Year: 2026
Type: article
Abstract: We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and in the absence of cross diffusion, the flow is exponentially contractive towards a compactly supported steady state. Our main result is that for small cross diffusion, the system still converges exponentially, at a slightly lower rate, to a deformed but still compactly supported steady state. Our approach relies on the interpretation of the PDE system as a gradient flow in a two-component Wasserstein metric. The energy consists of a uniformly convex part responsible for self-diffusion and non-local aggregation, and a totally non-convex part that generates cross diffusion; the latter is scaled by a coupling parameter $$\varepsilon >0$$ ε > 0 . The core idea of the proof is to perform an $$\varepsilon $$ ε -dependent modification of the convex/non-convex splitting and establish a control on the non-convex terms by the convex ones.
Authors: Daniel Matthes, Christian Parsch
Institution: Technical University of Munich
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Award: G6773318802
OpenAlex ID: W7129467924