A survey for the Muskat problem and a new estimate

A survey for the Muskat problem and a new estimate. Francisco Gancedo Abstract This paper shows a summary of mathematical results about the Muskat problem. The main concern is well-posed scenarios which include the possible formation of singularities in finite time or existence of solutions for all time. These questions are important in mathematical physics but also have a strong mathematical interest. Stressing some recent results of the author, we also give a new estimate for the problem in the last section. Initial data with L2decay and slope less than one provide weak solutions which satisfy a parabolic inequality as in the linear regime. 1 Introduction The mathematical analysis of fluid mechanics models in PDEs is a classical topic of research since Euler’s 1757 paper, where the evolution equation of an ideal flow was first derived. For the well established models, such as Navier-Stokes and Euler, the incompressible case presents basic and important open questions such as global regularity and finite time singularity formation of the solutions. It is a current area of mathematical research of fundamental interest in particular due to its relevance in Physics and wide applicability. In the analysis of PDEs from fluid mechanics, an outstanding class of problems are those in which the evolution of fluids of different nature are modeled. The interaction between the fluids provides the dynamics of their common free boundary that evolves with the flow. It gives rise to long standing problems such as vortex-patch [14], vortex-sheet [49, 4], water waves [44], viscous waves evolution [41], interface flows in porous media and Hele-Shaw cells [37, 9], as well as atmospheric front dynamics [23], among others. These free boundary dynamics problems are modeled by fluid mechanics PDEs such as Euler, NavierStokes, Darcy momentum equation and quasi-geostrophic systems. In all of them fundamental questions are local-in-time existence, global-in-time regularity of solutions or finite time singularity formation in well-posed scenarios. In this manuscript we focus on the classical Muskat problem [45]. It considers contour dynamics problems for incompressible fluids of different nature permeating a porous medium. Recently, computer evidence has shown how singularities may developed in Muskat [12]. With recent new techniques, it is now possible to prove different types of finite time singularity formation for 1 those scenarios [13, 10, 11]. These are the first analytic proofs of blow-up for incompressible fluid in well-posed situations. We introduce now the equations of the problem, considering an active scalar ρ(x, t), depending on time t≥0 and position x∈R2. Here we will pick the two dimensional case for simplicity of exposition. The fluid velocity is incompressible ∇ · u(x, t) = 0,(1) and the scalar ρ(x, t) satisfies a general transport evolution equation for incompressible flows ρt(x, t) + u(x, t)· ∇ρ(x, t) = 0.(2) That we are dealing with two different fluids is reflected in the configuration of ρ(x, t), which is a discontinuous function with constant values in two complementary connected sets D1(t) and D2(t) = R2\D1(t): ρ(x, t) = {ρ1, x ∈D1(t), ρ2, x ∈D2(t).(3) The constants ρ1and ρ2represent the density of each fluid that occupy the sets D1(t) and D2(t), respectively. Therefore equation (2) becomes the conservation of mass and it is understood in a weak sense. The main concern is about the dynamics of the free boundary of the fluids ∂Dj(t), j= 1,2, which is parameterized by the curve z(α, t) as follows: ∂Dj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α∈R}. Above the curve z(α, t) is asymptotically flat: z(α, t)−(α, 0) →0 as α→ ∞, and we will consider also the case of a 2π-periodic contour in the x1direction: z(α+ 2π, t) = z(α, t) + (2π, 0).The fluid with density ρ2essentially lies below the fluid of density ρ1in such a way that there is a constant M > 1 big enough so that R×(−∞,−M]⊂D2(t). We always assume that the initial velocity is in L2∫|u(x, 0)|2dx < ∞, the finite energy and physically relevant case. For the Muskat problem, the most common example for applications is the dynamics of water and oil [6]. This is a classical topic of investigation dating back to Muskat’s 1934 paper [45]. In consequence the fluids can also have different constant viscosities, given by µ(x, t) = {µ1, x ∈D1(t), µ2, x ∈D2(t).