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The movement of a solid in an incompressible perfect fluid as a geodesic flow


Authors: Olivier Glass and Franck Sueur
Journal: Proc. Amer. Math. Soc. 140 (2012), 2155-2168
MSC (2010): Primary 76B99, 74F10
DOI: https://doi.org/10.1090/S0002-9939-2011-11219-X
Published electronically: October 4, 2011
MathSciNet review: 2888201
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Abstract: The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain has recently been studied under its PDE formulation. In particular, classical solutions have been shown to exist locally in time. In this paper, following the celebrated result of Arnold concerning the case of a perfect incompressible fluid alone, we prove that these classical solutions are the geodesics of a Riemannian manifold of infinite dimension, in the sense that they are the critical points of an action, which is the integral over time of the total kinetic energy of the fluid-rigid body system.


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  • 1. V. I. Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 319-361. MR 0202082 (34:1956)
  • 2. Y. Brenier. Topics on hydrodynamics and volume preserving maps, Handbook of mathematical fluid dynamics, Vol. II (2003), 55-86. MR 1983589 (2004j:37162)
  • 3. J.-Y. Chemin. Fluides parfaits incompressibles, Astérisque 230 (1995). MR 1340046 (97d:76007)
  • 4. D. Ebin, J. Marsden. Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102-163. MR 0271984 (42:6865)
  • 5. O. Glass, F. Sueur, T. Takahashi. Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, preprint, 2010, arXiv:1003.4172, to appear in Ann. Sci. École Norm. Sup.
  • 6. J.-G. Houot, J. San Martin, M. Tucsnak. Existence and uniqueness of solutions for the equations modelling the motion of rigid bodies in a perfect fluid, Journal of Functional Analysis 259 (2010), no. 11, 2856-2885. MR 2719277
  • 7. A. Inoue, M. Wakimoto. On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 2, 303-319. MR 0481649 (58:1750)
  • 8. J. Ortega, L. Rosier, T. Takahashi. On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 1, 139-165. MR 2286562 (2007m:35201)
  • 9. J. H. Ortega, L. Rosier, T. Takahashi. Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid, M2AN Math. Model. Numer. Anal. 39 (2005), no. 1, 79-108. MR 2136201 (2006a:35253)
  • 10. C. Rosier, L. Rosier. Smooth solutions for the motion of a ball in an incompressible perfect fluid, Journal of Functional Analysis 256 (2009), no. 5, 1618-1641. MR 2490232 (2010d:35289)
  • 11. J. Vankerschaver, E. Kanso, J. E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech. 1 (2009), no. 2, 223-266. MR 2525759 (2010g:37086)
  • 12. J. Vankerschaver, E. Kanso, J. E. Marsden. The dynamics of a rigid body in potential flow with circulation, Regul. Chaotic Dyn. 15 (2010), no. 4-5, 606-629 MR 2679768
  • 13. V. A. Vladimirov, K. I. Ilin. On the Arnold stability of a solid in a plane steady flow of an ideal incompressible fluid, Theor. Comput. Fluid Dyn. 10 (1998), 425-437.
  • 14. V. A. Vladimirov, K. I. Ilin. On the stability of the dynamical system ``rigid body $ +$ inviscid fluid'', J. Fluid Mech. 386 (1999), 43-75. MR 1696733 (2000e:76069)

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Additional Information

Olivier Glass
Affiliation: Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: glass@ceremade.dauphine.fr

Franck Sueur
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie - Paris 6, 4 Place Jussieu, 75005 Paris, France
Email: fsueur@ann.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9939-2011-11219-X
Keywords: Perfect incompressible fluid, fluid-rigid body interaction, least action principle
Received by editor(s): February 7, 2011
Published electronically: October 4, 2011
Additional Notes: The authors were partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02.
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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