Mixing solutions for the Muskat problem with variable speed
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Journal of Evolution Equations
Mixing solutions for the Muskat problem with variable speed Florent Noisette and László Székelyhidi Jr.
Dedicated to the 60th birthday of Matthias Hieber. 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).
1. Introduction 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 also [25,30]). In this paper, 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 (see [9,24] and references therein) ∂t ρ + div (ρu) = 0,
(1)
div u = 0,
(2)
u + ∇ p = −(0, ρ),
(3)
ρ(x, 0) = ρ0 (x).
(4)
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, ρ + x2 > z 0 (x1 ), (5) ρ0 (x) = ρ − x2 < z 0 (x1 ). Thus, the interface separating the two fluids at the initial time is given by 0 := {(s, z 0 (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 . If the sheet
J. Evol. Equ.
F. Noisette and L. Székelyhidi
can be presented as a graph as above, one can show (see for example [9]) that the equation for z(s, t) is given by ρ − − ρ + ∞ (∂s z(s, t) − ∂s z(ξ, t))(s − ξ ) dξ. (6) ∂t z(s, t) = 2 2 2π −∞ (s − ξ ) + (z(s, t) − z(ξ, t)) +
−
Linearising (6) around the flat interface z = 0 reduces to ∂t f = ρ −ρ H(∂s f ), where 2 H denotes the classical Hilbert transform. Thus one distinguishes the following cases: The case ρ + > ρ − is called the unstable regime and amounts to the situation where the heavier fluid is on top. The case ρ + < ρ − is called the stable regime. In the stable case, this equation is locally well-posed in H 3 (R), see [7,9], as well as [1,2,19,23] for recent developments. In the unstable case, however, which is our focus in this article, we have an ill-posed problem (see [9,25]), and there are no general existence results for (6) known. Thus, the description of (1)–(4) as a free boundary problem seems not suitable for the unstable regime. Indeed, as shown in experiments [30], in this regime the sharp interface seems to break down and the two fluids start to mix on a mesoscopic scale. 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. 1.1. Mixing solutions and admissible subsolutions A notion of solution, which allows for a meaningful existence
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