2 1. INTRODUCTION

constant speed, a circular helix whose axis is parallel to b. Depending on the sym-

metry properties of the reference configuration Ω, the rotational-helical motion of B

can degenerate in simpler ones. For example, if Ω possesses the rotational property

similar to that of a multi-bladed propeller, the helicoidal trajectory degenerates

into a straight line, and the center of mass of the solid will simply translate (with

constant speed) along the direction of b. If Ω has spherical symmetry, then the

angular velocity is zero and B will only translate (no spin) with constant speed. As

far as the liquid, we show that in all cases it exhibits a conical wake region whose

axis is also parallel to b.

We shall next give a brief sketch of our method and comment on the assump-

tions that we need to make it work. For a full description we refer the reader to

Chapter 4. The starting point is to formulate the relevant equations for the solid-

liquid system in the frame F, assuming it exists. However, before doing this, we

need to rewrite the liquid equations in the exterior of the (undeformed) reference

configuration. This is done by means of a suitable smooth transformation intro-

duced in [19]. After this first step is achieved, the next step consists of linearizing

the obtained equations and proving the existence of a unique solution that obeys

appropriate estimates. This will then allow us to apply Tychonov’s fixed-point the-

orem and to prove, finally, the existence of a solution to the original problem. It

must be observed that these linearized equations are coupled through the unknown

kinematic parameters characterizing the frame F, that is, the angular velocity ω

and translational velocity ξ. A further unknown is the direction b of b in F that,

in a steady motion, must satisfy the requirement

(1.1) ω ∧ b = 0.

Therefore, even though the equations of the elastic solid and of the viscous liq-

uid are linearized, the complete system that we have uniquely to solve for is still

non-linear, due to the presence of equation (1.1). By using classical theorems on

linearized elasticity and Stokes equations, we then show that the above unique

solvability reduces to that of a nonlinear eigenvalue problem in the eigenvalue λ

and corresponding eigenvector b, where ω = λ b. Even though we prove that this

eigenvalue problem always has at least one solution, (λ0,b0), for this solution to be

(locally) unique we must require that λ0 is simple, namely, of algebraic multiplicity

equal to 1. This implies, in particular, that there are no other eigenvectors in a

suﬃciently small neighborhood of b0. For this reason, borrowing a nomenclature

introduced by Weinberger in [48], we call b0 an isolated orientation; see Section

4.2. However, imposing the simplicity of λ0 translates into certain geometric re-

strictions on the reference configuration: roughly speaking, Ω should not possess

“too much symmetry”; see Section 4.2 for details and examples. This requirement

excludes significant reference configurations such as rotationally symmetric ones

or, more generally, those having the same rotational symmetry of multi-bladed pro-

pellers. For this reason, we treat these cases separately, and show, for them also,

the existence of corresponding steady motions, even in the case when the body force

is not constant; see Chapter 7.

We wish to emphasize that, to date, we do not know if steady motions exist for

an arbitrary (smooth) reference configuration, even for small data, of course, and

we leave it to the interested reader as an intriguing open question.

What we have described so far gives an idea of the main strategy that we shall

follow. However, to make this strategy work requires much more effort. The basic