分野別セミナー

日時：2011年3月9日 (水) 10:00-12:00
　　　　　　3月10日 (木) 10:00-12:00

This mini-course addresses to graduate students and young
researchers in mathematics interested in applying both formal
and rigorous averaging methods to real-life problems described
by means of partial differential equations (PDEs). As
background application scenario we choose reaction, diffusion
and flow in porous media, but broadly speaking, a similar
procedure would apply if one would consider wave propagation
phenomena, for instance, in composite materials or in
electromagnetism.
We start off with the study of oscillatory elliptic PDEs
formulated firstly in fixed and, afterwards, in periodically-
perforated domains. We remove the oscillations by means of
a (formal) asymptotic homogenization method. The output of
this procedure consists of "guessed" averaged model equations
and explicit rules [based on cell problems] for computing the
effective coefficients.
As second step, we introduce the concept of two-scale convergence
(and correspondingly, two-scale compactness) in the sense of
Allaire and Nguetseng and derive rigorously the averaged PDE
models and coefficients obtained previously. This step relies
on the framework of Sobolev and Bochner spaces and uses basic
tools like weak convergence methods, compact embeddings as well
as extension theorems in Sobolev spaces. We particularly
emphasize the role the choice of microstructures (pores,
perforations, etc.) plays in performing the overall averaging
procedure.
The main objective of the course is to endow the participant
with this "mathematical homogenization" tool so that he/she
can apply it successfully further to other scenarios involving
physicochemical processes taking place within periodic arrays
of microstructures.