We consider a free surface problem of the incompressible Navier-Stokes system with non-flat initial surface. To obtain global well-posedness, we establish end-point maximal $L^1$-regularity for the initial-boundary value problems of the Stokes equations. The proof depends on the explicit expression of the fundamental integral kernel of the linearized Stokes equations and almost　orthogonal estimates with the space-time Littlewood-Paley dyadic decompositions. For nonlinear terms, we employ bilinear estimates both in the half-space and on the boundary. This talk is based on joint works with Prof. Takayoshi Ogawa (Tohoku Univ.)