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

This paper develops a new 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 ${\mathcal{B}}_{q,1}^s$-in-space framework with $(q,s)$ satisfying $1<q<\mathrm{\infty }$ and $1+1\mathrm{/}q<s<1\mathrm{/}q$, where ${\mathcal{B}}_{q,1}^s$ stands for either homogeneous or inhomogeneous Besov spaces. In particular, we establish a generalized semigroup theory within an $L_1$-in-time and ${\mathcal{B}}_{q,1}^s$-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 $\mathit{a}$ in $B_{q,1}^s({\mathbb{R}}_{+}^d{)}^d$, where $q$ and $s$ satisfy $d-1<q\le d$ and $-1+d\mathrm{/}q<s<1\mathrm{/}q$, respectively. If we assume that the initial data $\mathit{a}$ are small in ${\dot{B}}_{q,1}^{1+d\mathrm{/}q}({\mathbb{R}}_{+}^d{)}^d$, $d1<q<2d$, then the unique existence of a global strong solution to the system is proved.