A Representation of the Solution of the Stokes Equations in the Half Space \(\mathbb {R}^{3}_{+}\): Application to Spatial and Temporal Estimates of the Pressure

A solution formula of the Stokes problem in the half space \(\mathbb {R}^{3}_{+}\) is obtained by focusing on the normal derivative of the pressure at the boundary. This explicit formula can be used to estimate the \(L^\infty \) norm of the pressure and its normal derivative at the boundary that quantifies the initial layer depending on regularity and compatibility conditions of the initial data.