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Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid. (English) Zbl 1171.74337

The three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid is under consideration. The elastic structure and the fluid are contained in a fixed bounded connected set. The existence of a weak solution for regularized elastic deformations is showed as long as elastic deformations are not too important in order to avoid interpenetration and preserve orientation on the structure and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural to consider that its rigid motion (translation and rotation) may be large. The presented existence result has been announced in [1] M. Boulakia, Existence of a weak solution for the motion of an elastic structure in an incompressible viscous fluid, [C. R. Math. Akad. Sci. Paris 336, No 12, 985–990 (2003; Zbl 1129.74306)]. But the model considered in [1] is a simplified model where the structure motion is modeled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. Here some improvements have been provided on the model. On the structure it is considered a model which represents the motion of a structure with large rigid displacement and small elastic perturbations. Such model, introduced by C. Grandmont, Y. Maday and P. Métier, Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation, [C. R. Math. Akad. Sci. Paris 334, No. 6, 521–526 (2002; Zbl 1035.74027)] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35Q30 Navier-Stokes equations
37N15 Dynamical systems in solid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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