CHAPTER 1 Introduction Consider an elastic solid B fully submerged in a viscous liquid L that occupies the whole space outside B. We suppose the motion of the liquid is governed by the Navier-Stokes equations, and that the elastic solid is a St.Venant-Kirchhoff material. We are interested in the mathematical analysis of certain motions of the coupled system solid-liquid S. Consequently, the governing equations consist of a Navier- Stokes system coupled with a nonlinear system of elasticity equations. Since no constraints are imposed on the motion of B, the boundary values correspond to those of a free traction problem.1 We shall assume that, with respect to an inertial frame I, B moves under the action of a given constant body force b. No constraints are enforced on the motion of B, namely, the solid moves freely. Since b does not depend on time, it is natural to wonder whether there exists a frame F with respect to which B may be in equilibrium, that is, the deformation Φ evaluated in F from a stress-free reference configuration Ω is time independent. However, the liquid is also producing a traction on the surface of B, which, therefore, should be time-independent as well. This will be certainly the case if, with respect to F, the motion of the liquid is steady. If such a frame F exists, the corresponding motion of S in F will be referred to as steady motion. Besides their intrinsic mathematical interest, steady motions may be also viewed as “terminal states” of several important phenomena. A simple but significant ex- ample is the free fall of a deformable body in a viscous liquid when the density of the liquid is much smaller than that of the solid see Remark 3.5. A more so- phisticated example is furnished by the towing of airborne or underwater bodies, such as antennas, banners, gliders, and targets, by powered aircraft or boats see for example [10, 49]. Objective of this paper is to prove the existence of steady motions. More precisely, under suitable smoothness and geometric assumptions on the reference configuration—that will be described subsequently—we show that there is always a smooth steady motion, provided an appropriate dimensionless number, involving the density and the Lam´ e coeﬃcients of the solid, the shear viscosity coeﬃcient of the liquid, and the magnitude of b, see (3.30), is below a certain constant. Before detailing the basic assumptions, the methods, and the significant fea- tures of our approach, we would like first to comment on the physical character of our results. Specifically, the most general steady motion of S that we obtain can be described, with respect to the inertial frame I, as follows. The solid rotates, with constant angular velocity directed along b, while its center of mass describes, with 1 Also called pure traction problem, or simply traction problem. 1

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