Existence of Weak Solutions to a Constrained Aggregation-Diffusion-Reaction Model for Multiple Sclerosis
S. Fagioli, M. Kamath Katapady
TL;DR
This work proves the existence of weak solutions for a constrained nonlocal PDE modeling multiple sclerosis, coupling macrophage density $\rho$ with a constraint density $\beta$ via nonlocal chemotaxis governed by a kernel $K$ and saturation $h$. The authors employ a variational approach: first, a transport step implemented through a JKO-type minimizing movement scheme for the energy $\mathscr{F}[\rho|\nu]=\mathscr{A}[\rho]+\mathscr{K}[\rho]+\mathscr{S}[\rho|\nu]$ to handle aggregation-diffusion under the constraint, and then a splitting step to incorporate the reaction term $M(\rho)$, yielding a full transport-reaction scheme. A flow-interchange argument using the entropy $\mathscr{H}$ provides necessary compactness, yielding strong convergence for $\rho^{\gamma/2}$ and enabling passage to the limit in the nonlinear and nonlocal terms; the constraint yields $h(\rho)(1-\beta)=0$ and $\beta$ converges to a limit describing oligodendrocyte dynamics. The main result ensures the existence of a pair $(\rho,\beta)$ on $[0,T]$ with $\rho_0\in L^1\cap L^\gamma$, $\gamma>\max\{1,(d-2)/2\}$, that satisfies the weak formulation of the full constrained system.
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.