Weak solutions in nonlinear poroelasticity with incompressible constituents

Weak Solutions to the Stationary Incompressible Euler Equations

Multigrid method for nonlinear poroelasticity equations

In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme”...

Nonlinear Poroelasticity for a Layered Medium

The equations of motion and the nonlinear constitutive theory of fluid-filled poroelastic media are derived from the fundamental equations of elasticity and fluid mechanics for the constituents. A two-scale spatial expansion and the method of homogenization are employed. Explicit equations are obtained for the special case of a medium consisting of alternating solid and fluid layers. The linear...

Existence of Weak solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study the general case that the fluids have differen...

Global Weak Solutions for an Incompressible Charged Fluid with Multi-Scale Couplings: Initial-Boundary Value Problem

The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was investigated by the first author in [Transport Theory Statist. Phys. 31 (2002), 333–366], where a local existence-uniqueness theory was demonstrated, based upon Kato’s framework for examining evolution equations. In this article, the existence of a global weak solution is proved to hold for the model, in the case of the in...