  This paper develops a unified approach to show the maximal regularity theorem
of the Stokes equations with free boundary conditions in the half-space
$\mathbb R^d_+$, $d \ge 2$, within the $L_1$-in-time and $\mathcal B^s_{q,
1}$-in-space framework with $(q, s)$ satisfying $1 < q < \infty$ and $- 1 + 1 /
q < s < 1 / q$, where $\mathcal B^s_{q, 1}$ stands for either homogeneous or
inhomogeneous Besov spaces. In particular, we establish a generalized semigroup
theory within an $L_1$-in-time and $\mathcal B^s_{q,1}$-in-space framework,
which extends a classical $C_0$-analytic semigroup theory to the case of
inhomogeneous boundary conditions. The maximal $L_1$-regularity theorem is
proved by estimating the Fourier-Laplace inverse transform of the solution to
the generalized Stokes resolvent problem with inhomogeneous boundary
conditions, where density and interpolation arguments are used. The maximal
$L_1$-regularity theorem is applied to show the unique existence of a local
strong solution to the Navier-Stokes equations with free boundary conditions
for arbitrary initial data $\boldsymbol a$ in $B^s_{q, 1} (\mathbb R^d_+)^d$,
where $q$ and $s$ satisfy $d-1 < q \le d$ and $-1+d/q < s < 1/q$, respectively.
If we assume that the initial data $\boldsymbol a$ are small in $\dot{B}^{- 1 +
d / q}_{q, 1} (\mathbb R^d_+)^d$, $d - 1 < q < 2 d$, then the unique existence
of a global strong solution to the system is proved.
