  The interaction between a viscous fluid and an elastic solid is modeled by a
system of parabolic and hyperbolic equations, coupled to one another along the
moving material interface through the continuity of the velocity and traction
vectors. We prove the existence and uniqueness (locally in time) of strong
solutions in Sobolev spaces for quasilinear elastodynamics coupled to the
incompressible Navier-Stokes equations along a moving interface. Unlike our
approach for the case of linear elastodynamics, we cannot employ a fixed-point
argument on the nonlinear system itself, and are instead forced to regularize
it by a particular parabolic artificial viscosity term. We proceed to show that
with this specific regularization, we obtain a time interval of existence which
is independent of the artificial viscosity; together with a priori estimates,
we identify the global solution (in both phases), as well as the interface
motion, as a weak limit in srong norms of our sequence of regularized problems.
