Strong solutions of the incompressible Navier-Stokes equations in external domains: local existence and uniqueness

TL;DR

This paper establishes local existence and uniqueness of strong solutions to the incompressible Navier-Stokes equations in both bounded and unbounded external domains, using a novel inverse kinetic theory approach that does not require decay at infinity.

Contribution

It introduces a new inverse kinetic theory framework to prove local existence and uniqueness of strong solutions for INSE in unbounded domains without decay assumptions.

Findings

Proves local existence and uniqueness of strong solutions in external domains.

Develops a kinetic equation leading to a Navier-Stokes dynamical system.

Handles solutions that do not vanish at infinity.

Abstract

In this paper the problem of strong solvability of the incompressible Navier-Stokes equations (INSE) is revisited, with the goal of determining the minimal assumptions for the validity of a local existence and uniqueness theorem for the Navier-Stokes fluid fields (solutions of INSE). Emphasis is placed on fluid fields which, together with suitable derivatives, do not necessarily decay at infinity and hence do not belong to Sobolev spaces. For this purpose a novel approach based on a so-called inverse kinetic theory, recently developed by Tessarotto and Ellero, is adopted. This requires the construction of a suitable kinetic equation, advancing in time a suitably smooth kinetic distribution function and providing exactly, as its moment equations, the complete set of fluid equations. In turn, by proper definition of the kinetic equation, this permits the introduction of the so-called \textit{Navier-Stokes dynamical system}, i.e., the dynamical system which advances in time self-consistently the Navier-Stokes fluid fields. Investigation of the properties of this dynamical system is crucial for the establishment of an existence and uniqueness theorem for strong solutions of INSE. The new theorem applies both to bounded and unbounded domains and in the presence of generalized boundaries, represented by surfaces, curves or even sets of isolated points. In particular, for unbounded domains, solutions are considered, which do not necessarily vanish at infinity. Basic consequences for the functional setting of classical solutions are analyzed. \keywords{Navier-Stokes equations \and Kinetic theory \and Dynamical systems} PACS 47.10.ad,47.10.Fg,PACS 47.10.A- MSC 76D03,76D06