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

Gero Friesecke

TL;DR

The paper resolves whether the time-discretized, generalized Euler equations admit mass-splitting by constructing explicit mass-splitting examples in one dimension, both in continuous space and in a simple fully discrete setting. It frames the problem within multi-marginal optimal transport, clarifying Monge versus Kantorovich formulations and demonstrating a transparent mechanism that favors splitting under discretization. The results extend the understanding of when non-Monge optimal plans arise in MMOT problems related to fluid dynamics and illustrate how discretization choices influence whether mass splitting occurs. This has implications for modeling incompressible flows, numerical schemes, and the interpretation of Brenier's relaxation relative to classical Euler dynamics.

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.

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

TL;DR

The paper resolves whether the time-discretized, generalized Euler equations admit mass-splitting by constructing explicit mass-splitting examples in one dimension, both in continuous space and in a simple fully discrete setting. It frames the problem within multi-marginal optimal transport, clarifying Monge versus Kantorovich formulations and demonstrating a transparent mechanism that favors splitting under discretization. The results extend the understanding of when non-Monge optimal plans arise in MMOT problems related to fluid dynamics and illustrate how discretization choices influence whether mass splitting occurs. This has implications for modeling incompressible flows, numerical schemes, and the interpretation of Brenier's relaxation relative to classical Euler dynamics.

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.
Paper Structure (13 sections, 5 theorems, 98 equations, 5 figures)

This paper contains 13 sections, 5 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Different formulations of the Euler equations
  3. Euler's formulation
  4. Arnold's formulation
  5. Brenier's formulation
  6. Time-discretization and interpretation as optimal transport
  7. Mass-splitting in optimal transport
  8. Terminology; Monge versus Kantorovich
  9. Previous results
  10. Why does the mass want to split in the generalized Euler equations? A simple mechanism
  11. Refining the spatial mesh
  12. Example of mass-splitting on continuous space
  13. What is physically more correct, Euler (no mass splitting) or Brenier (mass splitting)?

Key Result

Lemma 2.1

For any closed and bounded subset $\Omega\subset{\mathbb R}^d$ and any $N\ge 2$, a probability measure $\gamma\in{\mathcal{P}}(\Omega^{N+1})$ solves MMOT'1--MMOT'2 if and only if its marginal $(\pi_0,...,\pi_{N-1})_\sharp \gamma\in{\mathcal{P}}(\Omega^N)$ with respect to the first $N$ copies of $\Om

Figures (5)

  • Figure 1: Left: each fluid particle follows a single path $\omega$ (Arnold's variational principle). Right: each fluid particle can follow multiple paths (Brenier's variational principle).
  • Figure 2: Solution to the fully discrete generalized Euler equations \ref{['discbren1']}--\ref{['discbren2']}
  • Figure 3: Numerical solutions of the generalized Euler equations on a space-time mesh. The thickness of the paths indicates the amount of mass transported.
  • Figure 4: Mass-splitting solution of the time-discrete generalized Euler equations \ref{["MMOT'1"]}--\ref{["MMOT'2"]}. Each fluid particle moves from its initial position to its end position via two paths.
  • Figure 5: The building blocks $\gamma_1$ (top) and $\gamma_2$ (bottom) of the optimal plan from Fig. \ref{['F:cts_example']}. The top plan can be described as "expand and mix, contract and de-mix, flip each block". The bottom plan corresponds to "flip each block, expand and mix, contract and de-mix".

Theorems & Definitions (5)

  • Lemma 2.1
  • Theorem
  • Proposition 4.1
  • Theorem 6.1
  • Lemma 6.2