  Time-dependent free surface problem for the incompressible Navier-Stokes
equations which describes the motion of viscous incompressible fluid nearly
half-space are considered. We obtain global well-posedness of the problem for a
small initial data in scale invariant critical Besov spaces. Our proof is based
on maximal $L^1$-regularity of the corresponding Stokes problem in the
half-space and special structures of the quasi-linear term appearing from the
Lagrangian transform of the coordinate.
