  This paper is devoted to proving the existence of time-periodic solutions of
one-phase or two-phase problems for the Navier-Stokes equations with small
periodic external forces when the reference domain is close to a ball. Since
our problems are formulated in time-dependent unknown domains, the problems are
reduced to quasiliner systems of parabolic equations with non-homogeneous
boundary conditions or transmission conditions in fixed domains by using the
so-called Hanzawa transform. We separate solutions into the stationary part and
the oscillatory part. The linearized equations for the stationary part have
eigen-value $0$, which is avoided by changing the equations with the help of
the necessary conditions for the existence of solutions to the original
problems. To treat the oscillatory part, we establish the maximal $L_p$-$L_q$
regularity theorem of the periodic solutions for the system of parabolic
equations with non-homogeneous boundary conditions or transmission conditions,
which is obtained by the systematic use of $\mathcal{R}$-solvers to the
resolvent problem for the linearized equations and the transference theorem for
the $L_p$ boundedness of operator-valued Fourier multipliers. These approaches
are the novelty of this paper.
