On a Cross-Diffusion System with Independent Drifts and no Self-Diffusion: The Existence of Totally Mixed Solutions
Alpár R. Mészáros, Guy Parker
TL;DR
This work proves global existence of weak solutions for a two-species cross-diffusion system on the 1D torus with diffusion acting on the sum through a logarithmic pressure and species-specific drifts, in the absence of self-diffusion. A key idea is the nonlinear change of variables $(\sigma, f(r))$ with $f(r)=\log\left(\frac{r}{1-r}\right)$, which exposes a BV-friendly transport structure; the authors implement a vanishing-viscosity regularisation and use Aubin–Lions–Simon compactness to pass to the limit, obtaining BV control and strong convergence for the individual densities. They show propagation of regularity and the necessary convergence to a weak solution, under a total-mixing initial condition that allows vacuum and does not enforce segregation. The results extend naturally to cross-diffusion-reaction systems, offering a robust framework for existence with mixed densities and differing drifts in 1D, and they highlight a gradient-flow perspective in the energy framework.
Abstract
We establish the global existence of weak solutions for a two-species cross-diffusion system, set on the 1-dimensional flat torus, in which the evolution of each species is governed by two mechanisms. The first of these is a diffusion which acts only on the sum of the species with a logarithmic pressure law, and the second of these is a drift term, which can differ between the two species. Our main results hold under a total mixing assumption on the initial data. This assumption, which allows the presence of vacuum, requires specific regularity properties for the ratio of the initial densities of the two species. Moreover, these regularity properties are shown to be propagated over time. In proving the main existence result, we also establish the spatial BV regularity of solutions. In addition, our main results naturally extend to similar systems involving reaction terms.