2022/02/24 by Jonas Bierler, Bierler, Jonas, Bogdan–Vasile Matioc +1
Mathematics · #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.2202.12004
In this paper we study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with general viscosities in a vertical homogeneous porous medium under the influence of gravity. Employing Rellich type identities in the regime where the fluids are ordered according to their viscosities, respectively a Neumann series argument when the fluids are not ordered by viscosity, we may recast the governing equations as a strongly coupled nonlinear and nonlocal evolution problem for the functions that parameterize the sharp interfaces that separate the fluids. This problem is of parabolic type if the Rayleigh-Taylor condition is satisfied at each interface. Based on this property, we then show that the multiphase Muskat problem is well-posed in all L2-subcritical Sobolev spaces and that it features some parabolic smoothing properties.