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

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.