# zbMATH — the first resource for mathematics

Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid. (English) Zbl 1087.35081
The authors investigate the motion of a rigid ball in an incompressible perfect fluid occupying $$\mathbb R^2$$. In fact, they prove the existence and uniqueness of the classical solution. The main difficulties of this problem are that the system is nonlinear, strongly coupled and that the domain of the fluid is variable. In addition, when the fluid is assumed to be perfect, there is no paper devoted to the existence of a (weak or classical) solution for the fluid-rigid body system.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35Q05 Euler-Poisson-Darboux equations
Full Text:
##### References:
 [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 0511.46001 · Zbl 0511.46001 [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 0954.35135 · Zbl 0954.35135 [3] J.-M. Coron , On the controllability of $$2$$-D incompressible perfect fluids . J. Math. Pures Appl. (9) 75 ( 1996 ) 155 - 188 . Zbl 0848.76013 · Zbl 0848.76013 [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 0954.76010 · Zbl 0954.76010 [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 0943.35063 · Zbl 0943.35063 [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 0953.35118 · Zbl 0953.35118 [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 ). Zbl 1090.35137 · Zbl 1090.35137 [8] E. Feireisl , On the motion of rigid bodies in a viscous compressible fluid . Arch. Ration. Mech. Anal. 167 ( 2003 ) 281 - 308 . Zbl 1090.76061 · Zbl 1090.76061 [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 1039.76071 · Zbl 1039.76071 [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 0957.76012 · Zbl 0957.76012 [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 1046.35084 · Zbl 1046.35084 [12] T. Gallay and C.E. Wayne , Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $$\mathbf{R}^2$$ . Arch. Ration. Mech. Anal. 163 ( 2002 ) 209 - 258 . Zbl 1042.37058 · Zbl 1042.37058 [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 1814364 | Zbl 1042.35002 · Zbl 1042.35002 [14] O. Glass , Exact boundary controllability of 3-D Euler equation . ESAIM: COCV 5 ( 2000 ) 1 - 44 (electronic). Numdam | Zbl 0940.93012 · Zbl 0940.93012 [15] C. Grandmont and Y. Maday , Existence for an unsteady fluid-structure interaction problem . ESAIM: M2AN 34 ( 2000 ) 609 - 636 . Numdam | Zbl 0969.76017 · Zbl 0969.76017 [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 0970.35096 · Zbl 0970.35096 [17] P. Hartman , Ordinary differential equations . Birkhäuser Boston, MA, second edition ( 1982 ). MR 658490 | Zbl 0476.34002 · Zbl 0476.34002 [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 0966.76016 · Zbl 0966.76016 [19] K.-H. Hoffmann and V.N. Starovoitov , Zur Bewegung einer Kugel in einer zähen Flüssigkeit . Doc. Math. 5 ( 2000 ) 15 - 21 (electronic). Zbl 0936.35125 · Zbl 0936.35125 [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 0166.45302 · Zbl 0166.45302 [22] K. Kikuchi , Exterior problem for the two-dimensional Euler equation . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 ( 1983 ) 63 - 92 . Zbl 0517.76024 · Zbl 0517.76024 [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 350177 | Zbl 0223.35039 · Zbl 0223.35039 [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 1422251 | Zbl 0866.76002 · Zbl 0866.76002 [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 pre01622637 · Zbl 1162.76339 [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 1018.76012 · Zbl 1018.76012 [27] D. Serre , Chute libre d’un solide dans un fluide visqueux incompressible . Existence. Japan J. Appl. Math. 4 ( 1987 ) 99 - 110 . Zbl 0655.76022 · Zbl 0655.76022 [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 1022.35041 · Zbl 1022.35041 [29] J. Simon , Compact sets in the space $$L^p(0,T;B)$$ . Ann. Mat. Pura Appl. (4) 146 ( 1987 ) 65 - 96 . Zbl 0629.46031 · Zbl 0629.46031 [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 1101.35356 · Zbl 1101.35356 [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 1054.35061 · Zbl 1054.35061 [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 769654 | Zbl 0568.35002 · Zbl 0568.35002 [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 1071.74017 · Zbl 1071.74017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.