Sur la dynamique de corps solides immergés dans un fluide incompressible

Franck Sueur[1]

  • [1] Laboratoire Jacques-Louis Lions Université Paris 6

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • page 1-20
  • ISSN: 2266-0607

Abstract

top
Cet exposé présente quelques résultats récents obtenus par l’auteur en collaboration avec Olivier Glass, Christophe Lacave, Ayman Moussa, Gabriela Planas et Takéo Takahashi, sur l’analyse théorique de la dynamique de corps solides immergśs dans un fluide incompressible.

How to cite

top

Sueur, Franck. "Sur la dynamique de corps solides immergés dans un fluide incompressible." Séminaire Laurent Schwartz — EDP et applications (2012-2013): 1-20. <http://eudml.org/doc/275713>.

@article{Sueur2012-2013,
abstract = {Cet exposé présente quelques résultats récents obtenus par l’auteur en collaboration avec Olivier Glass, Christophe Lacave, Ayman Moussa, Gabriela Planas et Takéo Takahashi, sur l’analyse théorique de la dynamique de corps solides immergśs dans un fluide incompressible.},
affiliation = {Laboratoire Jacques-Louis Lions Université Paris 6},
author = {Sueur, Franck},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {fre},
pages = {1-20},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Sur la dynamique de corps solides immergés dans un fluide incompressible},
url = {http://eudml.org/doc/275713},
year = {2012-2013},
}

TY - JOUR
AU - Sueur, Franck
TI - Sur la dynamique de corps solides immergés dans un fluide incompressible
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 20
AB - Cet exposé présente quelques résultats récents obtenus par l’auteur en collaboration avec Olivier Glass, Christophe Lacave, Ayman Moussa, Gabriela Planas et Takéo Takahashi, sur l’analyse théorique de la dynamique de corps solides immergśs dans un fluide incompressible.
LA - fre
UR - http://eudml.org/doc/275713
ER -

References

top
  1. L. Ambrosio, N. Gigli et G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Second edition. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, 2008. Zbl1090.35002MR2401600
  2. 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(1), 319-361, 1966. Zbl0148.45301MR202082
  3. W. Braun et K. Hepp. The Vlasov dynamics and its fluctuations in the 1 / N limit of interacting classical particles. Comm. Math. Phys. 56(2), 101-113, 1977. Zbl1155.81383MR475547
  4. Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25(3-4), 737-754, 2000. Zbl0970.35110MR1748352
  5. J.-Y. Chemin. Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel. Invent. Math. 103(3), 599-629, 1991. Zbl0739.76010MR1091620
  6. J.-Y. Chemin. Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace. J. Math. Pures Appl. 71(5), 407-417, 1992. Zbl0833.35112MR1191582
  7. J.-Y. Chemin. Fluides parfaits incompressibles. Astérisque, 230, 1995. Zbl0829.76003MR1340046
  8. S. Childress. An introduction to theoretical fluid mechanics. Courant Lecture Notes in Mathematics, 19. Courant Institute of Mathematical Sciences, New York ; American Mathematical Society, Providence, RI, 2009. Zbl1309.76001MR2546940
  9. C. Conca, J. A. San Martin et M. Tucsnak. Existence de solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations 25(5-6), 1019-1042, 2000. Zbl0954.35135MR1759801
  10. P. Degond. Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. Ann. Sci. École Norm. Sup. (4), 519-542, 1986. Zbl0619.35087MR875086
  11. J.-M. Delort. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4(3), 553-586, 1991. Zbl0780.35073MR1102579
  12. B. Desjardins et M. J. Esteban. On weak solutions for fluid-rigid structure interaction : compressible and incompressible models. Comm. Partial Differential Equations 25(7-8), 1399-1413, 2000. Zbl0953.35118MR1765138
  13. R. J. DiPerna et A. J. Majda. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40(3), 301-345, 1987. Zbl0850.76730MR882068
  14. R. L. Dobrushin. Vlasov equations. Funct. Anal. Appl. 13, 115-123, 1979. Zbl0422.35068MR541637
  15. E. Feireisl. On the motion of rigid bodies in a viscous incompressible fluid. Dedicated to Philippe Bénilan. J. Evol. Equ. 3(3), 419-441, 2003. Zbl1039.76071MR2019028
  16. E. Feireisl. On the motion of rigid bodies in a viscous fluid. Mathematical theory in fluid mechanics (Paseky, 2001). Appl. Math. 47(6), 463-484, 2002. Zbl1090.35137MR1948192
  17. E. Feireisl. On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167(4), 281-308, 2003. Zbl1090.76061MR1981859
  18. U. Frisch et V. Zheligovsky. A very smooth ride in a rough sea. Preprint 2012, http://arxiv.org/abs/1212.4333. Zbl1285.35072
  19. G. P. Galdi. On the motion of a rigid body in a viscous liquid : a mathematical analysis with applications. Handbook of mathematical fluid dynamics, Vol. I, 653-791, North-Holland, Amsterdam, 2002. Zbl1230.76016MR1942470
  20. P. Gamblin. Système d’Euler incompressible et régularité microlocale analytique. Ann. Inst. Fourier (Grenoble) 44(5), 1449–1475, 1994. Zbl0820.35111MR1313791
  21. M. Geissert, K. Götze et M. Hieber, Lp-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids. A paraître dans Trans. Amer. Math. Soc. Zbl1282.35273MR3003269
  22. D. Gérard-Varet et M. Hillairet. Regularity issues in the problem of fluid structure interaction. Arch. Ration. Mech. Anal. 195(2), 375-407, 2010. Zbl1192.35131MR2592281
  23. D. Gérard-Varet et M. Hillairet. Existence of weak solutions up to collision for viscous fluid-solid systems with slip. Preprint 2012, http://arxiv.org/abs/1207.0469. Zbl1307.35193MR2916378
  24. O. Glass. Exact boundary controllability of 3-D Euler equation. ESAIM Control Optim. Calc. Var. 5, 1-44, 2000. Zbl0940.93012MR1745685
  25. O. Glass et T. Horsin. Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patch. J. Math. Pures Appl. 93(1), 61-90, 2010. Zbl1180.35531MR2579376
  26. O. Glass et T. Horsin. Prescribing the motion of a set of particles in a 3D perfect fluid. SIAM J. on Control and Optimization 50(5), 2726-2742, 2012. Zbl1263.76018
  27. O. Glass, C. Lacave et F. Sueur. On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Preprint 2011, http://arxiv.org/abs/1104.5404. A paraître dans Bulletin de la SMF. Zbl1329.76048
  28. O. Glass et F. Sueur. The movement of a solid in an incompressible perfect fluid as a geodesic flow. Proceedings of the AMS. 140(6), 2155-2168, 2012. Zbl1261.76009MR2888201
  29. O. Glass et F. Sueur. On the motion of a rigid body in a two-dimensional irregular ideal flow. SIAM J. Math. Anal. 44(5), 3101-3126, 2012. Zbl1325.76026MR3023405
  30. O. Glass et F. Sueur. Uniqueness results for weak solutions of two-dimensional fluid-solid systems. Preprint 2012, http://arxiv.org/abs/1203.2894. Zbl06481060
  31. O. Glass et F. Sueur. Low regularity solutions for the two-dimensional “rigid body + incompressible Euler" system. Preprint 2012, http://hal.archives-ouvertes.fr/hal-00682976/. Zbl06433749MR3023405
  32. O. Glass, F. Sueur et T. Takahashi. Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid. Ann. Sci. École Norm. Sup. Volume 45(1), 1-51, 2012. Zbl1311.35217MR2961786
  33. F. Golse et L. Saint-Raymond. The Vlasov-Poisson system with strong magnetic field. J. Math. Pures Appl. 78(8), 791-817, 1999. Zbl0977.35108MR1715342
  34. C. Grotta Ragazzo, J. Koiller et W. M. Oliva. On the motion of two-dimensional vortices with mass. Nonlinear Sci., 4(5), 375-418, 1994. Zbl0808.76015MR1291115
  35. N. M. Günther. Über ein Hauptproblem der Hydrodynamik. (German) Math. Z. 24(1), 448-499, 1926. Zbl51.0659.02MR1544775
