Original Title: “Global Existence of QuasiStratified Solutions for the Confined IPM Equation”
Authors: Ángel Castro (ICMATCSIC), Diego Córdoba (ICMATCSIC) y Daniel Lear (ICMATCSIC)
Source: Archive For Rational Mechanics and Analysis 232 (2019), 437471
Date of online publication: 2019
The incompressible porous media (IPM) equation models the movement of incompressible fluids, such as water, through a medium that contains them and hampers their flow, as does sand, for example. The variables used to study this movement are the velocity of the flow, its density and its pressure. The resulting system is composed of three equations that express the principle of mass conservation, the incompressibility condition, and Darcy’s law, which states that in a porous medium the velocity of the fluid is proportional to the forces that act upon it. In this paper, the authors consider that the fluid is only subject to the forces of pressure and gravity, so that the three equation system can be reduced to a single transport equation for the density, the solutions of which determine what the density, velocity and pressure will be, depending on the current density at the time.
The paper addresses the problem of the global regularity of the solutions in a two-dimensional bounded medium. This medium consists of a periodic cell in the horizontal variable, a flat ceiling and a flat floor. It is shown that for PIM there exist regular solutions locally in time; that is, if the initial density is weak, then a weak solution exists during finite time. However, little is known about the global existence of solutions; in other words, it is not known whether it is true that for all the initial density there is a weak solution for all time, or if, on the other hand, an initial weak density exists such that the solution ceases to be weak in finite time. For example, if it can be demonstrated that stationary solutions exist that arise from densities that depend only on the vertical variable, and, considering the case when the fluid lies on all the plane, where Tarek Elgindi, demonstrated that small perturbations of the stationary equation, which takes the form d(y)=y, yield global solutions that converge in time to the stationary solution.
The result obtained by Ángel Castro, Diego Córdoba and Daniel Lear proves the global existence of solutions for small perturbations of d(y)=y in a bounded domain. To this end, they construct spaces adapted to the medium and invariant due to the flow of velocity, which include the boundary conditions that the initial density must satisfy in order to yield global solutions. The proof is based on a Galerkin method, energy estimates in these spaces, and on the Duchamel formula, which enables these researchers to take advantage of a dissipation hidden in the linear part of the equation.