Convergence to equilibrium for cross diffusion systems with nonlocal interaction

TL;DR

This paper investigates how solutions to a two-component non-linear diffusion-aggregation system with small cross diffusion effects tend to equilibrium, demonstrating convergence at a slightly reduced rate using a gradient flow approach in Wasserstein space.

Contribution

It extends the understanding of convergence to equilibrium for nonlocal interaction systems by analyzing the impact of small cross diffusion through a gradient flow framework.

Findings

Solutions converge to a steady state at a slightly lower rate with small cross diffusion.

The system remains compactly supported and deformed steady states are achieved.

The gradient flow approach effectively controls non-convex effects in the PDE system.

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, 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 $\varepsilon>0$. The core idea of the proof is to perform an $\varepsilon$-dependent modification of the convex/non-convex splitting and establish a control on the non-convex terms by the convex ones.