Quasi-maximum modulus principle for the Stokes equations

TL;DR

This paper extends maximum modulus estimates for solutions of the nonstationary Stokes equations to include time estimates, showing bounded velocity under specific boundary data conditions without requiring spatial continuity.

Contribution

It introduces a novel maximum modulus estimate for the nonstationary Stokes equations that incorporates time estimates and relaxes spatial continuity assumptions.

Findings

Velocity remains bounded with $L^$ boundary data and log-Dini continuity in time.

No spatial continuity assumption is needed for the maximum modulus estimate.

Completes the theoretical framework for maximum modulus estimates in nonstationary Stokes equations.

Abstract

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.