An inverse kinetic theory applying specifically to incompressible Newtonian fluids which
permits us to avoid the N2 algorithmic complexity of the Poisson equation for the fluid
pressure is presented. The theory is based on the construction of a suitable kinetic equation in
phase space, which permits us to determine exactly the fluid equations by means of the velocity
moments of the kinetic distribution function. It is found that the fluid pressure can also be
determined as a moment of the distribution function without solving the Poisson equation, as
is usually required in direct solution methods for the incompressible fluid equations. Finally,
the dynamical system, underlying the incompressible Navier–Stokes equations and advancing
in time the fluid fields, has been also identified and proven to produce an unique set of fluid
equations
