  In this paper, we study the inviscid limit of the free surface incompressible
Navier-Stokes equations with or without surface tension. By delicate estimates,
we prove the weak boundary layer of the velocity of the free surface
Navier-Stokes equations and the existence of strong or weak vorticity layer for
different conditions. When the limit of the difference between the initial
Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the
tangential projection on the free surface of the Euler strain tensor
multiplying by normal vector is nonzero, there exists a strong vorticity layer.
Otherwise, the vorticity layer is weak. We estimate convergence rates of
tangential derivatives and the first order standard normal derivative in energy
norms, we show that not only tangential derivatives and standard normal
derivative have different convergence rates, but also their convergence rates
are different for different Euler boundary data. Moreover, we determine
regularity structure of the free surface Navier-Stokes solutions with or
without surface tension, surface tension changes regularity structure of the
solutions.
