Preprint

Inserted: 15 jun 2026
Last Updated: 15 jun 2026

Year: 2026

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 $L^2 \cap BV_{loc}$. 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 à la Minty-Browder to obtain strong convergence and pass to the limit $\tau \to 0$ in the time-step of the discrete scheme. Our contribution simultaneously treats all porous medium degeneracies, the log-entropy, and fast diffusions of index $\alpha \in (\frac{1}{3},1)$, thereby complementing the recent results by Mészáros and Parker, Elbar and Santambrogio, Skrzeczkowski, and the prior work by Kim and Mészáros.

Tags: EYAWKAJKOS

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