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

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.