We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically analogous to the two-phase Hele-Shaw cell. We focus on a fluid interface given by a jump of densities, being the equation of the evolution obtained using Darcy's law. We prove local well-posedness when the smaller density is above (stable case) and in the unstable case we show ill-posedness.

The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the RayleighTaylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L 2 (R) maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy f 0 L ∞ < ∞ and ∂ x f 0 L ∞ < 1. We take advantage of the fact that the bound ∂ x f 0 L ∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance f 1 ≤ 1/5. Previous results of this sort used a small constant ǫ ≪ 1 which was not explicit [5,16,7,12].

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field u related to the scalar θ by u = ∇⊥Λ β−2 θ, where 1 < β ≤ 2 and Λ = (−∆)1/2 is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with u = ∇⊥(log(I − ∆))µθ for µ > 0, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani.National Research Foundation of KoreaNational Science FoundationMinisterio de Ciencia e InnovaciónEuropean Research Counci