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

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.

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)