Abstract
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 , , when neglecting surface tension, respectively in , , when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.
Citation
Bogdan-Vasile Matioc. "The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results." Anal. PDE 12 (2) 281 - 332, 2019. https://doi.org/10.2140/apde.2019.12.281
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