Navier-Stokes equations, determining forms, determining modes, inertial manifolds, dissipative dynamical systems

TL;DR

This paper introduces a new globally Lipschitz ODE called the determining form for the 2D Navier-Stokes equations, capturing the long-term dynamics and attractor projections via determining modes.

Contribution

It develops a novel infinite-dimensional ODE framework for the Navier-Stokes equations that encapsulates their long-term behavior and attractor structure.

Findings

Determining form is a globally Lipschitz ODE in a Banach space.

Traveling wave solutions correspond to projections of NSE attractor solutions.

The determining form is dissipative with an explicit absorbing ball estimate.

Abstract

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named {\it determining form}, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form $v(t,s)=v_0(t+s)$, correspond exactly to initial data $v_0$ that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter). Finally, a unified approach is outlined for an ODE satisfied by a variety of other determining parameters such as nodal values, finite volumes, and finite elements.