 1–272

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
h-principle for the 2-dimensional incompressible porous media equation with viscosity jump

Francisco Mengual

Vol. 15 (2022), No. 2, 429–476

We extend the results of Córdoba, Faraco and Gancedo (Arch. Ration. Mech. Anal. 200:3 (2011), 725–746) and Székelyhidi (Ann. Sci. Éc. Norm. Supér. (4) 45:3 (2012), 491–509) on the 2-dimensional incompressible porous media system with constant viscosity (Atwood number Aμ = 0) to the case of viscosity jump (|Aμ| < 1). We prove an h-principle whereby (infinitely many) weak solutions in CtLw are recovered via convex integration whenever a subsolution is provided. As a first example, nontrivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi (2012) for Aμ = 0, between the subsolution and the Lagrangian relaxed solution of Otto (Comm. Pure Appl. Math. 52:7 (1999), 873–915) holds for |Aμ| < 1 too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh–Taylor nor vorticity at the interface.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

hydrodynamics, unstable interface, convex integration
Mathematical Subject Classification
Primary: 35Q35, 76F25, 76S05
Received: 15 April 2020
Accepted: 6 October 2020
Published: 12 April 2022
Francisco Mengual
Departamento de Matemáticas
Universidad Autónoma de Madrid
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)