Segregated solutions of a degenerate cross-diffusion system with drifts

Abstract

We prove the global existence of segregated weak solutions of a one-dimensional degenerate cross-diffusion system with independent drifts, which is endowed with a Wasserstein gradient flow structure. We argue by a Lagrangian formulation written in terms of the (pseudo-)inverse for the cumulative mass function of the sum of the species, which we solve by a Minimising Movement Scheme in the setting of $L^2 \cap BV_{\mathrm{loc}}$. This Lagrangian problem gives rise to a parabolic PDE similar to a $p$-Laplace equation, with typical range $p \in (-\infty,1)$. We employ monotonicity methods \emph{à la} Minty--Browder to obtain strong convergence and pass to the limit $τ\to 0$ in the time-step of the discrete scheme. Our contribution simultaneously treats all porous medium degeneracies, the log-entropy, and fast diffusions of index $α\in (\frac{1}{3},1)$.