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Fluid Mixing and Incompressible Optimal Transport — arXiv:2610.09235

Source

arXiv (en)

Original

https://arxiv.org/abs/2610.09235

License (as archived)

CC0-1.0

Archived

2026-10-08T12:40:26.326Z

SHA-256 of the archived text

9a5fa6f5250198917fde9668a2f3cfddd4c1b4cd906c10e5744328dd7a2e0f52

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How to cite

Fluid Mixing and Incompressible Optimal Transport — arXiv:2610.09235 — arXiv (en), https://arxiv.org/abs/2610.09235 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_d54634891069 (Metraipolis archive dc_d54634891069, sha256 9a5fa6f5250198917fde9668a2f3cfddd4c1b4cd906c10e5744328dd7a2e0f52)

@misc{dcd54634891069,
  title = {Fluid Mixing and Incompressible Optimal Transport — arXiv:2610.09235},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.09235},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_d54634891069; sha256 9a5fa6f5250198917fde9668a2f3cfddd4c1b4cd906c10e5744328dd7a2e0f52},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.09235. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The problem of incompressible fluid mixing is long-studied, yet many questions remain open. This paper aims to address the question ``what do efficient flow fields for mixing look like, and how do they behave?'' Inspired by the dynamic and geometric approach to optimal mass transport, we formulate a version of the fluid mixing problem as an optimal control problem for transport of a passive scalar, in which the dynamics are given by the continuity equation together with an incompressibility constraint. We then develop a geometric framework around this problem which parallels that of classical optimal transport: the optimal control problem induces a metric on the set of reachable states (analogous to the Wasserstein metric), and this metric carries a formal Riemannian structure when the optimal control objective is induced by an inner product. Using this structure, we decompose our problem into a series of more fundamental geometric problems and derive (formal) necessary conditions for optimality, geodesic equations, and gradient flows for mixing. Specializing the control objective to the kinetic energy recovers the incompressible Euler equation, connecting our framework to the geometric hydrodynamics of Arnold. The resulting problem appears to be ill-posed, however, suggesting that the kinetic energy is an unnatural measure of mixing effort. Specializing instead to the enstrophy, which we argue is better suited to mixing, yields the Euler-Poincaré (EPDiff) geodesic equation for the right-invariant homogeneous $\dot{H}^1$ metric on the group of volume-preserving diffeomorphisms, and we present numerical simulations of both geodesics and gradient flows in this case. Along the way, we generalize the classical Helmholtz decomposition of vector fields, rederive the Clebsch representation of incompressible flows, and develop a parallel to the Otto calculus of optimal transport.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.OC, nuovi lavori del giorno).
Categorie arXiv: Optimization and Control (math.OC) ; Systems and Control (eess.SY); Fluid Dynamics (physics.flu-dyn).
Pagina del lavoro: https://arxiv.org/abs/2610.09235 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T12:16:00.175Z dalla pagina https://arxiv.org/list/math.OC/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).