  An important aspect of computational fluid dynamics is related to the
determination of the fluid pressure in isothermal incompressible fluids. In
particular this concerns the construction of an exact evolution equation for
the fluid pressure which replaces the Poisson equation and yields an algorithm
which is a Poisson solver, i.e., it permits to time-advance exactly the same
fluid pressure \textit{without solving the Poisson equation}% . In fact, the
incompressible Navier-Stokes equations represent a mixture of hyperbolic and
elliptic pde's, which are extremely hard to study both analytically and
numerically. In this paper we intend to show that an exact solution to this
problem can be achieved adopting the approach based on inverse kinetic theory
(IKT) recently developed for incompressible fluids by Ellero and Tessarotto
(2004-2007). In particular we intend to prove that the evolution of the fluid
fields can be achieved by means of a suitable dynamical system, to be
identified with the so-called Navier-Stokes (N-S) dynamical system. As a
consequence it is found that the fluid pressure obeys a well-defined evolution
equation. The result appears relevant for the construction of Lagrangian
approaches to fluid dynamics.
