  We prove that there exists a weak solution of the Stokes system with a
non-zero external force and no-slip boundary conditions in a half space of
dimensions three and higher so that its normal derivatives are unbounded near
boundary. A localized and divergence free singular force causes, via non-local
effect, singular behaviors of normal derivatives for the solution near
boundary, although such boundary is away from the support of the external
force. The constructed one is a weak solution that has finite energy globally,
and it can be comparable to the one in \cite{Seregin-Sverak10} as a form of a
shear flow that is of only locally finite energy. Similar construction is
performed for the Navier-Stokes equations as well.
