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Geometric hydrodynamics and infinite-dimensional Newton's equations

Boris Khesin, Gerard Misiolek, Klas Modin

Abstract

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

Geometric hydrodynamics and infinite-dimensional Newton's equations

Abstract

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

Paper Structure

This paper contains 49 sections, 58 theorems, 280 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Geodesics and Newton's equations: finite-dimensional examples
  3. Three motivating examples from hydrodynamics
  4. Riemannian metrics and their geodesics on spaces of diffeomorphisms and densities
  5. First examples of Newton's equations on diffeomorphism groups
  6. Other related equations
  7. An overview and main results
  8. Wasserstein-Otto geometry
  9. Newton's equations on Diff(M)
  10. Riemannian submersion over densities
  11. Hamiltonian setting
  12. Hamiltonian framework for the incompressible Euler equations
  13. Hamiltonian formulation and Poisson reduction
  14. Newton's equations on Dens(M)
  15. Wasserstein-Otto examples
  16. ...and 34 more sections

Key Result

proposition 1

Newton's equations with respect to the $L^2$-metric eq:L2met and the potential eq:quadratic_func take the following forms:

Figures (3)

  • Figure 1: Illustration of the Riemannian submersion in \ref{['thm:otto_riemannian_metric']}. Horizontal geodesics on $\mathrm{Diff}(M)$ (potential solutions) are transversal to the fibres and project to geodesics on $\mathrm{Dens}(M)$. Note that the point in $\mathrm{Dens}(M)$ denoted by $1$ corresponds to the reference volume form $\mu$, while $\rho$ corresponds to the volume density $\varrho$.
  • Figure 2: Relation between various phase space representations of Newton's equations on $\mathrm{Diff}(M)$ and $\mathrm{Dens}(M)$.
  • Figure 3: The bottom arrow is the isometry of the density space $\mathrm{Dens}(M)$ with the Fisher-Rao metric and a part $S^\infty_+$ of the infinite-dimensional sphere, while the top arrow corresponds to the Madelung transform.

Theorems & Definitions (146)

  • remark 1
  • definition 1
  • remark 2
  • remark 3
  • remark 4
  • proposition 1
  • remark 5
  • remark 6
  • remark 7
  • remark 8
  • ...and 136 more