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

Mengual, Francisco

doi:10.2140/apde.2022.15.429

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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

DOI: 10.2140/apde.2022.15.429

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 $A_{μ} = 0$ ) to the case of viscosity jump ( $| A_{μ} | < 1$ ). We prove an h-principle whereby (infinitely many) weak solutions in $C_{t} L_{w^{*}}^{\infty}$ 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 $A_{μ} = 0$ , between the subsolution and the Lagrangian relaxed solution of Otto (Comm. Pure Appl. Math. 52:7 (1999), 873–915) holds for $| A_{μ} | < 1$ 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

Vol. 15, No. 2, 2022