Vol. 15, No. 2, 2022

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h-principle for the 2-dimensional incompressible porous media equation with viscosity jump

Francisco Mengual

Vol. 15 (2022), No. 2, 429–476
Abstract

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.

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