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This paper develops a unified approach to show the maximal regularity theoremof 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 orinhomogeneous Besov spaces. In particular, we establish a generalized semigrouptheory 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 ofinhomogeneous boundary conditions. The maximal $L_1$-regularity theorem isproved by estimating the Fourier-Laplace inverse transform of the solution tothe generalized Stokes resolvent problem with inhomogeneous boundaryconditions, where density and interpolation arguments are used. The maximal$L_1$-regularity theorem is applied to show the unique existence of a localstrong solution to the Navier-Stokes equations with free boundary conditionsfor arbitrary initial data $\boldsymbol a$ in $B^s_{q, 1} (\mathbb R^d_+)^d$,where $q$ and $s$ satisfy $d-1 < q < \infty$ 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 theunique existence of a global strong solution to the system is proved.

Maximal $L_1$-regularity of the Navier-Stokes equations with free boundary conditions via a generalized semigroup theory