TY - JOUR

T1 - A unified approach to determining forms for the 2D Navier-Stokes equations - The general interpolants case

AU - Foias, C.

AU - Jolly, M. S.

AU - Kravchenko, R.

AU - Titi, E. S.

PY - 2014

Y1 - 2014

N2 - 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 C1b (R;RN) 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.

AB - 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 C1b (R;RN) 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.

KW - Determining forms

KW - Determining modes

KW - Dissipative dynamical systems

KW - Inertial manifold

KW - Navier-Stokes equation

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U2 - 10.1070/RM2014v069n02ABEH004891

DO - 10.1070/RM2014v069n02ABEH004891

M3 - Review article

AN - SCOPUS:84904336291

VL - 69

SP - 359

EP - 381

JO - Russian Mathematical Surveys

JF - Russian Mathematical Surveys

SN - 0036-0279

IS - 2

ER -