  We study the initial-boundary value problem of the Navier-Stokes equations
for incompressible fluids in a domain in $\R^3$ with compact and smooth
boundary, subject to the kinematic and Navier boundary conditions. We first
reformulate the Navier boundary condition in terms of the vorticity, which is
motivated by the Hodge theory on manifolds with boundary from the viewpoint of
differential geometry, and establish basic elliptic estimates for vector fields
subject to the kinematic and Navier boundary conditions. Then we develop a
spectral theory of the Stokes operator acting on divergence-free vector fields
on a domain with the kinematic and Navier boundary conditions. Finally, we
employ the spectral theory and the necessary estimates to construct the
Galerkin approximate solutions and establish their convergence to global weak
solutions, as well as local strong solutions, of the initial-boundary problem.
Furthermore, we show as a corollary that, when the slip length tends to zero,
the weak solutions constructed converge to a solution to the incompressible
Navier-Stokes equations subject to the no-slip boundary condition for almost
all time. The inviscid limit of the strong solutions to the unique solutions of
the initial-boundary value problem with the slip boundary condition for the
Euler equations is also established.
