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“Cauchy's almost forgotten Lagrangian
formulation of the Euler equation
for 3D incompressible flow and modern
perspectives “
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Thursday, 12. May 2016 |
Audio-only-Recording as MP3-File
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15:15 – 16:15 |
Speaker : Uriel Frisch (Observatoire de la Cote d'Azur)
Abstract :
Two prized papers, one by Augustin Cauchy in 1815,
presented to the
and the other by Hermann Hankel in 1861, presented to
major discoveries on vorticity dynamics, whose impact
in fluid dynamics and cosmology
is only beginning to be felt.
Cauchy found a Lagrangian formulation of 3D ideal
incompressible flow in terms of three
invariants, which generalizes to three dimensions the
law of conservation of vorticity along
fluid particle trajectories for two-dimensional flow.
The very concept of invariant, one
hundred years before the work of Emmy Noether relating
invariants and continuous
symmetries, was foreign to early 19th century
thinking. This probably explains why
Cauchy's invariants were quickly forgotten by almost
everybody.
Actually, only a corollary of the invariants, called
the Cauchy vorticity formula,
remained well-known.
Hankel showed that the Cauchy invariants formulation
gives a very simple Lagrangian
derivation of the Helmholtz vorticity-flux invariants
and, in the middle of the proof,
derived an intermediate result, which is actually the
conservation of the circulation of the
velocity around a closed contour moving with the
fluid. This circulation theorem was
to be rediscovered independently by William Thomson
(Kelvin) in 1869.
Cauchy's invariants were only occasionally cited in
the 19th century - besides Hankel,
foremost by George Stokes and Maurice Levy - and even
less so in the 20th until they
were rediscovered via Noether's theorem in the late
1960, but reattributed to Cauchy
only at the end of the 20th century by Russian
scientists.
Actually, Cauchy's Lagrangian formulation, combined
with a technique of recursion
relations for time-Taylor coefficients, first used by
cosmologists in the 1990s, gives a
powerful tool for tackling issues of time-analyticity
of fluid particle trajectories and of
numerical integration in Lagrangian coordinates. Such
techniques apply not only to the
incompressible Euler equations (with or without
boundaries) but to other equations
possessing similar invariants. This includes the
Euler-Poisson equation for
self-gravitating dark matter in an Einstein-de Sitter
or Lambda CDM Universe.
REFERENCES:
* Frisch, U.
and Villone, B. 2014. Cauchy's almost forgotten Lagrangian formulation of the
Euler equation
for 3D
incompressible flow, Europ. Phys. J. H 39, 325--351. arXiv:1402.4957 [math.HO]
* Frisch, U.
and Zheligovsky, V. 2014. A very smooth ride in a rough sea, Commun. Math.
Phys., 326,
499--505.
arXiv:1212.4333 [math.AP]
* Zheligovsky,
V. and Frisch, U. 2014. Time-analyticity of Lagrangian particle trajectories in
ideal fluid flow,
J. Fluid Mech., 749, 404--430.
arXiv:1312.6320 [math.AP]
* Rampf, C.,
Villone, B and Frisch, U. 2015. How smooth are particle trajectories in a
Lambda-CDM
Universe?,
Mon. Not. R. Astron. Soc., 452}, 1421--1436. arXiv:1504.00032 [astro-ph.CO]
* Podvigina,
O., Zheligovsky, V. and Frisch, U. 2015. The Cauchy-Lagrangian method for
numerical analysis
of Euler
flow. J. Comput. Phys., 306, 320--342. arXiv:1504.05030v1 [math.NA]
·
Besse, N. and Frisch, U. A constructive
approach to regularity of Lagrangian trajectories for incompressible
·
Euler flow in a bounded domain, submitted.
arXiv:1603.09219v1 [math.AP].