Green functions of conormal derivative problems for stationary Stokes system

TL;DR

This paper investigates Green functions for stationary Stokes systems with conormal boundary conditions, establishing their existence, uniqueness, estimates, and boundary bounds under certain regularity assumptions.

Contribution

It provides new results on the existence, uniqueness, and estimates of Green functions for the Stokes system with conormal boundary conditions, including boundary bounds under additional assumptions.

Findings

Established existence and uniqueness of Green functions.

Derived global pointwise bounds for Green functions.

Provided examples satisfying regularity and boundedness conditions.

Abstract

We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.