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 C² cylinders for the space variables in Chang and Choe (J Differ Equ 254(7):2682–2704, 2013) to time estimate. We show that if the boundary data is L^∞ and the normal part of the boundary data has log-Dini continuity with respect to the 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.