# The number of degrees of freedom for the 2D Navier-Stokes equation: a connection with Kraichnan's theory of turbulence

@inproceedings{Cheskidov2021TheNO, title={The number of degrees of freedom for the 2D Navier-Stokes equation: a connection with Kraichnan's theory of turbulence}, author={Alexey Cheskidov and Mimi Dai}, year={2021} }

The purpose of this paper is to estimate the number of degrees of freedom of solutions of the 2D NSE. More precisely, we prove that its mathematical analog, the number of determining modes, is bounded by the Kraichnan number squared κη, consistent with Kraichnan’s theory of two-dimensional turbulence [35]. The notion of determining modes was introduced by Foias and Prodi in the seminal work [25] where it was shown that high modes of a solution to the 2D NSE are controlled by low modes…

## One Citation

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## References

SHOWING 1-10 OF 46 REFERENCES

### A DETERMINING FORM FOR THE 2D NAVIER-STOKES EQUATIONS - THE FOURIER MODES CASE

- Mathematics
- 2012

. 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…

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

- Mathematics
- 2012

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,…

### 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…

### Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2019

Abstract Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a…

### Statistical Estimates for the Navier–Stokes Equations and the Kraichnan Theory of 2-D Fully Developed Turbulence

- Physics, Environmental Science
- 2002

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given…

### The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1991

§1. We shall denote by uα(P) = uα (x1, x2, x3, t), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x1, x2, x3. In considering the…

### 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…

### 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…