  In this paper, we establish the unique existence and some decay properties of
a global solution of a free boundary problem of the incompressible
Navier-Stokes equations in $L_p$ in time and $L_q$ in space framework in a
uniformly $H^2_\infty$ domain $\Omega\subset \BR^N$ for $N\geq4$. We assume the
unique solvability of the weak Dirichlet problem for the Poisson equation and
the $L_q$-$L_r$ estimates for the Stokes semigroup. The novelty of this paper
is that we do not assume the compactness of the boundary, which is essentially
used in the case of exterior domains proved by Shibata \cite{Shiba17CIME}. The
restriction $N\geq 4$ is required to deduce an estimate for the nonlinear term
$\bG(\bu)$ arising from $\divv\bv=0$. However, we establish the results in the
half space $\hsp$ for $N\geq 3$ by reducing the linearized problem to the
problem with $\bG=\fb0$, where $\bG$ is the right member corresponding to
$\bG(\bu)$.
