On the Muskat problem with viscosity jump: Global in time results

The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contou...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 345; pp. 552 - 597
Main Authors Gancedo, F., García-Juárez, E., Patel, N., Strain, R.M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 17.03.2019
Subjects
Fluid mechanics
Global regularity
Muskat problem
Viscosity jump
Fluid mechanics
Global regularity
Muskat problem
Viscosity jump
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Summary:The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in L∞. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2019.01.017