  We investigate the Navier-Stokes initial boundary value problem in the
half-plane $R^2_+$ with initial data $u_0 \in L^\infty(R^2_+)\cap J_0^2(R^2_+)$
or with non decaying initial data $u_0\in L^\infty(R^2_+) \cap J_0^p(R^2_+), p
> 2$ . We introduce a technique that allows to solve the two-dimesional
problem, further, but not least, it can be also employed to obtain weak
solutions, as regards the non decaying initial data, to the three-dimensional
Navier-Stokes IBVP. This last result is the first of its kind.
