Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid
ESAIM: Modélisation mathématique et analyse numérique, Volume 39 (2005) no. 1, pp. 79-108.

In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying 2 . We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.

DOI: 10.1051/m2an:2005002
Classification: 35Q35, 76B03, 76B99
Keywords: Euler equations, fluid-rigid body interaction, exterior domain, classical solutions
@article{M2AN_2005__39_1_79_0,
     author = {Ortega, Jaime H. and Rosier, Lionel and Takahashi, Tak\'eo},
     title = {Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {79--108},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {1},
     year = {2005},
     doi = {10.1051/m2an:2005002},
     mrnumber = {2136201},
     zbl = {1087.35081},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2005002/}
}
TY  - JOUR
AU  - Ortega, Jaime H.
AU  - Rosier, Lionel
AU  - Takahashi, Takéo
TI  - Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 79
EP  - 108
VL  - 39
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2005002/
DO  - 10.1051/m2an:2005002
LA  - en
ID  - M2AN_2005__39_1_79_0
ER  - 
%0 Journal Article
%A Ortega, Jaime H.
%A Rosier, Lionel
%A Takahashi, Takéo
%T Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 79-108
%V 39
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2005002/
%R 10.1051/m2an:2005002
%G en
%F M2AN_2005__39_1_79_0
Ortega, Jaime H.; Rosier, Lionel; Takahashi, Takéo. Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid. ESAIM: Modélisation mathématique et analyse numérique, Volume 39 (2005) no. 1, pp. 79-108. doi : 10.1051/m2an:2005002. http://www.numdam.org/articles/10.1051/m2an:2005002/

[1] H. Brezis, Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson, Paris (1983). Théorie et Applications. [Theory and applications]. | Zbl

[2] C. Conca, J. San Martín H. and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations 25 (2000) 1019-1042. | Zbl

[3] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. (9) 75 (1996) 155-188. | Zbl

[4] J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain. SIAM J. Control Optim. 37 (1999) 1874-1896 (electronic). | Zbl

[5] B. Desjardins and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71. | Zbl

[6] B. Desjardins and M.J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Partial Differential Equations 25 (2000) 1399-1413. | Zbl

[7] E. Feireisl, On the motion of rigid bodies in a viscous fluid. Appl. Math. 47 (2002) 463-484. Mathematical theory in fluid mechanics, Paseky (2001). | EuDML | Zbl

[8] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167 (2003) 281-308. | Zbl

[9] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equ. 3 (2003) 419-441. Dedicated to Philippe Bénilan. | Zbl

[10] G.P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148 (1999) 53-88. | Zbl

[11] G.P. Galdi and A.L. Silvestre, Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques. In Nonlinear problems in mathematical physics and related topics, I. Int. Math. Ser. (N.Y.), Kluwer/Plenum, New York 1 (2002) 121-144. | Zbl

[12] T. Gallay and C.E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on 𝐑 2 . Arch. Ration. Mech. Anal. 163 (2002) 209-258. | Zbl

[13] N.S. Gilbarg and D. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[14] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44 (electronic). | EuDML | Numdam | Zbl

[15] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN 34 (2000) 609-636. | EuDML | Numdam | Zbl

[16] M.D. Gunzburger, H.-C. Lee and 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 (2000) 219-266. | Zbl

[17] P. Hartman, Ordinary differential equations. Birkhäuser Boston, MA, second edition (1982). | MR | Zbl

[18] K.-H. Hoffmann and V.N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9 (1999) 633-648. | Zbl

[19] K.-H. Hoffmann and V.N. Starovoitov, Zur Bewegung einer Kugel in einer zähen Flüssigkeit. Doc. Math. 5 (2000) 15-21 (electronic). | EuDML | Zbl

[20] N.V. Judakov, The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid. Dinamika Splošn. Sredy, (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami) 255 (1974) 249-253.

[21] T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25 (1967) 188-200. | Zbl

[22] K. Kikuchi, Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983) 63-92. | Zbl

[23] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. | MR | Zbl

[24] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1, The Clarendon Press Oxford University Press, New York. Incompressible models, Oxford Science Publications. Oxford Lect. Ser. Math. Appl. 3 (1996). | MR | Zbl

[25] C. Rosier and L. Rosier, Well-posedness of a degenerate parabolic equation issuing from two-dimensional perfect fluid dynamics. Appl. Anal. 75 (2000) 441-465. | Zbl

[26] J. San Martín H., V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal. 161 (2002) 113-147. | Zbl

[27] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4 (1987) 99-110. | Zbl

[28] A.L. Silvestre, On the self-propelled motion of a rigid body in a viscous liquid and on the attainability of steady symmetric self-propelled motions. J. Math. Fluid Mech. 4 (2002) 285-326. | Zbl

[29] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. (4) 146 (1987) 65-96. | Zbl

[30] 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 (2003) 1499-1532. | Zbl

[31] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6 (2004) 53-77. | Zbl

[32] R. Temam, Navier-Stokes equations. North-Holland Publishing Co., Amsterdam, third edition (1984). Theory and numerical analysis, with an appendix by F. Thomasset. | MR | Zbl

[33] J.L. Vázquez and E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | Zbl

Cited by Sources: