University of Heidelberg, Applied Analysis Seminar
Thu 04.02.2021
14:15, posted in Applied Analysis
Dr. Markus Gahn (Heidelberg University)
Homogenization of reactive transport through evolving micro-domains
Online via HeiCONF

In this talk, we consider a reaction-diffusion equation in an
evolving micro-domain with a heterogeneous structure. The evolution of
the micro-domain depends on the solution of the problem, leading to a
free boundary value problem on the micro-scale. The evolving porous
medium includes periodically distributed spherical solid grains. Their
periodicity and their size is of order $\epsilon$, where the parameter
$\epsilon$ is small compared to the size of the whole domain. The radius
of every micro-grain depends on the concentration at its surface,
leading to a nonlinear problem. The aim is to pass to the limit
$\epsilon \to 0$ and rigorously derive a macroscopic model, the solution
of which approximates the solution of the microscopic model.
In a first step we transform the problem on the evolving micro-domain
to a problem on a fixed periodically perforated domain by using the
Hanzawa-transformation. This leads to a change in the coefficients of
the equations, which now depend on the radius of the grains and
therefore on the unknown concentration. We prove existence using the
Rothe-method and derive a priori estimates for the solutions uniformly
with respect to the parameter $\epsilon$. For the derivation of the
macroscopic model in the limit $\epsilon \to 0$ we use the method of
two-scale convergence, where we need strong compactness results to pass
to the limit in the nonlinear terms.

<< Back