Quasi-maximum modulus principle for the Stokes equations

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 $L^{\mathrm{\infty }}$ 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.