Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity
Markus Gahn, Maria Neuss‐Radu, Peter Knabner · Discrete and Continuous Dynamical Systems - S · 2017
We consider a system of non-linear reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order $\epsilon$, and the equations inside the layer depend on the parameter $\epsilon$ and an additional parameter $\gamma \in [-1,1)$, which describes the size of the diffusion in the layer. We derive effective models for the limit $\epsilon \to 0 $, when the layer reduces to an interface $\Sigma$ between the two bulk domains. The effective solution is continuous across $\Sigma$ for all $\gamma \in [-1,1)$. For $\gamma \in (-1,1)$, the jump in the normal flux is given by a non-linear ordinary differential equation on $\Sigma$. In the critical case $\gamma = -1$, a dynamic transmission condition of Wentzell-type arises at the interface $\Sigma$.