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.

62 pages. Revised version, accepted in Bull. AMS

References in corpus (3)

• Scaling Limit, Noise, Stability
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
• Hopf-Cole transformation via generalized Schrödinger bridge problem

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