J.-G. Liu, Jie Liu and R. Pego, *Estimates on the Stokes pressure by partitioning the energy of harmonic functions*,
in Kyoto Conference on the Navier-Stokes equations and their Applications, edited by Y. Giga, H. Kozono, H. Okamoto and Y. Shibta
(2007),
pp. 251--270, Kyoto Univ. .
**Abstract:**

*We show that in a tubular domain with sufficiently small width, the normal and tangential gradients of a harmonic
function have almost the same L2 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 Navier-
Stokes 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.*