4 1. INTRODUCTION

is an important contribution as it delivers existence of a solution to the free traction

problem with a physically meaningful live load, namely the surface forces applied

to the elastic body due to the fluid flow past it, forces that naturally depend on

the deformation of the body.

We would like to end this introductory chapter by quoting some related lit-

erature. Though the mathematical study of the interaction between a Navier-

Stokes liquid and elastic structures is a relatively young branch of analysis, over

the last few years there have been many significant contributions. For the case

of time-independent problems, after the pioneering work of Antman and Lanza

de Cristoforis [2, 28, 29], there has been an increasingly new interest. Among

the most relevant, we refer the reader to the articles [34, 24, 44] and to the ref-

erences cited therein. All the above works focus on a setting where the liquid is

contained in a (bounded) container with elastic walls. Recently, the (exterior) flow

of a Navier-Stokes liquid past an elastic body, fixed in space, has been studied in

[19], and some of the results there proved will be often employed in this paper.

For results on analogous unsteady problems (in a bounded domain), we refer to

[25, 6, 11, 12]. A number of interesting applications can be found in [22, 21].

Finally, we wish to observe that for a rigid body an analysis similar to the one

performed in this paper is carried out in [48, 35].

The plan of the paper is as follows. After collecting, in Chapter 2, the main

notation and some preliminary results, in Chapter 3 we furnish the mathemat-

ical formulation of our problem along with a corresponding suitable non-dimen-

sionalization. In Chapter 4 we state our assumptions and the main result in the

case of “non-symmetric” reference configurations. We also outline there the strat-

egy of proof and furnish some examples of reference configurations that do and some

other that do not satisfy our assumptions. In Chapter 5 we study the approximat-

ing problem in the bounded domains EN mentioned previously. In particular, we

prove existence of solutions and associated estimates. As we already emphasized,

the crucial point in the proof is that the constants involved in these estimates are

independent of N. Thanks to this latter, in Chapter 6, we may pass to the limit

N → ∞ and prove that the “approximating” solutions converge to a solution to

the original problem. The main theorem of Chapter 6 leaves out the case of refer-

ence configurations that possess rotational symmetry, like multi-bladed propellers,

spheroids, etc. This problem is taken up in Chapter 7, where we show existence (for

small data) of steady solutions also in the case of reference configurations having

rotational symmetry. Actually, in such a situation, the body force b is allowed to

have a (suitable) spatial dependence.