(4) Finally, the system is closed using Darcy’s law µ(x, t) κu(x, t) = −∇p(x, t)−g(0, ρ(x, t)),(5) 2 which relates the incompressible velocity with the pressure [31], considering that the fluids saturate the porous media. The permeability and the gravity constants are given by κand grespectively. In [47] a completely different physical scenario is studied, with comparable mathematical properties. This describes the flow evolution in Hele-Shaw cells, where the fluids are confined between two parallel plates that are close together. The evolution is essentially in 2D, and it is governed by the equation 12 b2µ(x, t)u(x, t) = −∇p(x, t)−g(0, ρ(x, t)), where bis the distance between the plates. Since these two pioneering works, these different physical phenomena have been extensively studied from a mathematical point of view. 2 Contour evolution equation The Muskat problem can be considered taking into account many more peculiarities as boundary effects [27] and three dimensional flows [2, 20]. The framework picked in this presentation allows us to reduce the problem from its original Eulerian variables formulation (eqs. (1,2,3,4,5)) to the self-evolution of an interface, hence the name contour evolution equation. It provides a simple way to linearize the system of equations to illustrate in a non-technical manner what is going on at the nonlinear level. Darcy’s law (5) shows that the velocity has to be irrotational ∂x1u2(x, t)−∂x2u1(x, t) = 0, in the interior of each domain Dj(t), j= 1,2. For that reason the vorticity is given by a measure on the free interface as follows ∂x1u2(x, t)−∂x2u1(x, t) = ω(α, t)δ(x=z(α, t)), defined in a distributional sense as follows: < u, (∂x2φ, −∂x1φ)>=∫ω(α, t)φ(z(α, t))dα, (6) with φ(x) a regular test function. Using the Biot-Savart law u(x, t) = (−∂x2, ∂x1)∆−1(∂x1u2−∂x2u1)(x, t), it is possible to recover the velocity from the vorticity. It is given by the partial derivatives of the Newton potential as follows: u(x, t) = 1 2πPV ∫(x−z(α, t))⊥ |x−z(α, t)|2ω(α, t)dα, for x=z(α, t) where PV denotes principal value (as it is necessary at infinity) and (x1, x2)⊥= (−x2, x1). Taking limits by approaching the free boundary 3 in the normal direction, we can obtain the velocity at the interface with a discontinuity. It reads u2(z(α, t), t) = BR(z, ω)(α, t) + 1 2 ω(α, t) |∂αz(α, t)|2∂αz(α, t), u1(z(α, t), t) = BR(z, ω)(α, t)−1 2 ω(α, t) |∂αz(α, t)|2∂αz(α, t), (7) where uj(z(α, t), t) denotes the limit obtained from inside Dj(t). Above BR stands for the Birkhoff-Rott integral, which is given by BR(z, ω)(α, t) = 1 2πPV ∫(z(α, t)−z(β, t))⊥ |z(α, t)−z(β, t)|2ω(β, t)dβ. (8) In the above contour operator it is easy to see the importance of the arc-chord condition in order for the Birkhoff-Rott integral to make sense. A one-to-one curve satisfies the arc-chord condition if |z(α, t)−z(β, t)| ≥ Cac(t)|α−β|,∀α, β ∈R, Cac(t)>0.(9) The discontinuity of the velocity at the free boundary, which is produced by the vorticity configuration (6), is given in the tangential direction and consequently it does not give any insight on the evolution of the shape for z(α, t). In fact, the normal velocity describes the dynamics and it is continuous on z(α, t) (see (7)). Darcy’s law implies ∆p(x, t) = −div (µ(x, t) κu(x, t) + g(0, ρ(x, t))), where ∆p(x, t) = RT(α, t)δ(x−z(α, t)), and the function RT(α, t) is given by RT(α, t) = µ2−µ1 κu(z(α, t), t)·∂αz⊥(α, t) + g(ρ2−ρ1)∂αz1(α, t).(10) Above u(z(α, t), t)·∂αz⊥(α, t) = u2(z(α, t), t)·∂αz⊥(α, t) = u1(z(α, t), t)·∂αz⊥(α, t), due to (7). Recovering the pressure through the Newton potential, p(x, t) = 1 2π∫ln |x−z(α, t)|RT(α, t)dα, for x=z(α, t), it is possible to obtain the continuity of the pressure at the free boundary p2(z(α, t), t) = p1(z(α, t), t), 4 which is just a mathematical consequence of Darcy’s law. Let us introduce the following notation: [µu](α, t) = (µ2u2(z(α, t), t)−µ1u1(z(α, t), t)) ·∂αz(α, t). Taking limits in Darcy’s law provides [µu](α, t) κ=−(∇p2(z(α, t), t)−∇p1(z1(α, t), t)) ·∂αz(α, t)−g(ρ2−ρ1)∂αz2(α, t) =−∂α(p2(z(α, t), t)−p1(z(α, t), t)) −g(ρ2−ρ1)∂αz2(α, t) =−g(ρ2−ρ1)∂αz2(α, t), which allows us to relate the vorticity amplitude ωwith the unknown curve through the following implicit identity ω(α, t)+2µ2−µ1 µ2+µ1BR(z, ω)(α, t)·∂αz(α, t) = −2gκ ρ2−ρ1 µ2+µ1∂αz2(α, t).