| Abstract: |
| Arnold made the celebrated observation that solutions to the incompressible Euler equations of fluid dynamics correspond to geodesics in the group of volume-preserving diffeomorphisms. A nontrivial fact is that minimizers of the corresponding variational principle may not exist. Brenier introduced a relaxation which he showed to be well-posed. Physically this formulation allows mass splitting, i.e., a fluid particle can move from A to B via an ensemble of trajectories.
Mathematically this formulation is an instance of multi-marginal optimal transport; for a simple introduction to this formulation see my recent textbook on optimal transport [1]. After revieving the different formulations of the Euler equations, we
- show that mass splitting still occurs after discretizing Brenier`s relaxation either in time or in both space and time
- provide a new argument for the mass splitting which is much simpler than previous analyses of the continuous case and reveals a transparent underlying mechanism.
We close with a brief discussion from a modeling point of view: what is physically more correct, Euler (no mass splitting) or Brenier (mass splitting)?
[1] G. Friesecke, Optimal Transport: a Comprehensive Introduction to Modeling, Analysis, Simulation, Applications, SIAM, 2025 |
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