On a constrained 2-D Navier-Stokes equation

TL;DR

This paper investigates a constrained 2-D Navier-Stokes equation to understand intermediate asymptotics and coherent structures, focusing on well-posedness and long-term behavior by neglecting energy and inertia variations.

Contribution

It introduces a novel constrained model of the 2-D Navier-Stokes equation that isolates specific dynamics related to intermediate asymptotics.

Findings

The constrained equation is well-posed under certain conditions.

It exhibits non-trivial intermediate asymptotic behavior.

The model helps explain the emergence of coherent structures.

Abstract

The planar Navier-Stokes equation exhibits, in absence of external forces, a trivial asymptotics in time. Nevertheless the appearence of coherent structures suggests non-trivial intermediate asymptotics which should be explained in terms of the equation itself. Motivated by the separation of the different time scales observed in the dynamics of the Navier-Stokes equation, we study the well-posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia.