  In this paper, we extend the maximum modulus estimate of the solutions of the
nonstationary Stokes equations in the bounded $C^2$ cylinders for the space
variables in \cite{CC} to time estimate. We show that if the boundary data is
$L^\infty$ and the normal part of the boundary data has log-Dini continuity
with respect to time, then the velocity is bounded. We emphasize that there is
no continuity assumption on space variables in the new maximum modulus
estimate. This completes the maximum modulus estimate.
