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Time-dependent free surface problem for the incompressible Navier-Stokesequations which describes the motion of viscous incompressible fluid nearlyhalf-space are considered. We obtain global well-posedness of the problem for asmall initial data in scale invariant critical Besov spaces. Our proof is basedon maximal $L^1$-regularity of the corresponding Stokes problem in thehalf-space and special structures of the quasi-linear term appearing from theLagrangian transform of the coordinate.

Maximal $L^1$-regularity and free boundary problems for the incompressible Navier-Stokes equations in critical spaces