What question did this study set out to answer?

The aim is to prove the global existence of segregated weak solutions for a specific one-dimensional degenerate system.

June 20, 2026Open Access

Segregated solutions of a degenerate cross-diffusion system with drifts

Key Points

The aim is to prove the global existence of segregated weak solutions for a specific one-dimensional degenerate system.
Utilized a Lagrangian formulation with the pseudo-inverse of the cumulative mass function.
Applied a Minimising Movement Scheme in the setting of L^2 ∩ BV_loc.
Employed monotonicity methods for obtaining strong convergence.
Demonstrated strong convergence as τ approaches 0 in the discrete scheme.
Established a parabolic PDE similar to a p-Laplace equation with p in (-∞, 1).
Simultaneously addressed all porous medium degeneracies and log-entropy conditions.

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 L² BV₋₎₂. 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 (13, 1).

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