# A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case

@article{Foias2012ADF, title={A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case}, author={Ciprian Foias and Michael S. Jolly and Rostyslav V. Kravchenko and Edriss S. Titi}, journal={Journal of Mathematical Physics}, year={2012}, volume={53}, pages={115623-115623} }

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form dv/dt = F(v), in the Banach space, X, of all bounded continuous functions of the variable s∈R with values in certain finite-dimensional linear space. This new evolution ODE, named 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…

## 37 Citations

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

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## References

SHOWING 1-10 OF 31 REFERENCES

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

- Mathematics
- 2014

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…

### Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems

- MathematicsMath. Comput.
- 1997

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equations on the finite-dimensional space of interpolant…

### Determination of the solutions of the Navier-Stokes equations by a set of nodal values

- Mathematics
- 1984

We consider the Navier-Stokes equations of a viscous incompresible fluid, and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions. The…

### Asymptotic numerical analysis for the Navier-Stokes equations, 1

- Mathematics
- 1982

Abstract : Our aim in this work is to show that, in a 'permanent regime', the behaviour of a viscous incompressible fluid can be, in principle, determined by the study of a finite number of modes. It…

### Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

- Mathematics
- 1989

In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial…

### Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations

- Mathematics
- 2005

AbstractWe examine how the global attractor
$$\mathcal {A}$$ of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time…

### Feedback Control of Nonlinear Dissipative Systems by Finite Determining Parameters - A Reaction-diffusion Paradigm

- Mathematics
- 2013

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the…