On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, p. 401-417

A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a bounded Radon measure with distinguished sign and lies in the Sobolev space H -1 . In this paper we are interested in the case where there is a rigid body immersed in the fluid moving under the action of the fluid pressure. We succeed to prove the existence of solutions à la Delort in a particular case with a mirror symmetry assumption already considered by Lopes Filho et al. (2006) [11], where it was assumed in addition that the rigid body is a fixed obstacle. The solutions built here satisfy the energy inequality and the body acceleration is bounded. When the mass of the body becomes infinite, the body does not move anymore and one recovers a solution in the sense of Lopes Filho et al. (2006) [11].

@article{AIHPC_2013__30_3_401_0,
     author = {Sueur, Franck},
     title = {On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     pages = {401-417},
     doi = {10.1016/j.anihpc.2012.09.001},
     zbl = {06295426},
     mrnumber = {3061429},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_3_401_0}
}
Sueur, Franck. On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, pp. 401-417. doi : 10.1016/j.anihpc.2012.09.001. http://www.numdam.org/item/AIHPC_2013__30_3_401_0/

[1] J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 no. 3 (1991), 553-586 | MR 1102579

[2] R.J. Diperna, A.J. Majda, Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math. 40 no. 3 (1987), 301-345 | MR 882068 | Zbl 0850.76730

[3] P. Gérard, Résultats récents sur les fluides parfaits incompressibles bidimensionnels (dʼaprès J.-Y. Chemin, J.-M. Delort), Séminaire Bourbaki, vol. 1991/92 Astérisque 206 (1992), 411-444 | Numdam | MR 1206075

[4] O. Glass, C. Lacave, F. Sueur, On the motion of a small body immersed in a two dimensional incompressible perfect fluid, preprint, arXiv:1104.5404, 2011, Bull. Soc. Math. Fr., in press.

[5] O. Glass, F. Sueur, On the motion of a rigid body in a two-dimensional irregular ideal flow, SIAM J. Math. Anal. 44 no. 5 (2012), 3101-3126, arXiv:1107.0575 (2011) | MR 3023405 | Zbl 1325.76026

[6] O. Glass, F. Sueur, Low regularity solutions for the two-dimensional “rigid body + incompressible Euler” system, arXiv:1203.5988 (2012) | MR 3200756

[7] D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes, F. Sueur, Vortex sheets in exterior domains and Kelvinʼs Circulation Theorem, in preparation.

[8] K. Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 no. 1 (1983), 63-92 | MR 700596 | Zbl 0517.76024

[9] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford Lecture Ser. Math. Appl. vol. 3 (1996) | MR 1422251 | Zbl 0866.76002

[10] M.C. Lopes Filho, H.J. Nussenzveig Lopes, Z. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal. 158 no. 3 (2001), 235-257 | MR 1842346 | Zbl 1058.35176

[11] M.C. Lopes Filho, H.J. Nussenzveig Lopes, Z. Xin, Vortex sheets with reflection symmetry in exterior domains, J. Differential Equations 229 no. 1 (2006), 154-171 | MR 2265622 | Zbl 1102.76009

[12] 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 no. 1 (2007), 139-165 | Numdam | MR 2286562 | Zbl 1168.35038

[13] 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 no. 1 (2005), 79-108 | Numdam | MR 2136201 | Zbl 1087.35081

[14] G. Planas, F. Sueur, On the inviscid limit of the system “viscous incompressible fluid + rigid body” with the Navier conditions, arXiv:1206.0029 (2012) | Numdam | MR 3165279

[15] L. Saint-Raymond, Un résultat générique dʼunicité pour les équations dʼévolution, Bull. Soc. Math. France 130 no. 1 (2002), 87-99 | Numdam | MR 1906194 | Zbl 0996.35002

[16] S. Schochet, The weak vorticity formulation of the 2D Euler equations and concentration-cancellation, Comm. Partial Differential Equations 20 no. 5–6 (1995), 1077-1104 | MR 1326916 | Zbl 0822.35111

[17] F. Sueur, A Kato type theorem for the inviscid limit of the Navier–Stokes equations with a moving rigid body, preprint, arXiv:1110.6065, 2011, Comm. Math. Phys., http://dx.doi.org/10.1007/s00220-012-1516-x, in press.