Maximum modulus estimate for the solution of the nonstationary Stokes equations

TL;DR

This paper establishes a maximum modulus estimate for solutions to the nonstationary Stokes equations in smooth domains, analyzing boundary kernel components and boundary data conditions to bound velocity behavior.

Contribution

It provides a novel maximum modulus estimate for nonstationary Stokes solutions, analyzing boundary kernel singularities and conditions for velocity boundedness.

Findings

Normal velocity is bounded near the boundary if boundary data is bounded.

Maximum velocity modulus is bounded in the domain under Dini continuity of boundary data.

Analysis of singular and regular parts of the Poisson kernel enhances understanding of boundary effects.

Abstract

A maximum modulus estimate for the nonstationary Stokes equations in $C^2$ domain is found. The singular part and regular part of Poisson kernel are analyzed. The singular part consists of the gradient of single layer potential and the gradient of composite potential defined on only normal component of the boundary data. Furthermore, the normal velocity near the boundary is bounded if the boundary data is bounded. If the normal component of the boundary data is Dini-continuous and the tangential component of the boundary data is bounded, then the maximum modulus of velocity is bounded in whole domain.