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We study the initial-boundary value problem of the Navier-Stokes equationsfor incompressible fluids in a domain in $\R^3$ with compact and smoothboundary, subject to the kinematic and Navier boundary conditions. We firstreformulate the Navier boundary condition in terms of the vorticity, which ismotivated by the Hodge theory on manifolds with boundary from the viewpoint ofdifferential geometry, and establish basic elliptic estimates for vector fieldssubject to the kinematic and Navier boundary conditions. Then we develop aspectral theory of the Stokes operator acting on divergence-free vector fieldson a domain with the kinematic and Navier boundary conditions. Finally, weemploy the spectral theory and the necessary estimates to construct theGalerkin approximate solutions and establish their convergence to global weaksolutions, 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 incompressibleNavier-Stokes equations subject to the no-slip boundary condition for almostall time. The inviscid limit of the strong solutions to the unique solutions ofthe initial-boundary value problem with the slip boundary condition for theEuler equations is also established.

A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions