Segregated solutions of a degenerate cross-diffusion system with drifts

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 L² ∩ BVloc. This Lagrangian problem gives rise to a parabolic PDE similar to a p-Laplace equation, with typical range p ∈ (-∞,1). We employ monotonicity methods à la Minty--Browder to obtain strong convergence and pass to the limit τ→ 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∈ (1/3,1).