Elliptic and parabolic equations in fractured media

Abstract

The elliptic and the parabolic equations with Dirichlet boundary conditions in fractured media are considered. The fractured media consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let epsilon is an element of (0, 1] denote the size ratio of the matrix blocks to the whole domain and let omega(2) is an element of (0, 1] denote the permeability ratio of the disconnected subset to the connected sub-region. It is proved that the W-1,W-P norm of the elliptic and the parabolic solutions in the high permeability sub-region are bounded uniformly in omega, epsilon. However, the W-1,W-P norm of the solutions in the low permeability subset may not be bounded uniformly in omega, epsilon. For the elliptic and the parabolic equations in periodic perforated domains, it is also shown that the W-1,W-P norm of their solutions are bounded uniformly in epsilon. (C) 2015 Elsevier Inc. All rights reserved.