A cross-diffusion system with independent drifts and fast diffusion
Charles Elbar, Filippo Santambrogio
TL;DR
The paper analyzes a two-species cross-diffusion system on the one-dimensional torus with fast-diffusion type pressure $S^{α}$, allowing independent drifts $V$ and $W$ for the species. By recasting the problem in terms of the total density $S$ and the log-contrast $r=\log\rho-\log\mu$, the authors derive coupled equations that admit robust BV and gradient controls, enabling compactness. An approximation scheme with viscosity, together with a carefully designed shift $u=\\partial_x r-\\gamma y(S)$, yields uniform BV bounds and strong convergence of the densities, which allows passage to the limit in the nonlinear fluxes and proves global existence of weak solutions for $0<α≤1$ when $V,W∈C^3$ and the initial data satisfy mild entropy and BV conditions. This extends prior results from the linear diffusion case to fast diffusion, using an approach that remains robust under general diffusion laws and distinct environmental drifts. The results provide a rigorous mathematical foundation for two-species territorial dynamics with heterogeneous environments and degenerate diffusion, with potential applications in ecology and tissue modeling.
Abstract
We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a fast-diffusion law with exponent $0< α\le 1$ and different external potentials. For arbitrary non-negative $L^{1}$ initial data with bounded entropy and a mixing condition we prove the existence of global weak solutions. This extends the recent result of Mészáros, Parker from the linear diffusion ($α=1$) to the fast-diffusion.