Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains

Abstract

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation-fragmentation-diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng-Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.

A toy model of misfolded protein aggregation and neural damage propagation in neurodegenerative diseases

“…They later move into other cells, where they act as seeds and induce the self-association among monomers and polymers of different sizes for describing the elongation of fibrils by end-to-end formation [5][6][7][8]. Further models include the role of prions [9], the growth kinetics of amyloids [10][11][12], and the use of network-approaches to understanding the behaviour of different brain regions [13,14]. At the macroscopic level, the spreading of neural damages is typically modelled through a nonlinear reaction-diffusion mechanism [15,16], that can be effectively coupled with nucleation-aggregation-fragmentation models for the dynamics in the brain connectome [17,18].…”

Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

“…In [5], a thermal transfer was considered in a two-phase domain with an imperfect interface, where both the temperature and the flux are discontinuous across the interface. Coupled multi-component reaction-diffusion systems were treated with respect to non-linear reaction terms over the domain in [10] and examined for degenerate asymptotic behaviour in [32] using the two-scale convergence. Finally, in [11], a non-linear transmission condition was treated with respect to the homogenisation procedure with the help of Minty's argument.…”