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

TL;DR

This paper explores the mathematical properties of a finite-dimensional dynamical system associated with the Navier-Stokes equations, offering an exact kinetic theory approach that could impact fluid dynamics research.

Contribution

It introduces an inverse kinetic theory framework to analyze the Navier-Stokes equations as a finite-dimensional dynamical system, providing a new exact representation.

Findings

Establishment of a Navier-Stokes dynamical system via inverse kinetic theory

Implications for mathematical analysis of fluid equations

Potential applications in fluid dynamics and applied sciences

Abstract

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