Abstract
We provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).
Highlights
The mathematical model for the evolution of two incompressible fluids moving in a porous medium, such as oil and water in sand, was introduced by Morris Muskat in his treatise [22] and is based on Darcy’s law
We focus on the case of constant permeability under the action of gravity so that, after non-dimensionalizing, the equations describing the evolution of density ρ and velocity u are given by
We assume that at the initial time the two fluids, with densities ρ+ and ρ−, are separated by an interface which can be written as the graph of a function over the horizontal axis is, ρ0(x) =
Summary
The mathematical model for the evolution of two incompressible fluids moving in a porous medium, such as oil and water in sand, was introduced by Morris Muskat in his treatise [22] and is based on Darcy’s law (see [25,30]). We assume that at the initial time the two fluids, with densities ρ+ and ρ−, are separated by an interface which can be written as the graph of a function over the horizontal axis is, ρ0(x) =. The interface separating the two fluids at the initial time is given by 0 := {(s, z0(s))|s ∈ R}. Assuming that ρ(x, t) remains in the form (5) for positive times, the system reduces to a non-local evolution problem for the interface. The case ρ+ < ρ− is called the stable regime. In a number of applications [22,30], it is precisely this mixing process in the unstable regime which turns out to be highly relevant, calling for an amenable mathematical framework
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