Existence of Weak Solutions to a Constrained Aggregation-Diffusion-Reaction Model for Multiple Sclerosis

Abstract

We establish an existence result for weak solutions to an aggregation-diffusion-reaction equation with a constraint, arising in the modelling of multiple sclerosis. The model is derived from a general chemotaxis-type framework and describes the time evolution of the density of activated macrophages, which is subject to attraction by oligodendrocytes. The latter are governed by a constraint equation. The proof relies on a variational splitting scheme that isolates the transport (aggregation-diffusion) and reaction contributions. The structure of the constraint makes it possible to recover the oligodendrocyte density as the limit of a sequence of characteristic functions.