(11) The dynamics is given by the velocity with the following evolution equation zt(α, t) = BR(z, ω)(α, t) + c(α, t)∂αz(α, t),(12) where the subscript tdenote partial derivative in time and c(α, t) is the function which provides parametrization freedom. It is worth mentioning that considering different c(α, t) the geometry of the curve is the same, as the evolution is described by the normal direction of the velocity [42]. It is usual to pick c as the function zero, but different choices provide different advantages in the analysis of the evolution equation. In particular, it could be chosen in such a way that the interface is parameterized as a graph (see formula (18) below). In conclusion, the contour dynamics equation is now closed and given by (8,11,12). 3 Mathematical results The Muskat problem has a rich variety of features which have been studied with a wide diversity of techniques. Interesting scenarios consider 3D fluids, multiphase flows, boundary effects or permeability discontinuities (see for example [5, 43]), etc. Different methods interact, raging from analytic to computerassisted proofs [38]. In what follows, the permeability κis considered to be equal to one. A very significant peculiarity of the problem is that Muskat develops instabilities [46]. If the system of equations (1,2,3,4,5) is satisfied in a week sense, some scenarios yield non-uniqueness of solutions [50]. In the contour evolution setting (8,11,12), those unstable cases give rise to ill-possed equations. These phenomena can be understood through the Rayleigh-Taylor condition. Considering the jump across the normal direction of the gradient pressures, it is possible to find −(∇p2(z(α, t), t)− ∇p1(z(α, t), t))=RT (α, t), 5 with RT the Rayleigh-Taylor function given in (10). The Rayleigh-Taylor condition is said to be satisfied if RT (α, t)>c>0. By linearizing the equations (8,11,12) near the steady state z(α, t) = (α, 0), it is possible to obtain fL t(α, t) = −(µ2+µ1)−1RT LΛfL(α, t),(13) where (α, fL(α, t)) represents the linearized free boundary, and the constant RTLis the linear version of the Rayleigh-Taylor function. The operator Λ is the minus square root of the Laplacian, Λ = (−∆)1/2, also given by a kernel representation and using the Fourier transform as follows: ΛfL(α) = 1 πPV ∫fL(α)−fL(β) (α−β)2dβ, d ΛfL(ξ) = |ξ|c fL(ξ).(14) Now the importance of the Rayleigh-Taylor is disclosed. The case RTL>0 turns the Muskat problem into a parabolic system at the linear level. For RTL<0 the character of the equation changes dramatically, giving an ill-posed system. This fact is easy to understand by using the Fourier transform in space to solve (13) obtaining c fL(ξ, t) = c fL(ξ, 0) exp(−(µ2+µ1)−1RTL|ξ|t). At the nonlinear level, the Rayleigh-Taylor function (see equation (10)) implicates the normal velocity of the fluids with viscosity jump and the geometry of the contour for different densities. Basically, the unstable case arises in the viscosity jump situation when a less viscous fluid pushes a more viscous one. This case was studied in [48], where the contour dynamic equation is proved to be ill-possed. In the density jump case (µ2=µ1), the unstable regime holds when the more dense fluid lies above the interface and the less dense fluid lies below it. The contour dynamics equation is shown to be ill-posed in this scenario [21]. On the other hand, the lost of derivative in the contour equation is of order one, so that it is possible to find solutions of the system with analytic initial data even in the unstable case [32, 12]. At the linear level this fact can be checked using equation (13) and the theory of the Fourier transform for analytic functions. Besides gravity, the evolution Muskat problem can be driven by capillary force. In that case surface tension effects are considered, and the discontinuity of the pressure on the interface is proportional