Stokes flow with kinematic and dynamic boundary conditions

TL;DR

This paper reviews boundary value problems for the Stokes system in Lipschitz domains, focusing on mixed boundary conditions and establishing the uniqueness of both pressure and velocity solutions.

Contribution

It provides a detailed analysis of the first and second boundary value problems for the Stokes system with mixed boundary conditions, including necessary modifications to standard theory.

Findings

Pressure and velocity are proven to be unique under mixed boundary conditions.

The paper extends existing theory to accommodate Lipschitz domains with mixed boundary conditions.

Minor modifications to standard theory are introduced to handle the mixed boundary conditions.

Abstract

We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in $\mathbb{R}^n.$ Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the velocity on one part of the boundary while a Neumann condition for the stress tensor is imposed on the remaining part. Some minor modifications to the standard theory are therefore required. The most noteworthy result is that both pressure and velocity are unique.