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Classical solutions for the equations modelling the motion ofa ball in a bidimensional incompressible perfect fluid

Published online by Cambridge University Press:  15 March 2005

Jaime H. Ortega ,
Lionel Rosier  and
Takéo Takahashi
Jaime H. Ortega
Affiliation:
Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Casilla 170/3, Correo 3, Santiago, Chile and Universidad del Bío-Bío, Facultad de Ciencias, Departamento de Ciencias Básicas, Casilla 447, Campus Fernando May, Chillán, Chile. jortega@dim.uchile.cl
Lionel Rosier
Affiliation:
Institut Elie Cartan, Université Henri Poincaré Nancy 1, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France. rosier@iecn.u-nancy.fr; takahash@iecn.u-nancy.fr
Takéo Takahashi
Affiliation:
Institut Elie Cartan, Université Henri Poincaré Nancy 1, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France. rosier@iecn.u-nancy.fr; takahash@iecn.u-nancy.fr
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Abstract

In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying ${\mathbb R}^2$. 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.

Keywords

Euler equationsfluid-rigid body interactionexterior domainclassical solutions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 79 - 108
Copyright
© EDP Sciences, SMAI, 2005

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References

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].
Conca, C., San Martín, J. 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. CrossRef
Coron, J.-M., On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. (9) 75 (1996) 155–188.
Coron, J.-M., 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). CrossRef
Desjardins, B. and Esteban, M.J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59–71. CrossRef
Desjardins, B. and Esteban, M.J., On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Partial Differential Equations 25 (2000) 1399–1413.
Feireisl, E., On the motion of rigid bodies in a viscous fluid. Appl. Math. 47 (2002) 463–484. Mathematical theory in fluid mechanics, Paseky (2001). CrossRef
Feireisl, E., On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167 (2003) 281–308. CrossRef
Feireisl, E., On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equ. 3 (2003) 419–441. Dedicated to Philippe Bénilan. CrossRef
Galdi, G.P., On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148 (1999) 53–88. CrossRef
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.
Gallay, T. and Wayne, C.E., 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. CrossRef
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.
Glass, O., Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1–44 (electronic). CrossRef
Grandmont, C. and Maday, Y., Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN 34 (2000) 609–636. CrossRef
Gunzburger, M.D., Lee, H.-C. and Seregin, G.A., 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. CrossRef
P. Hartman, Ordinary differential equations. Birkhäuser Boston, MA, second edition (1982).
Hoffmann, K.-H. and Starovoitov, V.N., On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9 (1999) 633–648.
Hoffmann, K.-H. and Starovoitov, V.N., Zur Bewegung einer Kugel in einer zähen Flüssigkeit. Doc. Math. 5 (2000) 15–21 (electronic).
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.
Kato, T., On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25 (1967) 188–200. CrossRef
Kikuchi, K., Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983) 63–92.
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.
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).
Rosier, C. and Rosier, L., Well-posedness of a degenerate parabolic equation issuing from two-dimensional perfect fluid dynamics. Appl. Anal. 75 (2000) 441–465. CrossRef
San Martín, J. 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.
Serre, D., Chute libre d'un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4 (1987) 99–110. CrossRef
Silvestre, A.L., 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. CrossRef
Simon, J., Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. (4) 146 (1987) 65–96. CrossRef
Takahashi, T., 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.
Takahashi, T. and Tucsnak, M., Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6 (2004) 53–77. CrossRef
R. Temam, Navier-Stokes equations. North-Holland Publishing Co., Amsterdam, third edition (1984). Theory and numerical analysis, with an appendix by F. Thomasset.
Vázquez, J.L. and Zuazua, E., Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705–1738. CrossRef