Related papers: Localized mixing zone for Muskat bubbles and turne…
In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb{R}^2$ or a bounded strip…
We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non…
We consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by…
The Muskat problem involves filtration of two incompressible fluids throughout a porous medium. In this paper we shall discuss in 3-D the relevance of the Rayleigh-Taylor condition, and the topology of the initial interface, in order to…
We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each…
We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with $C^{3,\alpha}$-regularity of the interface in the unstable regime and for all…
In this work we study the evolution of the interface between two different fluids in a porous media with two different permeabilities. We prove local existence in Sobolev spaces, when the free boundary is given by the discontinuity among…
We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable…
The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where…
It was shown recently by Cordoba, Faraco and Gancedo that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework, uses ideas from the theory of laminates, in…
We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for…
The inhomogeneous Muskat problem models the dynamics of an interface between two fluids of differing characteristics inside a non-uniform porous medium. We consider the case of a porous media with a permeability jump across a horizontal…
We consider the Muskat problem with surface tension for one fluid or two fluids, with or without viscosity jump, with infinite depth or Lipschitz rigid boundaries, and in arbitrary dimension $d$ of the interface. The problem is nonlocal,…
We study the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock).…
In this paper we study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with general viscosities in a vertical homogeneous porous medium under the influence of gravity. Employing Rellich type…
We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space…
We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to…
We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic…
The Muskat problem models the evolution of the interface given by two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition…
In this paper, we prove the asymptotic stability of the incompressible porous media (IPM) equation near a stable stratified density, for initial perturbations in the Sobolev space $H^k$ with any $2<k \in\mathbb{R}$. While it is known that…