Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 2 (2019), 281-332.

The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

Bogdan-Vasile Matioc

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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 ( 3 2 , 2 ) , when neglecting surface tension, respectively in H s , s ( 2 , 3 ) , when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.

Article information

Anal. PDE, Volume 12, Number 2 (2019), 281-332.

Received: 18 October 2016
Revised: 17 January 2018
Accepted: 7 May 2018
First available in Project Euclid: 9 October 2018

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Zentralblatt MATH identifier

Primary: 35R37: Moving boundary problems 35K59: Quasilinear parabolic equations 35K93: Quasilinear parabolic equations with mean curvature operator 35Q35: PDEs in connection with fluid mechanics 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Muskat problem surface tension singular integral


Matioc, Bogdan-Vasile. The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12 (2019), no. 2, 281--332. doi:10.2140/apde.2019.12.281.

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