A cross-diffusion system describing ion transport through biological membranes or nanopores in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. The ion concentrations solve strongly coupled diffusion equations with a drift term involving the electric potential which is coupled to the concentrations through a Poisson equation. The global-in-time existence of bounded weak solutions and the uniqueness of weak solutions under moderate regularity assumptions are shown. The main difficulties of the analysis are the cross-diffusion terms and the degeneracy of the diffusion matrix, preventing the use of standard tools. The proofs are based on the boundedness-by-entropy method, extended to nonhomogeneous boundary conditions, and the uniqueness technique of Gajewski. A finite-volume discretization in one space dimension illustrates the large-time behavior of the numerical solutions and shows that the equilibration rates may be very small. (C) 2018 Elsevier Inc. All rights reserved.

In this paper, we study the quantization dimension of a random self-similar measure mu supported on the random self-similar set K(omega). We establish a relationship between the quantization dimension of mu and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension. (C) 2009 Elsevier Inc. All rights reserved.

This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2. (C) 2007 Elsevier Inc. All rights reserved.

It is the purpose of this short note to discuss some aspects of the validity question concerning the Korteweg-de Vries (KdV) approximation for periodic media. For a homogeneous model possessing the same resonance structure as it arises in periodic media we prove the validity of the KdV approximation with the help of energy estimates. (C) 2011 Elsevier Inc. All rights reserved.