On solutions of the 2D Navier-Stokes equations with constant energy and enstrophy

TL;DR

This paper investigates the theoretical existence and geometric properties of special solutions called ghost solutions in the 2D Navier-Stokes equations, providing conditions and computational methods to identify such solutions.

Contribution

It introduces geometric structures of ghost solutions, analyzes their stability, and proposes a computational approach to verify their existence.

Findings

Shared geometric structures of ghost solutions identified

Stability properties of a subclass of ghost solutions analyzed

A computational method to detect ghost solutions developed

Abstract

It is not yet known if the global attractor of the space periodic 2D Navier-Stokes equations contains nonstationary solutions $u(x,t)$ such that their energy and enstrophy per unit mass are constant for every $t \in (-\infty, \infty)$. The study of the properties of such solutions was initiated in \cite{CMM13}, where, due to the hypothetical existence of such solutions, they were called "ghost solutions". In this work, we introduce and study geometric structures shared by all ghost solutions. This study led us to consider a subclass of ghost solutions for which those geometric structures have a supplementary stability property. In particular, we show that the wave vectors of the active modes of this subclass of ghost solutions must satisfy certain supplementary constraints. We also found a computational way to check for the existence of these ghost solutions.