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Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. (English) Zbl 1120.76064

Summary: We consider the evolution of interface of two incompressible fluids in a porous medium, which is known as Muskat problem and in two dimensions is mathematically analogous to the two-phase Hele-Shaw cell. We focus on a fluid interface given by a jump of densities, being the equation of the evolution obtained by using Darcy’s law. We prove local well-posedness when the smaller density is above (stable case), and in the unstable case we show ill-posedness.

MSC:

76S05 Flows in porous media; filtration; seepage
76D27 Other free boundary flows; Hele-Shaw flows
35R35 Free boundary problems for PDEs
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