Viscous displacement in porous media: the Muskat problem in 2D

2017/01/04 by Matioc, Bogdan-Vasile · 1 citation
#35K55 #35K59 #35Q35 #35R37 #42B20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.00992

Abstract

We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh-Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary large initial data in H^s, s>2, if surface tension is taken into account, respectively for arbitrary large initial data in H² that additionally satisfy the Rayleigh-Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.

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