  We consider the stationary Navier-Stokes equations in a three-dimensional
curved thin domain around a given closed surface under the slip boundary
conditions. Our aim is to show that a solution to the bulk equations is
approximated by a solution to limit equations on the surface appearing in the
thin-film limit of the bulk equations. To this end, we take the average of the
bulk solution in the thin direction and estimate the difference of the averaged
bulk solution and the surface solution. Then we combine an obtained difference
estimate on the surface with an estimate for the difference of the bulk
solution and its average to get a difference estimate for the bulk and surface
solutions in the thin domain, which shows that the bulk solution is
approximated by the surface one when the thickness of the thin domain is
sufficiently small.
