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arXiv · 2606.25781

Interfaces and non-uniqueness in a cross-diffusion system with independent drifts

Charles Elbar·Guy Parker
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Abstract

We study a one-dimensional cross-diffusion system of two populations. Their densities are diffused with a common pressure that depends on the total density, but are transported by two independent external potentials. Starting from segregated initial data with a single interface, we show that the way the two densities meet is not given by the equation alone: it depends on the notion of solution one chooses. When the drifts push the two phases towards each other at the interface, the vanishing-viscosity solution creates an overlap, whereas a segregated weak solution also exists. The two solutions are distinct, so the Cauchy problem is not well posed in the class of weak solutions. We first prove the result for an explicit stationary segregated solution, and then extend the non-uniqueness to general segregated data.

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Charles Elbar, Guy Parker. 2026-07-20. Interfaces and non-uniqueness in a cross-diffusion system with independent drifts. https://arxiv.org/abs/2606.25781

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