Least action principles for incompressible flows and geodesics between shapes. (English) Zbl 1428.35376
Summary: As V. I. Arnol’d observed in the 1960s [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)], the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: any two shapes of equal volume can be approximately connected by an Euler spray – a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 35J96 | Monge-Ampère equations |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 53C22 | Geodesics in global differential geometry |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 86A05 | Hydrology, hydrography, oceanography |
| 35R35 | Free boundary problems for PDEs |
Citations:
Zbl 0148.45301Software:
EMDReferences:
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