Imanuvilov, Oleg; Takahashi, Takéo Exact controllability of a fluid–rigid body system. (English) Zbl 1124.35056 J. Math. Pures Appl. (9) 87, No. 4, 408-437 (2007). A two-dimensional fluid-structure system consisting of a rigid body in the form of a circle and a viscous incompressible fluid modelled by the Navier-Stokes equations is considered. The exact controllability of the position of the circle is shown. In the proof, a new Carleman inequality is used. Reviewer: Martin Gugat (Erlangen) Cited in 2 ReviewsCited in 22 Documents MSC: 35Q30 Navier-Stokes equations 93B05 Controllability 76D05 Navier-Stokes equations for incompressible viscous fluids 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) Keywords:fluid-structure interaction; Navier-Stokes equations; Carleman inequality PDFBibTeX XMLCite \textit{O. Imanuvilov} and \textit{T. Takahashi}, J. Math. Pures Appl. (9) 87, No. 4, 408--437 (2007; Zbl 1124.35056) Full Text: DOI References: [1] M. Boulakia, A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., submitted for publication; M. Boulakia, A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., submitted for publication · Zbl 1149.35068 [2] Conca, C.; San Martín H., J.; Tucsnak, M., Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25, 5-6, 1019-1042 (2000) · Zbl 0954.35135 [3] Desjardins, B.; Esteban, M. J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146, 1, 59-71 (1999) · Zbl 0943.35063 [4] Desjardins, B.; Esteban, M. J., On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations, 25, 7-8, 1399-1413 (2000) · Zbl 0953.35118 [5] Doubova, A.; Fernández-Cara, E., Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., 15, 5, 783-824 (2005) · Zbl 1122.93008 [6] Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17, 413-415 (1966) · Zbl 0146.12503 [7] Fabre, C.; Lebeau, G., Prolongement unique des solutions de l’équation de Stokes, Comm. Partial Differential Equations, 21, 3-4, 573-596 (1996) · Zbl 0849.35098 [8] Feireisl, E., On the motion of rigid bodies in a viscous fluid, Mathematical Theory in Fluid Mechanics (Paseky, 2001). Mathematical Theory in Fluid Mechanics (Paseky, 2001), Appl. Math., 47, 6, 463-484 (2002) · Zbl 1090.35137 [9] Feireisl, E., On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal., 167, 4, 281-308 (2003) · Zbl 1090.76061 [10] Fernández-Cara, E.; Guerrero, S.; Imanuvilov, O. Y.; Puel, J.-P., Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83, 12, 1501-1542 (2004) · Zbl 1267.93020 [11] Galdi, G., On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148, 1, 53-88 (1999) · Zbl 0957.76012 [12] Galdi, G.; Silvestre, A., Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, (Nonlinear Problems in Mathematical Physics and Related Topics, I. Nonlinear Problems in Mathematical Physics and Related Topics, I, Int. Math. Ser. (N.Y.), vol. 1 (2002), Kluwer/Plenum: Kluwer/Plenum New York), 121-144 · Zbl 1046.35084 [13] Grandmont, C.; Maday, Y., Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34, 3, 609-636 (2000) · Zbl 0969.76017 [14] Gunzburger, M.; Lee, H.-C.; 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, 3, 219-266 (2000) · Zbl 0970.35096 [15] Hoffmann, K.-H.; Starovoitov, V., Zur Bewegung einer Kugel in einer zähen Flüssigkeit, Doc. Math., 5, 15-21 (2000), (electronic) · Zbl 0936.35125 [16] Hoffmann, K.-H.; Starovoitov, V., On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9, 2, 633-648 (1999) · Zbl 0966.76016 [17] Imanuvilov, O. Y., Controllability of parabolic equations, Mat. Sb., 186, 6, 109-132 (1995) [18] Imanuvilov, O. Y.; Puel, J.-P., Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Internat. Math. Res. Notices, 16, 883-913 (2003) · Zbl 1146.35340 [19] Inoue, A.; Wakimoto, M., On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24, 2, 303-319 (1997) · Zbl 0381.35066 [20] Judakov, N. V., The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid, Dinamika Splošn. Sredy, Dinamika Zidkost. so Svobod. Granicami, 18, 249-253 (1974), 255 [21] Munnier, A.; Zuazua, E., Large time behavior for a simplified \(N\)-dimensional model of fluid-solid interaction, Comm. Partial Differential Equations, 30, 1-3, 377-417 (2005) · Zbl 1080.35088 [22] Ortega, J.; Rosier, L.; Takahashi, T., Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid, M2AN Math. Model. Numer. Anal., 39, 1, 79-108 (2005) · Zbl 1087.35081 [23] Raymond, J.-P.; Vanninathan, M., Exact controllability in fluid-solid structure: the Helmholtz model, ESAIM Control Optim. Calc. Var., 11, 2, 180-203 (2005), (electronic) · Zbl 1125.93007 [24] San Martín H., J.; Starovoitov, V.; Tucsnak, M., Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Rational Mech. Anal., 161, 2, 113-147 (2002) · Zbl 1018.76012 [25] Serre, D., Chute libre d’un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4, 1, 99-110 (1987) · Zbl 0655.76022 [26] Silvestre, A., 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, 4, 285-326 (2002) · Zbl 1022.35041 [27] 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, 12, 1499-1532 (2003) · Zbl 1101.35356 [28] Takahashi, T.; Tucsnak, M., Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6, 1, 53-77 (2004) · Zbl 1054.35061 [29] Vázquez, J.; Zuazua, E., Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations, 28, 9-10, 1705-1738 (2003) · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.