Related papers: Porous media: the Muskat problem in 3D
We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip $\RR\times[-l,l]$. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The…
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…
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 consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous to the Hele-Shaw cell [26]. We establish, for a class of…
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,…
This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top…
We study the Rayleigh-Taylor problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows, with zero resistivity, surface tension (or without surface tenstion) and special initial magnetic field, evolving with a free…
For the water waves equations, the existence of splat singularities has been shown in [3], i.e., the interface self-intersects along an arc in finite time. The aim of this paper is to show the absence of splat singularities for the…
We consider the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. After constructing Rayleigh-Taylor steady-state solutions…
We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across…
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 work we study the inhomogeneous Muskat problem, \emph{i.e.} the evolution of an internal wave between two different fluids in a porous medium with discontinuous permeability. In particular, under precise conditions on the initial…
We show the solvability of a multidimensional Muskat type initial boundary value problem. The proposed mathematical model describing the transport phenomena of non-homogeneous flow in porous media, relies on a generalized formulation of the…
This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant…
We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface…
Rayleigh-Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present linearized theory for arbitrary 3D initial disturbances that grow in time, and…
We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface…
We investigate the Couette-Taylor problem for a steady incompressible viscous fluid in a 3D cylindrical annulus, where one of the two cylinders is still, under both Dirichlet and boundary conditions involving the vorticity that naturally…
We review some recent results on the Muskat problem modelling multiphase flow in porous media. Furthermore, we prove a new regularity criterion in terms of some norms of the initial data in critical spaces ($\dot{W}^{1,\infty}$ and…
In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb{R})$, where ${p\in(1,2]}$ and ${s\in(1+1/p,2)}$. This is achieved by showing…