# Green functions for pressure of Stokes systems

## TL;DR

This paper constructs and analyzes Green functions for the pressure in stationary Stokes systems with measurable and Dini mean oscillation coefficients, providing bounds and extending to flow velocity Green functions.

## Contribution

It introduces new construction methods for Green functions under minimal regularity conditions and establishes global bounds in complex domains.

## Key findings

- Constructed Green functions with measurable and Dini mean oscillation coefficients.
- Established global pointwise bounds for Green functions and derivatives.
- Extended analysis to Green functions for flow velocity in Stokes systems.

## Abstract

We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $\Omega\subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions, and $\Omega$ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $\Omega$ has a $C^{1,\rm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considered.