arXiv subjects

Guy Parker

Publications and source records attributed to Guy Parker.

4 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2026

Existence Theory for a Cross-Diffusion System with Independent Drifts: Mixing Dynamics

We consider a cross-diffusion system for which the diffusion of each species is governed solely by the aggregate density through a pressure law of logarithmic or fast diffusion type. The model is set over a one dimensional bounded interval, equipped with no-flux boundary conditions, and accommodates for the presence of potential drifts which are allowed to differ across each species. We establish the global existence of solutions without having to assume the total mixing of solutions. As a consequence, we give a full resolution of the PDE systems recently studied by the authors and by Elbar--Santambrogio, by allowing a general class of initial data with finite bounded variation, with no further structural assumptions on their supports.

math.AP↗

arXiv2025

On a Cross-Diffusion System with Independent Drifts and no Self-Diffusion: The Existence of Totally Mixed Solutions

We establish the global existence of weak solutions for a two-species cross-diffusion system, set on the 1-dimensional flat torus, in which the evolution of each species is governed by two mechanisms. The first of these is a diffusion which acts only on the sum of the species with a logarithmic pressure law, and the second of these is a drift term, which can differ between the two species. Our main results hold under a total mixing assumption on the initial data. This assumption, which allows the presence of vacuum, requires specific regularity properties for the ratio of the initial densities of the two species. Moreover, these regularity properties are shown to be propagated over time. In proving the main existence result, we also establish the spatial BV regularity of solutions. In addition, our main results naturally extend to similar systems involving reaction terms.

math.AP↗

arXiv2023

Some Convexity Criteria for Differentiable Functions on the 2-Wasserstein Space

We show that a differentiable function on the 2-Wasserstein space is geodesically convex if and only if it is also convex along a larger class of curves which we call `acceleration-free'. In particular, the set of acceleration-free curves includes all generalised geodesics. We also show that geodesic convexity can be characterised through first and second-order inequalities involving the Wasserstein gradient and the Wasserstein Hessian. Subsequently, such inequalities also characterise convexity along acceleration-free curves.

math.FA↗