Conormal derivative problems for stationary Stokes system in Sobolev spaces

TL;DR

This paper establishes the solvability of conormal derivative problems for the stationary Stokes system in Sobolev spaces on Reifenberg flat domains with irregular coefficients, advancing the mathematical understanding of fluid dynamics in complex geometries.

Contribution

It introduces new solvability results for the Stokes system with irregular coefficients in Sobolev spaces on Reifenberg flat domains, utilizing a novel local Poincaré inequality.

Findings

Proved solvability of the conormal derivative problem in Sobolev spaces.

Developed a local Poincaré inequality for Reifenberg flat domains.

Extended analysis to systems with coefficients measurable in one direction.

Abstract

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincar\'e inequality on Reifenberg flat domains, the proof of which is of independent interest.