  We study an incompressible viscous flow around an obstacle with an
oscillating boundary that moves by a translational periodic motion, and we show
existence of strong time-periodic solutions for small data in different
configurations: If the mean velocity of the body is zero, existence of
time-periodic solutions is provided within a framework of Sobolev functions
with isotropic pointwise decay. If the mean velocity is non-zero, this
framework can be adapted, but the spatial behavior of flow requires a setting
of anisotropically weighted spaces. In the latter case, we also establish
existence of solutions within an alternative framework of homogeneous Sobolev
spaces. These results are based on the time-periodic maximal regularity of the
associated linearizations, which is derived from suitable R-bounds for the
Stokes and Oseen resolvent problems. The pointwise estimates are deduced from
the associated time-periodic fundamental solutions.
