  We study the initial-boundary value problem of the Navier-Stokes equations
for incompressible fluids in a general domain in $\R^n$ with compact and smooth
boundary, subject to the kinematic and vorticity boundary conditions on the
non-flat boundary. We observe that, under the nonhomogeneous boundary
conditions, the pressure $p$ can be still recovered by solving the Neumann
problem for the Poisson equation. Then we establish the well-posedness of the
unsteady Stokes equations and employ the solution to reduce our
initial-boundary value problem into an initial-boundary value problem with
absolute boundary conditions. Based on this, we first establish the
well-posedness for an appropriate local linearized problem with the absolute
boundary conditions and the initial condition (without the incompressibility
condition), which establishes a velocity mapping. Then we develop
\emph{apriori} estimates for the velocity mapping, especially involving the
Sobolev norm for the time-derivative of the mapping to deal with the
complicated boundary conditions, which leads to the existence of the fixed
point of the mapping and the existence of solutions to our initial-boundary
value problem. Finally, we establish that, when the viscosity coefficient tends
zero, the strong solutions of the initial-boundary value problem in $\R^n (n\ge
3)$ with nonhomogeneous vorticity boundary condition converges in $L^2$ to the
corresponding Euler equations satisfying the kinematic condition.
