The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results

Abstract

In this paper we consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class $H^s,$ $s\in (3/2,2)$, when neglecting surface tension, respectively in $H^s,$ $s\in (2,3),$ when surface tension effects are included. Besides, we provide new criteria for the global existence of solutions.

49 pages, revision, to appear in Analysis & PDE

References in corpus (4)

• Absence of splash singularities for SQG sharp fronts and the Muskat problem
• On the Muskat problem: global in time results in 2D and 3D
• On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension
• On the Muskat flow

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• The vanishing surface tension limit of the Muskat problem
• Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime
• H-principle for the 2D incompressible porous media equation with viscosity jump
• Regularity of solutions to the Muskat equation
• Entropy solutions to macroscopic IPM
• The Second Iterate of the Muskat Equation in Supercritical Spaces