to its curvature as follows: p2(z(α, t), t)−p1(z(α, t), t) = −τ∂2 αz(α, t)·∂⊥ αz(α, t) |∂αz(α, t)|3,(15) where τ > 0 is the surface tension coefficient. In this case equation (11) is replaced by ω(α, t) + 2µ2−µ1 µ2+µ1BR(z, ω)(α, t)·∂αz(α, t) = −2gκ ρ2−ρ1 µ2+µ1∂αz2(α, t)−τ∂α(∂2 αz·∂⊥ αz |∂αz|3)(α, t). (16) 6 The linearization of the system in this case is given by fLτ t(α, t) = −(µ2+µ1)−1RT LΛfLτ (α, t) + τΛ∂2 αfLτ (α, t),(17) showing a high parabolic regularizing effect for the graph (α, fLτ (α, t)). The local-in-time existence for the nonlinear problem without gravity and surface tension (g= 0 and RTL= 0 in above linear interpretation) and in the one fluid case (µ2= 0 = ρ2) was given in [33]. See also [29] for the boundary value problem. Same type of results for the two fluids case were given in [35]. Besides this surface tension regularizing mechanism, Rayleigh-Taylor instabilities still play a crucial role considering the force of gravity. In [40] initial small perturbation are shown to be unstable under small time evolution for low order norms. In [34] finger shaped stationary-states are found using bifurcation theory which are unstable. On the other hand, when the Rayleigh-Taylor condition is satisfied initially, surface tension solutions approach to solutions without surface tension effects as the coefficient τvanishes [3]. Without surface tension (τ= 0), the positivity of the nonlinear RayleighTaylor function have been shown to be crucial to disclose a local-in-time existence result [1, 51]. The system is proved to be well-posed in the case of equal viscosity µ1=µ2=µfor the stable case [21] by using energy estimates on the contour dynamics equation through the chain of Sobolev norms. In this situation, the Rayleigh-Taylor condition is satisfied only if the free boundary is represented by the graph of a function (α, f(α, t)) and ρ2> ρ1. The contour evolution equation is given in this situation by ft(α, t) = g(ρ2−ρ1) 2µπ ∫β(∂αf(α, t)−∂αf(α−β, t)) β2+ (f(α, t)−f(α−β, t))2dβ, (18) and RT(α, t) = g(ρ2−ρ1) due to ∂αz1(α, t) = 1. The case with different viscosities and densities was shown to be well-posed in [19]. In that proof it is crucial to get control of the norm of the implicit operator given in (11) involved in the definition of the amplitude of the vorticity ω. The arguments rely upon quantitative bounds of Hilbert transforms in variable domains in the plane. It requires a harmonic analysis approach involving the Hopf maximum principle, conformal mappings and Harnack inequalities. A local-in-time control of the positivity of the Rayleigh-Taylor sign condition is indispensable to reach legitimate energy estimates, as for a general parametrization RT (α, t) does not need to be positive. Finally we would like to quote some recent articles where local-in-time existence is shown of classical solution for large and low regular initial data. For the one fluid case (µ1=ρ1= 0) see [30] and [17] for the density jump case. If µ2=µ1=µand τ= 0, it is possible to obtain decay of the L∞norm of the interface for arbitrary initial data (see [22]). The graph interface evolves by (18) giving ∥f−1 2π∫π −π f0dα∥L∞(t)≤ ∥f0−1 2π∫π −π f0dα∥L∞e−Ct, 7 for 2π-periodic fand ∥f∥L∞(t)≤ ∥f0∥L∞(1 + Ct)−1, in the asymptotically flat case, where f(α, 0) = f0(α) and C=C(f0)>0. These maximum principles are sharp as they provide the same rate of decay as equation (13) for fL. On the other hand, the L2norm evolution allows to control half a derivative of fLin (13) due to the identity ∥fL∥2 L2(t) + g(ρ2−ρ1) µ∫t 0 ∥Λ1/2fL∥2 L2(s)ds =∥fL 0∥2 L2, or equivalently ∥fL∥2 L2(t)+ g(ρ2−ρ1) 2µπ ∫t 0∫R∫R(fL(α, s)−fL(β, s) α−β)2 dβdαds =∥f0∥2 L2,(19) using the integral formula (14) and that RTL=g(ρ2−ρ1). For the nonlinear problem, the identity ∥f∥2 L2(t)+g(ρ2−ρ1) 2µπ ∫t 0∫R∫R ln (1+(f(α, s)−f(β, s) α−β)2)dαdβds=∥f0∥2 L2(20) holds [16], which does not give a chance of gaining any regularity at the level of f. This can be easily shown by the bound ∫R∫R ln (1+(f(α, s)−f(β, s) α−β)2)dαdβ ≤C∥f∥L1(s),(21) which allows to control the nonlinear term with zero derivatives. In the case with small initial data, it is possible to use the parabolic character of the equation in the stable