Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 198

shell is completely filled up with a viscous liquid, and its height is much bigger than
the diameter of the drumhead.
From the mathematical viewpoint, investigating a problem by these simplified
models may have an important bearing on the resolution of the same problem
for the original complete model. For instance, strong solutions to liquid-structure
interaction problems are constructed around the solution to a set of “approximate
equations” obtained by disregarding the nonlinearity and fixing the interface as
the reference configuration for the structure; see, e.g., [ 7 , 19 ] and the bibliography
therein enclosed. The models we shall consider are exactly described by these
“approximate equations.”
There is a vast amount of literature dedicated to the mathematical analysis of
such models, too long to be included here. We limit ourselves to refer the reader to
[ 4 , 12 ] and the bibliography therein.
The first liquid-structure problem we investigate is presented in Sect. 3.5.3 and
consists of a viscous liquid (within the Stokes approximation) occupying a three-
dimensional bounded domain, F , and surrounding a linearized elastic material
occupying another bounded and simply connected domain S . Combining the
result of Sect. 3.4 with those of [ 4 ] regarding the existence of a strongly continuous
semigroup of contractions for the problem, we prove necessary and sufficient
conditions for the occurrence of resonance. More precisely we show that resonance
cannot “generically” occur if and only if S is such that a suitable overdetermined
boundary-value problem in S has only the trivial solution.
As we already remarked, in this type of problems the dissipative interaction
occurs at the interface liquid-structure, . In mathematical terms this can be
described as follows. Because of the adherence condition imposed on the liquid at
, the velocity fields of both liquid .v/ and structure . u t / must there coincide. Now,
by the effect of viscosity , the coupled system liquid-solid dissipates its energy at
arate D WD R F jD.v/j

2 ,whereD is the stretching tensor. By the trace theorem
and Korn's inequality, we can show, in particular,

D R j u t j

2 . As a result, the
liquid “induces” into the structure some sort of dissipation, but only at the interface,
which, however, need not diffuse inside the solid in such a way as to rule out the
occurrence of resonance.
In view of the above observations, one may wonder whether this type of
dissipation could prevent resonance at least in the case of “thin” structures, like
plate or membranes. The answer to this question is positive, and the corresponding
analysis is carried out in Sects. 3.5.4 and 3.5.5 . The typical situation here [ 12 , 37 ]
is that the liquid is contained in a simply connected, smooth three-dimensional
domain with a part, , of its boundary that is elastic. The displacement field
on is assumed to be directed in the normal direction. In the case flat, in [ 12 ]itis
shown, among other things, that the relevant system of equations defines a strongly
continuous semigroup of contractions that is also uniformly stable. Consequently,
we conclude the absence of resonance for this model (Theorem 3.12 ). Successively,
we analyze the case when may be “curved.” In this situation, the methods of
[ 12 ] are seemingly not applicable. Nevertheless, using some arguments developed