A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case

TL;DR

This paper introduces a unified method using interpolant operators to embed the long-term dynamics of 2D Navier-Stokes equations into a finite-dimensional ODE framework, ensuring convergence to steady states.

Contribution

It develops a general interpolant-based approach to construct determining forms for 2D Navier-Stokes equations, linking global attractors to finite-dimensional ODEs.

Findings

Determining form has a Lyapunov function, ensuring convergence to steady states.

Steady states of the determining form correspond one-to-one with trajectories on the global attractor.

Method applies broadly to various dissipative dynamical systems.

Abstract

In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named {\it determining form}, in a space of trajectories which is isomorphic to $C^1_b(\bR; \bR^N)$, for $N$ large enough depending on the physical parameters of the Navier-Stokes equations. We present a unified approach based on interpolant operators that are induced by any of the determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, etc... There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, thus its solutions converge, as time goes to infinity, to the set of steady states of the determining form. The second is that these steady states of the determining form are identified, one-to-one, with the trajectories on the global attractor of the Navier-Stokes equations. It is worth adding that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.