state (see (13) and (17) for the lineal interpretation) to prove global in time regularity in different situations. For purely surface tension driven fluids (g= 0) see results in [29, 18]. Without surface tension (τ= 0), global existence for the viscosity jump case was proven in [48] and extended to the density jump case in [21], showing in both papers instant analyticity of the solutions. For gravity and surface tension interaction with boundary values see [34]. Those global existence results have been extended in some situations assuming initial smallness for critical norms with respect to the scaling [16, 39], and showing instant analyticity in [7]. In works [16, 15] some results of global in time regularity of classical solutions are shown with µ1=µ2,τ= 0 and medium-size initial slope in the Wiener algebra, i.e ∫|ξ|| ˆ f(ξ)|dξ ≤c0 with c0an explicit constant. In particular, the terminology medium-size is used to emphasize that the constant c0is of size O(1) and independent of the physical 8 constants g,κ,ρj, and µj(j= 1,2). Those papers show global existence of Lipschitz weak solutions with initial slope less than 1 and gradient less than 1/3 in 3D. Using equation (18), multiplying by a test function and integrating by parts, it is possible to find a weak formulation of the system. In fact, we say that the graph (α, f(α, t)) is a weak solution of the Muskat problem if the following identity is satisfied ∫T 0∫R ηt(α, t)f(α, t)dαdt +∫R η(α, 0)f0(α)dα =∫T 0∫R ∂αη(α, t)g(ρ2−ρ1) 2µπ PV ∫R arctan (f(α, t)−f(β, t) α−β)dβdαdt, (22) for any η∈C∞ c([0, T)×R). A fascinating behavior of Muskat solution, which can be proved analytically, is finite time singularity formation starting from regular stable initial data. In [13] it is proved that in the case µ1=µ2and τ= 0 there are solutions of the Muskat equation with initial interfaces being certain smooth stable graphs, which enter the unstable regime, where the interface is no longer a graph, in finite time. In particular there exists a time tpin which lim t→t+ p ∥∂αf∥L∞(t) = +∞, for solutions of equation (18). In other words, the interface evolves into a nongraph in finite time. For some contour dynamics problems these “wave-turning” effects are not dramatic, it is just a breakdown in the parametrization as a graph. But for the Muskat problem this is a strong change in the character of the equation. In particular the significance of a wave-turning is that the Rayleigh-Taylor condition breaks down. At some branch in the interface it is possible to localize the heavy fluid on top of the lighter one. An important reason why this phenomenon arises is that, even for large initial data, Muskat solutions become instantly analytic [13]. So that, despite the interface is about to reach an unstable regime, the analyticity remains by the time the wave-turning occurs. In fact, the Muskat curve solution exists and remains analytic for some time after the turnover, even in the unstable regime. Furthermore, global existence can be false for certain scenarios with large initial data. In [10] it is shown that some of these smooth initial interfaces in the stable regime turn to the unstable regime and later blow-up; i.e. for some time ts> tpthere is a lost of regularity in the interface. Therefore Muskat develops finite time singularities starting from well-posed scenarios. This is the first case of singularity formation in contour dynamics of incompressible fluids in an initially well-posed problem. The pattern of these initial data is far from trivial: numerical simulations performed in [25] show that there exists initial data with steep slopes for which a regularizing effect appears. Even more, some analytic unstable solutions can reach a stable regime and some later time become unstable [26]. If the contour evolution remains regular in the stable regime is not known, but a finite time singularity formation characterization is given in [17] in terms of the interface slope. 9 [21] D. C´ordoba and F. Gancedo. Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 273, 2, 445-471, 2007. [22] D. C´ordoba and F. Gancedo. A maximum principle for the Muskat problem for fluids with different densities. Comm. Math. Phys., 286, no. 2, 681-696, 2009. [23] D. C´ordoba and F. Gancedo. Contour dynamics for 2D active scalars. Newsletter of the European Mathematical Society, no. 71 (March 2009). [24] D. C´ordoba and F. Gancedo. Absence of squirt singularities for the multiphase Muskat problem. Comm. Math. Phys., 299(2):561–575, 2010. [25] D. C´ordoba, F. Gancedo and R. Orive. A note on the interface dynamics for convection in porous media. Physica D, 237, 1488-1497, 2008. [26] D. C´ordoba, J. G´omez-Serrano and A. Zlatos. A note in stability shifting for the Muskat problem. Phil. Trans. R. Soc. A, 373, 2050, 2015. [27] D. C´ordoba, R. Granero and R. Orive. The confined Muskat problem: differences with the deep water regime. Commun. Math. Sci., 12(3):423–455, 2014. [28] D. C´ordoba and T. Pernas-Casta˜no. Non-splat singularity for the one-phase Muskat problem. To appear in Trans. Amer. Math. Soc. 2015. [29] X. Chen. The hele-shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal., 123(2):117–151, 1993. [30] C.H. Cheng, R. Granero-Belinch´on, and S. Shkoller. Well-posedness of the Muskat problem with H2initial data. To appear in Adv. in Math., 2015. [31] H. Darcy. Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris, 1856. [32] T. Dombre, A. Pumir and E. Siggia. On the interface dynamics for convection in porous media. Physica D, 57, 311-329, 1992. [33] J. Duchon, R. Robert. E´volution d’une interface par capillarit´e et diffusion de volume. I. Existence locale en temps. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 1, no. 5, 361-378, 1984. [34] J. Escher and B.V. Matioc. On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results. Z. Anal. Anwend. 30, no. 2, 193-218, 2011. [35] J. Escher and G. Simonett. Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations, no. 2, 619-642, 1997. 16 [36] F. Gancedo and R.M. Strain. Absence of splash singularities for SQG sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci., 111, no. 2, 635-639, 2014. [37] K.A. Gillow and S.D. Howison. A bibliography of free and moving boundary problems for Hele-Shaw and Stokes flow. http://people.maths.ox.ac.uk/howison/Hele-Shaw/. Accessed August 19, 2015. [38] J. G´omez-Serrano and R. Granero-Belinch´on. On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity, 27(6):1471–1498, 2014. [39] R. Granero-Belinch´on. Global existence for the confined Muskat problem. SIAM J. Math. Anal., 46(2):1651–1680, 2014. [40] Y. Guo, C. Hallstrom and D. Spirn. Dynamics near unstable, interfacial fluids. Comm. Math. Phys. 270, no. 3, 635-689, 2007. [41] Y. Guo and I. Tice. Decay of viscous surface waves without surface tension. ArXiv:1011.5179, 2011. [42] T.Y. Hou, J.S. Lowengrub and M.J. Shelley. Removing the Stiffness from Interfacial Flows with Surface Tension. J. Comput. Phys., 114: 312-338, 1994. [43] H. Kn¨upfer and N. Masmoudi. Darcys flow with prescribed contact angle: Well-posedness and lubrication approximation. Arch. Ration. Mech. Anal., pages 1–58, 2015. [44] D. Lannes. The water waves problem: Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, 188. American Mathematical Society, 2013. [45] M. Muskat. Two Fluid systems in porous media. The encroachment of water into an oil sand. Physics, 5, 250-264, 1934. [46] F. Otto. Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. 52, 873-915, 1999. [47] P.G. Saffman and G. Taylor. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A 245, 312-329, 1958. [48] M. Siegel, R. Caflisch and S. Howison. Global Existence, Singular Solutions, and Ill-Posedness for the Muskat Problem. Comm. Pure and Appl. Math., 57, 1374-1411, 2004. [49] S. Wu. Recent progress in mathematical analysis of vortex sheets. Proceedings of the International Congress of Mathematicians, Vol. III, 233-242, Higher Ed. Press, Beijing, 2002. 17 [50] L. Jr. Sz´ekelyhidi. Relaxation of the incompressible porous media equation. Ann. Sci. ´ Ec. Norm. Sup´er. (4) 45, no. 3, 491-509, 2012. [51] F. Yi. Local classical solution of Muskat free boundary problem. J. Partial Diff. Eqs., 9, 84-96, 1996. Francisco Gancedo Departamento de An´alisis Matem´atico & IMUS Universidad de Sevilla C/ Tarfia s/n, Campus Reina Mercedes 41012 Sevilla, Spain Email: [email protected] 18