Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Olivier Glass; Franck Sueur; Takéo Takahashi

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 1, page 1-51
  • ISSN: 0012-9593

Abstract

top
We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Hölder space . In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and in particular till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.

How to cite

top

Glass, Olivier, Sueur, Franck, and Takahashi, Takéo. "Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid." Annales scientifiques de l'École Normale Supérieure 45.1 (2012): 1-51. <http://eudml.org/doc/272145>.

@article{Glass2012,
abstract = {We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Hölder space $C^\{1,r\}$. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and in particular till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.},
author = {Glass, Olivier, Sueur, Franck, Takahashi, Takéo},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {fluid-solid interaction; regularity properties; perfect incompressible fluid},
language = {eng},
number = {1},
pages = {1-51},
publisher = {Société mathématique de France},
title = {Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid},
url = {http://eudml.org/doc/272145},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Glass, Olivier
AU - Sueur, Franck
AU - Takahashi, Takéo
TI - Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 1
SP - 1
EP - 51
AB - We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Hölder space $C^{1,r}$. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and in particular till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.
LA - eng
KW - fluid-solid interaction; regularity properties; perfect incompressible fluid
UR - http://eudml.org/doc/272145
ER -

References

top
  1. [1] L. V. Ahlfors, Complex analysis, third éd., McGraw-Hill Book Co., 1978. Zbl0395.30001MR510197
  2. [2] C. Bardos & E. S. Titi, Loss of smoothness and energy conserving rough weak solutions for the Euler equations, Discrete Contin. Dyn. Syst. Ser. S3 (2010), 185–197. Zbl1191.76057MR2610558
  3. [3] J.-P. Bourguignon & H. Brezis, Remarks on the Euler equation, J. Functional Analysis15 (1974), 341–363. Zbl0279.58005MR344713
  4. [4] J.-Y. Chemin, Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel, Invent. Math.103 (1991), 599–629. Zbl0739.76010MR1091620
  5. [5] J.-Y. Chemin, Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace, J. Math. Pures Appl.71 (1992), 407–417. Zbl0833.35112MR1191582
  6. [6] J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995). Zbl0829.76003MR1340046
  7. [7] A. Dutrifoy, Precise regularity results for the Euler equations, J. Math. Anal. Appl.282 (2003), 177–200. Zbl1048.35079MR2000337
  8. [8] P. Gamblin, Système d’Euler incompressible et régularité microlocale analytique, in Séminaire sur les Équations aux Dérivées Partielles, 1992–1993, École Polytech., 1993, exp. no 20. Zbl0876.35089MR1240561
  9. [9] P. Gamblin, Système d’Euler incompressible et régularité microlocale analytique, Ann. Inst. Fourier (Grenoble) 44 (1994), 1449–1475. Zbl0820.35111MR1313791
  10. [10] P. W. Gross & P. R. Kotiuga, Electromagnetic theory and computation: a topological approach, Mathematical Sciences Research Institute Publications 48, Cambridge Univ. Press, 2004. Zbl1096.78001MR2067778
  11. [11] J. G. Houot, J. San Martin & M. Tucsnak, Existence of solutions for the equations modeling the motion of rigid bodies in an ideal fluid, J. Funct. Anal.259 (2010), 2856–2885. Zbl1200.35222MR2719277
  12. [12] T. Kato, On the smoothness of trajectories in incompressible perfect fluids, in Nonlinear wave equations (Providence, RI, 1998), Contemp. Math. 263, Amer. Math. Soc., 2000, 109–130. Zbl0972.35102MR1777638
  13. [13] T. Kato, Two manuscripts left by late Professor Tosio Kato in his personal computer, Sūrikaisekikenkyūsho Kōkyūroku1234 (2001), 260–274. MR1906059
  14. [14] H. Koch, Transport and instability for perfect fluids, Math. Ann.323 (2002), 491–523. Zbl1006.76008MR1923695
  15. [15] 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), 79–108. Zbl1087.35081MR2136201
  16. [16] J. H. 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 Linéaire24 (2007), 139–165. Zbl1168.35038MR2286562
  17. [17] C. Rosier & L. Rosier, Smooth solutions for the motion of a ball in an incompressible perfect fluid, J. Funct. Anal.256 (2009), 1618–1641. Zbl1173.35105MR2490232
  18. [18] P. Serfati, Équation d’Euler et holomorphies à faible régularité spatiale, C. R. Acad. Sci. Paris Sér. I Math.320 (1995), 175–180. Zbl0834.34077MR1320351
  19. [19] P. Serfati, Solutions en temps, - Lipschitz bornées en espace et équation d’Euler, C. R. Acad. Sci. Paris Sér. I Math.320 (1995), 555–558. Zbl0835.76012MR1322336
  20. [20] P. Serfati, Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. Math. Pures Appl.74 (1995), 95–104. Zbl0849.35111MR1325824
  21. [21] A. Shnirelman, Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid, Amer. J. Math.119 (1997), 579–608. Zbl0874.93055MR1448216
  22. [22] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30, Princeton Univ. Press, 1970. Zbl0207.13501MR290095
  23. [23] V. Thilliez, On quasianalytic local rings, Expo. Math.26 (2008), 1–23. Zbl1139.32003MR2384272

NotesEmbed ?

top

You must be logged in to post comments.