Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 214

with c 4 >0independent of n and k. The theorem then follows from ( 3.44 )to( 3.47 )
with the help of Plancherel theorem. t

Remark 3.3. A significant consequence of the argument adopted in the proof of
the previous theorem is that, under the stated periodic boundary conditions, even
though—in absence of external loads and heat sources—the total energy of the
elastic body is strictly decreasing, the energy of the “divergence-free” part of the
displacement field remains constant in time . This can be immediately seen from
( 3.37 ) with F 0, ‰ 0 which after some simple manipulation furnishes

d E 1

dt

d
dt Πkcurlv t .t/k 2 C kr.curlv.t//k 2 D 0;

1
2

1
2

dt kr' t .t/k

2 Dkr.t/k

d E 2

dt

d

1
2

1
2

2 C . C 2/k'.t/k 2 Ck.t/k 2 :

Notice also that, since one can prove that E 2 .t/ ! 0 as t !1, the energy remains

stored only in the rotational part of the displacement field. As we shall see in details
in the next section, “stored energy” is one of the main reasons (but not the only one)
for the occurrence of resonance.

Remark 3.4. A result qualitatively analogous to that of Theorem 3.2 could be
proved also in the case of a bounded and sufficiently smooth domain , provided
we use appropriate boundary conditions on the displacement field u , as proposed in
[ 32 ]; see also [ 42 ]. The latter require that on @ the tangential component of u as
well as div u identically vanish.

3.4 An Abstract Approach

From what we have seen in the previous section by analyzing the linear ther-
moelastic model, we may state that, roughly speaking, resonance in a mechanical
system does not occur if and only if whatever the period of the applied driving
mechanism can be, the system allows for a corresponding time-periodic motion of
the same period. As a consequence, in order to furnish a general approach to the
problem, it seems appropriate to investigate existence of time-periodic solutions
for a sufficiently broad class of linear evolution equations, that include the relevant
hyperbolic-parabolic systems as a special case. This will be the object of the present
section, where we shall analyze this question with the help of the mean ergodic
theorem and some of its main consequences, within a class of problems whose
dynamics is governed by a strongly continuous semigroup of contractions.
Let X be a reflexive Banach space with associated norm kk,andletX denote
its dual with norm kk . For a given operator A W X 7! X we indicate by D .A/ its
domain of definition and by R .A/ its range. Moreover, we set N .A/ WD fx 2 X W
Ax D 0g. The identity operator on X is indicated by I.
Let f W R 7! X be an assigned periodic function of period T>0.Our
objective is to provide necessary and sufficient conditions for the occurrence of