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Entropy solutions to
the macroscopic incompressible porous media equation
Ángel Castro, Daniel Faraco and Björn Gebhard |
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Vol. 18 (2025), No. 9, 2241–2292
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Abstract |
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We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarse- grained solutions) in the setting of the unstable Muskat problem. We show that both (a) imposing this criterion on the level of convex integration subsolutions and (b) the strategy of Otto based on a relaxation via minimizing movements lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface. |
Keywords
incompressible porous media, Saffman–Taylor instability,
relaxation, subsolutions, nonlocal hyperbolic conservation
law, mixing zone, level sets
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Mathematical Subject Classification
Primary: 76E17, 76S05
Secondary: 35L65, 35Q35
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Milestones
Received: 20 December 2023
Revised: 20 August 2024
Accepted: 29 October 2024
Published: 5 September 2025
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Authors |
| Ángel Castro | |
| Consejo Superior de Investigaciones
Científicas Instituto de Ciencias Matemáticas-CSIC ICMAT-CSIC-UAM-UCM-UC3M Madrid Spain |
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| Daniel Faraco | |
| Departamento de Mathemátics Universidad Autónoma de Madrid Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Madrid Spain |
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| Björn Gebhard | |
| Angewandte Mathematik Münster Universität Münster Münster Germany |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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