Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 231

subject to the action of a constant magnetic field H 0 , and whose motion is governed
by the equations of magnetoelasticity. If we neglect the displacement current,
assume that the material is homogeneous and isotropic and, finally, disregard
nonlinear couplings, the above equations can be written as follows [ 6 , Chap. 9]

9
=

1
0 curlh H 0 C f

u tt u . C /r.div u / D

1

h t

0 curl curlh D curl. u t H 0 /
divh D 0

in .0; 1/;

;

(3.80)

where h is the magnetic field, >0is the conductivity of the material, and 0 >0
the magnetic permeability of the vacuum. We shall consider the above equation with
the following boundary conditions

u .x;t/ D 0; nh.x;t/ D 0; ncurlh.x;t/ D 0;.x;t/2 @.0; 1/; (3.81)

where n is the unit outer normal to @.
We shall assume that is of class C 2 and, for simplicity, also simply connected.
As in the thermoelastic case, also in the case at hand the relevant Eqs. ( 3.80 )-
( 3.81 ) are partially dissipative. In fact, if we set f 0, the total energy

2 k u t .t/k

2 ;

1 2

2 Ckh.t/k

2
2 C kr u k

2
2 C . C /kdiv u k

2

E WD

is shown to be a decreasing function of time as a consequence of the following
equation

d dt D 1

0 kcurlhk

2

2 :

The latter is easily established by dot-multiplying ( 3.80 ) 1 by u ,( 3.80 ) 2 by h and
integrating the resulting equations by parts over . Moreover, in [ 39 ]itisshown
that ( 3.80 )-( 3.81 ) generates a strongly continuous semigroup of contractions, U .t/,

in the (Hilbert) space, , of functions having finite energy, namely

H WD ǚ . u ; u t ;h/ 2 ŒH 0 ./ 3

H

H./ ;

ŒL 2 ./ 3 (3.82)

with

H./ WD ǚ h 2 L 2 ./ W divh D 0 h nj @ D 0 ;

(3.83)

(where the trace is meant in the sense of H 1=2 .@/). In [ 39 ] it is also shown that
E

.0/ < 1 and ; satisfy C ¤ 0 (in addition
to ( 3.3 )). This result provides a rigorous proof of the damping effect of the magnetic
field over the free vibration of the elastic material (within the model adopted). In
addition, in view of Theorem 3.4 and Corollary 3.1 it also ensures the existence of
T -periodic solutions corresponding to “generic” T -periodic loads f .

.t/ ! 0, provided only that E