Green’s functions for the Stokes equations

Green’s functions are an indispensable tool in the study of boundary value problems and linear partial differential equations. In highly viscous fluid flows, the Navier-Stokes equations can be linearized to obtain the Stokes equations, which are amenable to Green’s function analysis. Due to its relevance in fluid dynamics, the Stokes equations are widely discussed in physics and engineering literature. Many of these resources use formal calculations and avoid rigorous mathematical justification. To address this gap in the literature, we will derive the fundamental solution of the Stokes equations in free space and prove some of its key properties by drawing from the theory of distributions. We will then find the Green’s function for the Stokes equations on a half-space by using the Papkovich-Neuber representation. The formula obtained is simpler and easier to implement compared to the classical solution originally developed by J.R. Blake.