Muskat Problem Dynamics in Porous Media
Summary
The Muskat problem investigates the evolution of an interface separating two immiscible fluids moving through a porous medium under Darcy’s law. Depending on the density and viscosity contrast, the interface may remain smooth or develop fingering instabilities characteristic of the Rayleigh–Taylor regime. Mathematical analysis centres on well-posedness of the governing equations, stability and eventual singularity formation, and the regularising role of surface tension. Recent advances blend functional-analytic techniques in Sobolev spaces with variational formulations, providing insight into long-term behaviour and criteria for global existence. Applications span enhanced oil recovery, groundwater remediation and carbon sequestration, where control of interface dynamics is crucial for efficiency and safety.
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Muskat Problem Dynamics in Porous Media publication trend
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Technical terms
Darcy’s law: empirical relation between fluid velocity and pressure gradient in a porous medium under laminar flow conditions.
Rayleigh–Taylor instability: interface instability arising when a denser fluid overlies a lighter one, leading to finger-like protrusions.
Weak solution: generalised solution concept in which governing equations hold in an integral or distributional sense, accommodating discontinuities or limited smoothness.
Sobolev space: functional space characterised by integrability and differentiability properties of functions, used to quantify solution regularity.
Wasserstein space: metric space of probability measures equipped with an optimal-transport distance, enabling gradient-flow formulations of evolution equations.
References
- On the Parabolicity of the Muskat Problem: Well-Posedness, Fingering, and Stability Results. Zeitschrift für Analysis und ihre Anwendungen (2011).
- Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport. Archive for Rational Mechanics and Analysis (2020).
- Self-similar solutions for the Muskat equation. Advances in Mathematics (2022).
- Regularity of Solutions to the Muskat Equation. Archive for Rational Mechanics and Analysis (2023).
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