CASA Minisymposium

In honour of Prof. Willi Jäger, Visiting Professor of the KNAW

Speakers: Willi Jäger, Andro Mikelić, Ben Schweizer

 Theme :   Multiscales

14.00 - 14.50:  Willi Jäger (University of Heidelberg)

Homogenization of a Parabolic Variational Inequality with Unilateral Constraints on the Time Derivative Periodically Distributed on the Boundary

A homogenization problem for a parabolic variational inequality is considered arising as a weak formulation of an initial-boundary value problem for a solution u to a parabolic equation in a domain, restricted by unilateral constraints on the time variation of u. The time derivative is restricted on subsets G_ε (perforations), periodically distributed with period of order ε on the boundary of the domain. That means, a problem with rapidly oscillating nonlinear boundary condition is arising. The scale limit ε tending to 0 is studied under various conditions on the scaling of G_ε, the asymptotic behaviour of the solutions to the variational inequality is analyzed and an effective limit problem is derived. The main difficulties arise in the case where the perforations are of scale ε².

This analysis is joint work with M. Neuss-Radu, T. A. Shaposhnikova.

15.00 - 15.50: Andro Mikelić (University of Lyon 1, France)

Upscaling of the differential equations modeling the reactive flow through a deformable system of biological cells

Experimental research on the physiology of living cells and tissues is providing more and more detailed information on the nano- and micro-scale and this work was motivated by questions asked by physiologists interested in perfusion and transport through tissue under varying mechanical and chemical conditions. There is an urgent demand for mathematical modeling of reactive flow and transport and its interaction with elastic cell structures.

Here we consider equations on the fine scale modeling the real processes occurring in the cells, the intercellular space and in the membranes. We are including

1.     Fluid flow in the extracellular space, diffusion, transport and reactions of substances in the fluid.

2.     Exchange of fluid and substances at the membranes. Small deformation of the structures.

3.     Diffusion and transport of fluid and substances, chemical reactions inside the cells.

4.     Changes of the structures and their mechanical properties with the flow and with the substances concentrations.

Detailed study of the model equations in the biological tissue is in the article [1].

Our goal is to obtain the upscaled system modeling reactive flow through biological tissue on the macroscopic scale, starting from a system on the cell level. Using multiscale techniques, we preform the scale limit and derive a macroscopic (effective) model system, preserving relevant information on the processes on the microscopic level. One obtains in the limit a system similar to the Biot-law in the theory of dynamic poroelasticity, however, due to the scaling resulting from the analysis of the real data, the macroscopic velocities are solving a differential equation containing only its spatial derivatives.

The interaction of fluid with solid structures has been studied in the literature in several papers and after passing to the homogenization limit the macroscopic law known as Biot-law could be derived. The system, which is analyzed here, represents a larger class of problems coupling fluid flow, solid structure and chemical reaction for slow flow velocity and their interaction with mechanics leads to new obstacles requiring new ideas and methods developed in this work.

The modeling novelty in our paper is dependence of the Young modules, of the elastic structure, on the concentration. Consequently, the cell chemistry causes the deformation. Adding diffusion, transport and reactions of chemical substances and their interaction with mechanics leads to new mathematical difficulties requiring new ideas and methods.

In addition to the formal upscaling, we will give some hints about the rigorous homogenization result.

[1]  W. Jäger, A. Mikelić, M. Neuss-Radu: Analysis of differential equations modelling the reactive flow through a deformable system of cells, Archive for Rational Mechanics and Analysis, Vol. 192, no. 2 (2009), p. 331-374.

16.00 - 16.50: Ben Schweizer (University of Dortmund)

Negative index of optical media and a theory for meta-materials

We discuss materials with a negative index for the refraction of light. We will explain some physical background, properties of negative index materials and the possibility to construct such materials. The break-through in this area of research was the design of smart "meta-materials" -- heterogeneous media composed of small structures. The length scale of the structure is so small that the medium behaves like a homogeneous block, but the effect of the small structures is visible in the effective properties of the homogeneous block. The mathematical tool for the analysis of meta-materials is the homogenization theory. We present fundamentals of this theory and an application to the three-dimensional Maxwell equations. We can present a positive result on negative index media. The analysis is based on two-scale convergence and several singular limits, a cross-link to Differential Geometry will also appear. The results were obtained together with Guy Bouchitte (Toulon).