We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important partial differential equations of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schrodinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a Kahler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results.

https://www.pnas.org/content/pnas/115/24/6165.full.pdf

Geometric hydrodynamics via Madelung transform.

Incompressible limits of the Navier-Stokes equations for all time

Low Mach number limit of viscous polytropic fluid flows

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

Low Mach number limit for the non-isentropic Navier–Stokes equations

We show that the motion of a slightly compressible fluid is near that of an incompressible fluid. That is, for a given initial velocity field, the motion of a compressible fluid with large sound speed is near to that of an idealized incompressible fluid. We consider the compressible fluid motion in Lagrangian coordinates and show that it can be defined by two functions giving the kinetic and potential energies. The minimal set for the potential energy is the configuration space of incompressible fluid motion. If the equation of state of the compressible fluid is changed so that the sound speed increases, the minimum of the potential energy function becomes sharper. The compressible fluid motion approaches a curve in the minimal set and this curve defines an incompressible fluid motion.

https://www.pnas.org/content/pnas/72/2/539.full.pdf

David G. Ebin

Papers

Motion of a slightly compressible fluid.