Steady fluid flows and symplectic geometry. (English) Zbl 0805.58023
Summary: We show that for an even-dimensional fluid there exists a strong relation, via the Morse theory and symplectic geometry, between the topology of the vorticity function and the existence of a stationary solution of Euler’s equation. In particular, it turns out that there is no smooth steady flow on the disc provided that the vorticity function is Morse, positive and has both a local maximum and minimum in the interior of the disc.
As we show, the structure of four-dimensional steady flows is similar to that of three-dimensional flows described in Arnold’s theorem. Namely, under certain hypotheses, an analytic four-dimensional steady flow is fibered into invariant tori and annuli, for such a flow gives rise to an integrable Hamiltonian system on a symplectic four-manifold. It is also proved that an odd-dimensional chaotic steady flow is always a Beltrami flow, i.e., its velocity and vorticity fields are proportional.
As we show, the structure of four-dimensional steady flows is similar to that of three-dimensional flows described in Arnold’s theorem. Namely, under certain hypotheses, an analytic four-dimensional steady flow is fibered into invariant tori and annuli, for such a flow gives rise to an integrable Hamiltonian system on a symplectic four-manifold. It is also proved that an odd-dimensional chaotic steady flow is always a Beltrami flow, i.e., its velocity and vorticity fields are proportional.
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 76B47 | Vortex flows for incompressible inviscid fluids |
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