In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying . 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 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] 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] 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] On the controllability of -D incompressible perfect fluids. J. Math. Pures Appl. (9) 75 (1996) 155-188. | Zbl
,[4] 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] Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71. | Zbl
and ,[6] On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Partial Differential Equations 25 (2000) 1399-1413. | Zbl
and ,[7] 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] On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167 (2003) 281-308. | Zbl
,[9] On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equ. 3 (2003) 419-441. Dedicated to Philippe Bénilan. | Zbl
,[10] On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148 (1999) 53-88. | Zbl
,[11] 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
and ,[12] Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on . Arch. Ration. Mech. Anal. 163 (2002) 209-258. | Zbl
and ,[13] Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl
and ,[14] Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44 (electronic). | EuDML | Numdam | Zbl
,[15] Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN 34 (2000) 609-636. | EuDML | Numdam | Zbl
and ,[16] 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
, and ,[17] Ordinary differential equations. Birkhäuser Boston, MA, second edition (1982). | MR | Zbl
,[18] On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9 (1999) 633-648. | Zbl
and ,[19] Zur Bewegung einer Kugel in einer zähen Flüssigkeit. Doc. Math. 5 (2000) 15-21 (electronic). | EuDML | Zbl
and ,[20] 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] On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25 (1967) 188-200. | Zbl
,[22] Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983) 63-92. | Zbl
,[23] 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
and ,[24] 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] Well-posedness of a degenerate parabolic equation issuing from two-dimensional perfect fluid dynamics. Appl. Anal. 75 (2000) 441-465. | Zbl
and ,[26] 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] Chute libre d'un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4 (1987) 99-110. | Zbl
,[28] 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] Compact sets in the space . Ann. Mat. Pura Appl. (4) 146 (1987) 65-96. | Zbl
,[30] 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] 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
and ,[32] Navier-Stokes equations. North-Holland Publishing Co., Amsterdam, third edition (1984). Theory and numerical analysis, with an appendix by F. Thomasset. | MR | Zbl
,[33] Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | Zbl
and ,Cited by Sources: