The pressure term has always created difficulties in treating the
Navier-Stokes equations of incompressible flow, reflected in the
lack of a useful evolution equation or boundary conditions to
determine it. In joint work with Bob Pego and Jie Liu, we
show that in bounded domains with no-slip boundary conditions,
the Navier-Stokes pressure can be determined in a such way that
it is strictly dominated by viscosity. As a consequence, in a
general domain with no-slip boundary conditions, we can treat the
Navier-Stokes equations as a perturbed vector diffusion equation
instead of as a perturbed Stokes system. We illustrate the
advantages of this view by providing simple proofs of (i) the
stability of a difference scheme that is implicit only in
viscosity and explicit in both pressure and convection terms,
requiring no solutions of stationary Stokes systems or inf-sup
conditions, and (ii) existence and uniqueness of strong solutions
based on the difference scheme.

A preprint is available at http://arxiv.org/abs/math.AP/0502549