  36. M. Gunzburger, H.-C. Lee et G. A. Seregin. Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2(3), 219–266, 2000. Zbl0970.35096MR1781915
  37. M. Hauray et P. E. Jabin. N -particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183(3), 489-524, 2007. Zbl1107.76066MR2278413
  38. T. I. Hesla, Collisions of Smooth Bodies in Viscous Fluids : A Mathematical Investigation. PhD thesis, University of Minnesota, revised version. 2005. 
  39. M. Hillairet. Lack of collision between solid bodies in a 2D incompressible viscous flow. Comm. Partial Differential Equations 32(7-9), 1345-1371, 2007. Zbl1221.35279MR2354496
  40. K.-H. Hoffmann et V. N. Starovoitov. On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9(2), 633-648, 1999 Zbl0966.76016MR1725677
  41. J.-G. Houot, J. San Martin et M. Tucsnak. Existence and uniqueness of solutions for the equations modelling the motion of rigid bodies in a perfect fluid. J. Funct. Anal. 259(11), 2856-2885, 2010. Zbl1200.35222MR2719277
  42. A. Inoue et 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(2), 303–319, 1977. Zbl0381.35066MR481649
  43. N. V. Judakov. The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid (in Russian). Dinamika Splošn. Sredy 18, 249-253, 1974. MR464811
  44. T. Kato. On the smoothness of trajectories in incompressible perfect fluids. In Nonlinear wave equations (Providence, RI, 1998), volume 263 of Contemp. Math., 109–130. Amer. Math. Soc., Providence, RI, 2000. Zbl0972.35102MR1777638
  45. J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193-248, 1933. Zbl59.0763.02MR1555394
  46. J. Leray. Étude de diverses équations intégrales non linéaires et de quelques problèmes de l’hydrodynamique. J. Maths Pures Appl. 12, 1-82, 1933. Zbl0006.16702
  47. L. Lichtenstein. Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze. (German) Math. Z. 23(1), 89-154, 1925. Zbl51.0658.01MR1544733
  48. L. Lichtenstein. Über einige Existenzprobleme der Hydrodynamik. (German) Math. Z. 32(1), 608-640, 1930. Zbl56.1249.01MR1545189
  49. P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications 3, 1996. Zbl0908.76004MR1422251
  50. G. Loeper. Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. 86(1), 68-79, 2006. Zbl1111.35045MR2246357
  51. C. Marchioro et M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, 1994. Zbl0789.76002MR1245492
  52. A. Moussa et F. Sueur. A 2d spray model with gyroscopic effects. Asymptotic Analysis. 81(1), 53-91, 2013. Zbl1284.35440
  53. H. Neunzert. The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles. Trans. Fluid Dynamics 18, 663-678, 1977. 
  54. J. H. Ortega, L. Rosier et 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(1), 79-108, 2005. Zbl1087.35081MR2136201
  55. J. H. Ortega, L. Rosier et T. Takahashi. On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24(1), 139-165, 2007. Zbl1168.35038MR2286562
  56. G. Planas et F. Sueur. On the “viscous incompressible fluid + rigid body” system with Navier conditions A paraître aux Ann. Inst. H. Poincaré Anal. Non Linéaire. Zbl1288.35375
  57. C. Rosier et L. Rosier. Smooth solutions for the motion of a ball in an incompressible perfect fluid, Journal of Functional Analysis 256(5), 1618-1641, 2009. Zbl1173.35105MR2490232
  58. J. A. San Martin, V. Starovoitov et M. Tucsnak. Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161(2), 113-147, 2002. Zbl1018.76012MR1870954
  59. P. Serfati. Équation d’Euler et holomorphies à faible régularité spatiale. C. R. Acad. Sci. Paris Sér. I Math. 320(2), 175-180, 1995. Zbl0834.34077MR1320351
  60. P. Serfati. Solutions C en temps, n - log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris Sér. I Math., 320(5), 555-558, 1995. Zbl0835.76012MR1322336
  61. P. Serfati. Structures holomorphes à faible régularité spatiale en mécanique des fluides. J. Math. Pures Appl. 74(2), 95-104, 1995. Zbl0849.35111MR1325824
  62. D. Serre. Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4(1), 99-110, 1987. Zbl0655.76022MR899206
  63. A. Shnirelman. On the analyticity of particles trajectories in the ideal incompressible fluid. Global and stochastic analysis 2(1), 2012. Zbl1296.35133
  64. V. N. Starovoitov. Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid. J. Math. Sci. 130(4), 4893–4898, 2005. Zbl1148.35347MR2065504
  65. F. Sueur. On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data. A paraître aux Ann. Inst. H. Poincaré Anal. Non Linéaire. http://dx.doi.org/10.1016/j.anihpc.2012.09.001. Zbl06295426
  66. F. Sueur. Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain. J. of Differential Equations 251(12), 3421-3449, 2011. Zbl1248.76011MR2837690
  67. F. Sueur. A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body. Comm. in Math. Physics 316(3), 783-808, 2012. kato3d3 Zbl1253.35106MR2993933
  68. T. Takahashi. Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations 8(12), 1499-1532, 2003. Zbl1101.35356MR2029294
  69. C. Villani. Optimal transport, old and new. Fundamental Principles of Mathematical Sciences 338, Springer-Verlag. Zbl1156.53003MR2459454
  70. Y. Wang et A. Zang. Smooth solutions for motion of a rigid body of general form in an incompressible perfect fluid. J. Differential Equations, 252, 4259-4288, 2012. Zbl1241.35151MR2879731
  71. W. Wolibner. Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698-726, 1933. Zbl0008.06901MR1545430
  72. V. I. Yudovich. Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3, 1032-1066, 1963. Zbl0129.19402MR158189

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.