Mass splitting in the time-discrete generalized Euler equations and non-Monge solutions in multi-marginal optimal transport

TL;DR

This paper demonstrates the existence of mass-splitting solutions in time-discrete generalized Euler equations, revealing non-Monge solutions in multi-marginal optimal transport through explicit examples.

Contribution

It provides the first explicit example of mass-splitting in one-dimensional generalized Euler equations and introduces a simple fully discrete example illustrating the underlying mechanism.

Findings

Mass-splitting solutions exist in one-dimensional generalized Euler equations.

Explicit examples demonstrate non-Monge solutions in multi-marginal optimal transport.

A transparent mechanism for mass-splitting is identified.

Abstract

The time-discretized, spatially continuous generalized Euler equations are a prototype example of multi-marginal optimal transport, yet the question whether they exhibit mass-splitting (or equivalently, whether they have solutions that are not of Monge form) has remained open. Here we resolve this question by giving a mass-splitting example in one spatial dimension. Moreover we present a related and very simple fully discrete example of mass-splitting which reveals a transparent underlying mechanism.