Steady-state solutions for the Muskat problem

In this paper we study the existence of stationary solutions for the Muskat problem with a large surface tension coefficient. Ehrnstrom, Escher and Matioc studied in Mats Ehrnström (Methods Appl Anal 20:33-46, 2013) that there exists solutions to this problem for surface tensions below a finite value. In these notes we go beyond this value considering large surface tension. Also by numerical simulation we show some examples that explains the behavior of solutions.


Introduction
We consider two incompressible fluids with different densities and equal viscosity in a porous medium, separated by a curve z = (z 1 , z 2 ). The dynamic of this two phase-flow can be modelled by Incompressible Porous Media equation (IPM) which is given by the Darcy's law, the mass conservation equation and the incompressibility condition μ κ v = −∇ p − (0, g ρ), where v is the incompressible velocity, p is the pressure, μ is the dynamic viscosity, κ is the permeability of the medium, ρ is the liquid density and g is the acceleration due to gravity. Without loss of generality, we set μ/κ = 1. The initial density, which takes only two constant   Fig. 1). The research on the Muskat problem has been very intense due to the many applications (see [16,20,23] and [18]), one of the characteristic of this problem is that some scenarios are unstable and leads to an ill-posed problem, this scenario is when the heavier fluid is above the other one. In the stable scenario when the heavier fluid is in the lower part, the problem is locally well-posed. This arises from the linear stability analysis of the equation for the interface evolution (see [2][3][4] and [22]). Such linear stability is characterized by the sign of the Rayleigh-Taylor function for a curve z = (z 1 , z 2 ) that parameterizes the interface between the fluids. In the stable case as mathematical case is a challenging example of a free boundary problem driven by a nonlinear and non-local parabolic equation. In the unstable case recent investigations have show that the method of convex integration yields infinitely many weak solutions (see [5] and [6]). There is little doubt that the most important open questions is to understand a selection principle for such weak solutions. Recently the case of stationary IPM has been studied from this perspective in [15] by Hitruhin and Lindberg. If we consider surface tension, there is a jump discontinuity of the pressure across the interface which is modeled to be equal to the local curvature times the surface tension coefficient In this case we have viscous fingering, a phenomenon known in porous media as a result of an instability of the interface when a lower density fluid penetrates a more viscous fluid, the result is the formation of patterns of type finger in the interface. The free boundary problem is well posed (see [7] and [11]), since capillarity eliminates instabilities in Rayleigh-Taylor condition and this problem has been studied extensively (see [1,10,14,17,21] and [13]). For further results on the Muskat problem with and without surface tension see the survey [12].
In this work we give a description of the stationary solutions for the Muskat problem with surface tension, looking for 2π periodic solutions, this can be achieved by reducing the Muskat equation (2) to an ordinary differential equation (5). Using the Darcy's Law we compute the vorticity

The Muskat equation is given by
(2) Therefore steady-state solutions of (2) are solutions of the equation and a solution of (3) is a steady-state solutions of (2) and therefore a steady-state solutions of the Muskat problem. In this work we take by simplicity equal viscosities, if we consider different viscosities the stationary equation obtained is the same as in the case of equal viscosities. For a treatment of the case with different viscosities see [19], Sect. 6. From here, a curve z : R → R 2 is a steady-state solution of (2) if satisfies where we consider that z 2 , the second component of the curve, is a odd function. Ehrnstrom, Escher and Matioc in [8] found a finite limit value with the property that if the surface tension coefficient remains below this value the curve remains 2π periodic and when the surface tension coefficient approaches to this finite value from below, the maximal slope of the curve tends to infinity. The equation (3) in the stable case ρ − > ρ + has only trivial solution (see [9]). In this notes we consider the unstable case, when ρ + > ρ − , that is, when the heavier fluid occupies the upper part and we are interesting in describing a solution with the following initial data We take and recall the definition of the beta function The main result of this notes is the following theorem: there exists λ * > 0 greater than λ * such that for each λ ∈ (λ * , λ * ] there exists a unique α(λ) ∈ [0, ∞) and a periodic solution curve of the steady-state equation (3) that does not self-intersect.
In [8], Ehrnstrom, Escher and Matioc consider the parameter λ > λ * and in this case the curve is locally the graph of a function (x, f (x)), instead of that we consider a curve (g(y), y) and we are able to show that there exists 2π periodic solutions for the case λ * < λ < λ * , moreover we show that if λ < λ * the solutions there are no longer periodic. The proof in the main theorem is obtained by analyzing a explicit formula (7) for the period. Moreover we describe some numerical examples that indicates λ * ∼ λ * /7.

Steady-state solutions
In this section we study the period of the steady-state solution with the conditions (4), under such conditions the curve z is a graph respect to a function h, that is, z(y) = (h(y), y) and the curve is a steady-state solution if the function h satisfies To prove the main theorem we have three previous lemmas, the first about the existence of 2π periodic solutions, the second related to the function λ → α and the third on the intersection of the solution curves. We leave at the end of the section the proof of the main theorem.
The idea is integrate the equation (5) directly over the interval [0, y] to determine conditions in the parameters λ and α. After the integration we have the next equation In order to simplify the notations we put β := α (1 + α 2 ) 1/2 . Taking squares and since h and (λ/2)y 2 − β has the same sign, we have We observe from equation (6) that there exists a zero for h and a positive value where h explodes, that is also at this point the curve is no longer the graph of the function h. We define the period of the solution curve as the integral Lemma 1 If λ ∈ (0, λ * ], the period (7) of the solution curve satisfies Proof Taking the change of variable τ = s/ 2λ −1 (1 + β), we get the next expression for the period where g τ (α) = (1 + β)τ 2 − β. The derivatives respect to β and α of g τ and β respectively are ∂ ∂β g τ (α) = τ 2 − 1 < 0 and ∂β ∂α Then from the chain rule the derivative respect to α is and therefore the period T (λ, α) is decreasing with respect to α. Now, if we want to determine the pair (λ, α) such that T (λ, α) = π/2, we will determine conditions on λ observing its behavior as α goes to zero and infinity. We will see explicitly that the following inequalities are satisfied lim For the first inequality, we compute T (λ, 0) from (8), that is When α = 0, β = 0 and we have Therefore, if we take The next step is compute the limit when α → ∞ in equation (7). We observe for a fixed λ < λ * the limit in the integral satisfies (11) We can see in the last integral, that for a fixed λ < λ * lim α→∞ T (λ, α) < 0 moreover T (λ, α) → −∞, because the first integral goes to −∞.
To complete the proof of the main theorem we need to know if the curves solutions has intersections. We know from (7) that the function h has a minimum at the point y = 2λ −1 β, define which is the value of h at 2λ −1 β. Now we want to determine if the fact that λ ∈ (0, λ * ] is enough to have this property is important because we want that the solution curve has not intersections.
Taking the change of variable τ = s/ 2λ −1 β we can rewrite the integral as we have the next lemma.

Numerical examples
In this section we show some numerical examples of steady-state solutions for different values of λ. The Figs. 2, 3, 4 and 5 explains the behavior of the solution curves when λ approaches to the value λ * . The limit curve z λ * (Fig. 4) remains 2π periodic but has self-intersections, also for λ * /16 (Fig. 5) we observe that the curve solution has self-intersections, then we can infer that λ * /16 < λ * .  Remark 1 It remains to have a analytical proof of the value λ * that lead us to an explicit value for λ * .