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

The connection between fluid dynamics and classical statistical mechanics has motivated in the past mathematical investigations of the incompressible Navier–Stokes (NS) equations (INSE) by means of an asymptotic kinetic theory. This feature has suggested the search for possible alternative exact approaches, based on the construction of a suitable inverse kinetic theory (IKT), which can avoid the asymptotic character and the intrinsic mathematical difficulty of direct kinetic theories. In this paper the fundamental mathematical properties of the NS phase-space dynamical system underlying INSE and determined by IKT are investigated. In particular, an equivalence theorem with the INSE problem and a global existence theorem are proved to hold for the NS dynamical system.