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A remarkable feature of fluid dynamics is its relationship with classicaldynamics and statistical mechanics. This has motivated in the past mathematicalinvestigations concerning, in a special way, the "derivation" based on kinetictheory, and in particular the Boltzmann equation, of the incompressibleNavier-Stokes equations (INSE). However, the connection determined in this wayis usually merely asymptotic (i.e., it can be reached only for suitable limitfunctions) and therefore presents difficulties of its own. This feature hassuggested the search of an alternative approach, based on the construction of asuitable inverse kinetic theory (IKT; Tessarotto et al., 2004-2007), which canavoid them. IKT, in fact, permits to achieve an exact representation of thefluid equations by identifying them with appropriate moment equations of asuitable (inverse) kinetic equation. The latter can be identified with aLiouville equation advancing in time a phase-space probability density function(PDF), in terms of which the complete set of fluid fields (prescribing thestate of the fluid) are determined. In this paper we intend to investigate themathematical properties of the underlying \textit{finite-dimensional}phase-space classical dynamical system, denoted \textit{Navier-Stokes dynamicalsystem}, which can be established in this way. The result we intend toestablish has fundamental implications both for the mathematical investigationof Navier-Stokes equations as well as for diverse consequences and applicationsin fluid dynamics and applied sciences.

Mathematical properties of the Navier-Stokes dynamical system for incompressible Newtonian fluids