Navier and Stokes meet Poincar\' e and Dulac

TL;DR

This paper explores the long-term behavior of Navier-Stokes solutions with potential forces, revealing a surprising connection to Poincaré-Dulac normal forms and discussing related open problems.

Contribution

It uncovers a novel link between asymptotic expansions of Navier-Stokes solutions and Poincaré-Dulac normal forms, expanding understanding of their long-time dynamics.

Findings

Asymptotic expansions lead to Poincaré-Dulac normal forms.

Survey of long-time asymptotic results for Navier-Stokes.

Discussion of open problems in the field.

Abstract

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that that the asymptotic expansion leads surprisingly to a Poincar\' e-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open issues.