University of Heidelberg, **Applied Analysis Seminar**

Dr. Markus Gahn (Heidelberg University)

Homogenization of reactive transport through evolving micro-domains

*Online via HeiCONF*

Homogenization of reactive transport through evolving micro-domains

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.

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