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
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