  In this work, we investigate the existence and uniqueness properties of a
composite structure (multilayered) fluid interaction PDE system which arises in
multi-physics problems, and particularly in biofluidic applications related to
the mammalian blood transportation process. The PDE system under consideration
consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics
which gives rise to an additional 2D elastic equation on the boundary interface
between these 3D PDE systems.
  By means of a nonstandard mixed variational formulation we show that the PDE
system generates a $C_0$-semigroup on the associated finite energy space of
data. In this work, the presence of the pressure term in the 3D Stokes equation
adds a great challenge to our analysis. To overcome this difficulty, we follow
a methodology which is based on the necessarily non-Leray based elimination of
the associated pressure term, via appropriate nonlocal operators. Moreover,
while we express the fluid solution variable via decoupling of the Stokes
equation, we construct the elastic solution variables by solving a mixed
variational formulation via a Babuska-Brezzi approach.
