# American Institute of Mathematical Sciences

September  2014, 19(7): 2353-2364. doi: 10.3934/dcdsb.2014.19.2353

## Mathematical study of the small oscillations of a floating body in a bounded tank containing an incompressible viscous liquid

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria-Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome, Italy 2 Professor Emeritus of Theoretical Mechanics, University of Franche-Compté, 2 B rue des Jardins, F - 25000 Besançon, France

Received  March 2013 Revised  August 2013 Published  August 2014

The authors study the small oscillations of a floating body in a bounded tank containing an incompressible viscous fluid.
Using the variational formulation of the problem, they obtain an operator equation from which they can study the spectrum of the problem.
The small motions are strongly and weakly damped aperiodic motions and, if the viscosity is sufficiently small, there is also at most finite number of damped oscillatory motions.
The authors give also an existence and uniqueness theorem for the solution of the associated evolution problem.
Citation: Doretta Vivona, Pierre Capodanno. Mathematical study of the small oscillations of a floating body in a bounded tank containing an incompressible viscous liquid. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2353-2364. doi: 10.3934/dcdsb.2014.19.2353
##### References:
 [1] P. Appell, Traité de Mécanique Rationelle, 3 (1952) Gauthier Villars -Paris. [2] N. K. Askerov, S. G. Krein and G. I. Laptev, The problem on oscillations of a viscous fluid and related operator equations, Functional Analysis and Its applications, 2 (1969), 21-32. [3] R. Dautray and J. L. Lion, Mathematical Analysis and Numerical Methods for Science and Technology, 8, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61566-5. [4] N. D. Kopachevskii and S. G. Krein, Operator Approach in Linear Problems of Hydrodynamics, 128 Birkhäuser-Verlag, 2001. doi: 10.1007/978-3-0348-8342-9. [5] N. D. Kopachevskii and S. G. Krein, Operator Approach in Linear Problems of Hydrodynamics, 146 Birkhäuser-Verlag, 2003. doi: 10.1007/978-3-0348-8063-3. [6] S. G. Krein, Oscillations of a viscous fluid in a container, DAN SSR, 159 (1964), 262-265. [7] J. L. Lions, Equations Differentielle Opérationelles et Problémes Aux Limites, Springer-Verlag, Berlin, 1961. [8] N. N. Moiseyev, On the oscillations of a body floating in a bounded volume of fluid, Moskov. Fiz Tekh. Inst. Issled. Mekh. Prikl. Mat., 1 (1958), 145-166 (in Russian). [9] N. N. Moiseyev and V. V. Rumiantsev, Dynamic Stability of Bodies Containing Fluid, Springer-Verlag, Berlin, 1968. [10] F. Riesz and B. Nagy, Leçons D'analyse Fonctionelle, Gauthier Villars -Paris, Gauthier Villars -Paris, 1968.

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##### References:
 [1] P. Appell, Traité de Mécanique Rationelle, 3 (1952) Gauthier Villars -Paris. [2] N. K. Askerov, S. G. Krein and G. I. Laptev, The problem on oscillations of a viscous fluid and related operator equations, Functional Analysis and Its applications, 2 (1969), 21-32. [3] R. Dautray and J. L. Lion, Mathematical Analysis and Numerical Methods for Science and Technology, 8, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61566-5. [4] N. D. Kopachevskii and S. G. Krein, Operator Approach in Linear Problems of Hydrodynamics, 128 Birkhäuser-Verlag, 2001. doi: 10.1007/978-3-0348-8342-9. [5] N. D. Kopachevskii and S. G. Krein, Operator Approach in Linear Problems of Hydrodynamics, 146 Birkhäuser-Verlag, 2003. doi: 10.1007/978-3-0348-8063-3. [6] S. G. Krein, Oscillations of a viscous fluid in a container, DAN SSR, 159 (1964), 262-265. [7] J. L. Lions, Equations Differentielle Opérationelles et Problémes Aux Limites, Springer-Verlag, Berlin, 1961. [8] N. N. Moiseyev, On the oscillations of a body floating in a bounded volume of fluid, Moskov. Fiz Tekh. Inst. Issled. Mekh. Prikl. Mat., 1 (1958), 145-166 (in Russian). [9] N. N. Moiseyev and V. V. Rumiantsev, Dynamic Stability of Bodies Containing Fluid, Springer-Verlag, Berlin, 1968. [10] F. Riesz and B. Nagy, Leçons D'analyse Fonctionelle, Gauthier Villars -Paris, Gauthier Villars -Paris, 1968.
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