Estimates on the Stokes Pressure by Partitioning the Energy of Harmonic Functions

We show that in a tubular domain with sufficiently small width, the normal and tangential gradients of a harmonic function have almost the same L norm. This estimate yields a sharp estimate of the pressure in terms of the viscosity term in the Navier-Stokes equation with no-slip boundary condition. By consequence, one can analyze the NavierStokes equations simply as a perturbed vector diffusion equation instead of as a perturbed Stokes system. As an application, we describe a rather easy approach to establish a new isomorphism theorem for the non-homogeneous Stokes system.