  We study the Stokes system with the localized boundary data in the
half-space. We are concerned with the local regularity of its solution near the
boundary away from the support of the given boundary data which are product
forms of each spatial variable and the temporal variable. We first show that if
the boundary data are smooth in time, the corresponding solutions are also
smooth in space and time near the boundary, even if the boundary data are only
spatially integrable. Secondly, if the normal component of the boundary data is
absent, we are able to construct a solution such that its second normal
derivatives of the tangential components become singular near the boundary.
Perturbation argument enables us to construct solutions of the Navier-Stokes
equations with similar singular behaviors near the boundary in the half-space
as the case of Stokes system. Lastly, we provide specific types of the
localized boundary data to obtain the pointwise bounds of the solutions up to
second derivatives. It turns out that such solutions are globally strong, and
the second normal derivatives are, however, unbounded near the boundary. These
results can be compared to the previous works in which only the normal
component is present. In fact, the temporally non-smooth tangential boundary
data can also cause spatially singular behaviors near the boundary, although
such behaviors are milder than those caused by the normal boundary data of the
same type.
