  The motion of an elastic solid inside of an incompressible viscous fluid is
ubiquitous in nature. Mathematically, such motion is described by a PDE system
that couples the parabolic and hyperbolic phases, the latter inducing a loss of
regularity which has left the basic question of existence open until now.
  In this paper, we prove the existence and uniqueness of such motions (locally
in time), when the elastic solid is the linear Kirchhoff elastic material. The
solution is found using a topological fixed-point theorem that requires the
analysis of a linear problem consisting of the coupling between the
time-dependent Navier-Stokes equations set in Lagrangian variables and the
linear equations of elastodynamics, for which we prove the existence of a
unique weak solution. We then establish the regularity of the weak solution;
this regularity is obtained in function spaces that scale in a hyperbolic
fashion in both the fluid and solid phases. Our functional framework is
optimal, and provides the a priori estimates necessary for us to employ our
fixed-point procedure.
