  We study the interaction of an incompressible fluid in two dimensions with an
elastic structure yielding the moving boundary of the physical domain. The
displacement of the structure is described by a linear viscoelastic beam
equation. Our main result is the existence of a unique global strong solution.
Previously, only the ideal case of a flat reference geometry was considered
such that the structure can only move in vertical direction. We allow for a
general geometric set-up, were the structure can even occupy the complete
boundary.
  Our main tool -- being of independent interest -- is a maximal regularity
estimate for the steady Stokes system in domains with minimal boundary
regularity. In particular, we can control the velocity in $W^{2,2}$ in terms of
a forcing in $L^2$ provided the boundary belongs roughly to $W^{3/2,2}$. This
is applied to the momentum equation in the moving domain (for a fixed time)
with the material derivative as right-hand side. Since the moving boundary
belongs a priori only to the class $W^{2,2}$, known results do not apply here
as they require a $C^2$-boundary.
