  Navier-Stokes equations establish the hydrodynamical problem by definition.
The importance of these equations is quite natural to understand if we focus on
the role they assume in a large spectrum of dynamical problems which involve
'fluids'. Neverthless, they are an undeniable source of pure mathematical
problems in PDE's theory. The essential core of their formulation was primarily
well structured on the simple concept that the infinitesimal portions of a
continuous medium, which flows locally in some manner, must obey in a 'bounded'
domain to the same fundamental rules we use to describe the evolution of
isolated lagrangian systems, basically momentum and mass conservation laws, so
that the consequent architecture of the mathematical implant appears very clear
and understandable. Looking to the framework of the numerical solvers, taking
in care the richness of their differential structure and the correlated
existence of complex dynamics which are mathematically coherent, I try to put
in light in the most simple way the fundamental difficulty that arises when we
have to impose a reasonable 'initial values problem' in order to simulate
numerically well known fluid-dynamical scenarios, trying at the same time to
offer a possible method to avoid such an obstacle in determining simulation
parameters from which starting in respect of the essential Hadamard's point of
view of the Cauchy problem.
