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We prove that there exists a weak solution of the Stokes system with anon-zero external force and no-slip boundary conditions in a half space ofdimensions three and higher so that its normal derivatives are unbounded nearboundary. A localized and divergence free singular force causes, via non-localeffect, singular behaviors of normal derivatives for the solution nearboundary, although such boundary is away from the support of the externalforce. 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 ashear flow that is of only locally finite energy. Similar construction isperformed for the Navier-Stokes equations as well.

Singular weak solutions near boundaries in a half space away from localized force for the Stokes and Navier-Stokes equations