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

C. Foias ; M. S. Jolly ; R. Kravchenko ; E. S. Titi

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 2, pp. 359-381

Résumé

It is shown that the long-time dynamics (the global attractor) of the 2D Navier–Stokes system is embedded in the long-time dynamics of an ordinary differential equation, called a determining form, in a space of trajectories which is isomorphic to $C^1_b(\mathbb{R};\mathbb{R}^N)$ for sufficiently large $N$ depending on the physical parameters of the Navier–Stokes equations. A unified approach is presented, based on interpolant operators constructed from various determining parameters for the Navier–Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, and so on. There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, and thus its solutions converge to the set of steady states of the determining form as the time goes to infinity. The second is that these steady states of the determining form can be uniquely identified with the trajectories in the global attractor of the Navier–Stokes system. It should be added that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems. Bibliography: 23 titles.

Détail

Keywords: Navier–Stokes equation, inertial manifold, determining forms, determining modes, dissipative dynamical systems.