# Limiting Behavior for an Iterated Viscosity

Ciprian Foias; Michael S. Jolly; Oscar P. Manley

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 2, page 353-376
- ISSN: 0764-583X

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topFoias, Ciprian, Jolly, Michael S., and Manley, Oscar P.. "Limiting Behavior for an Iterated Viscosity." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 353-376. <http://eudml.org/doc/197415>.

@article{Foias2010,

abstract = {
The behavior of an ordinary differential equation for the low wave number velocity
mode is analyzed. This equation was derived in [5]
by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It
resembles the NSE in form, except
that the kinematic viscosity is replaced by an iterated viscosity
which is a partial sum, dependent on the low-mode velocity. The convergence of
this sum as the number of iterations is taken to be arbitrarily large is explored.
This leads to a limiting dynamical system which displays
several unusual mathematical features.
},

author = {Foias, Ciprian, Jolly, Michael S., Manley, Oscar P.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Navier-Stokes.; ordinary differential equation; low wave number velocity mode; iterative process; two-dimensional Navier-Stokes equations; kinematic viscosity; iterated viscosity; limiting dynamical system},

language = {eng},

month = {3},

number = {2},

pages = {353-376},

publisher = {EDP Sciences},

title = {Limiting Behavior for an Iterated Viscosity},

url = {http://eudml.org/doc/197415},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Foias, Ciprian

AU - Jolly, Michael S.

AU - Manley, Oscar P.

TI - Limiting Behavior for an Iterated Viscosity

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 2

SP - 353

EP - 376

AB -
The behavior of an ordinary differential equation for the low wave number velocity
mode is analyzed. This equation was derived in [5]
by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It
resembles the NSE in form, except
that the kinematic viscosity is replaced by an iterated viscosity
which is a partial sum, dependent on the low-mode velocity. The convergence of
this sum as the number of iterations is taken to be arbitrarily large is explored.
This leads to a limiting dynamical system which displays
several unusual mathematical features.

LA - eng

KW - Navier-Stokes.; ordinary differential equation; low wave number velocity mode; iterative process; two-dimensional Navier-Stokes equations; kinematic viscosity; iterated viscosity; limiting dynamical system

UR - http://eudml.org/doc/197415

ER -

## References

top- P. Constantin and C. Foias, Navier-Stokes Equations, Univ. Chicago Press, Chicago, IL (1988). Zbl0687.35071
- N. Dunford and J.T. Schwartz, Book Linear Operators, Wiley, New York (1958) Part II.
- C. Foias, What do the Navier-Stokes equations tell us about turbulence? in Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995). Contemp. Math.208 (1997) 151-180. Zbl0890.76030
- C. Foias, O.P. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. RAIRO Modél. Math. Anal. Numér.22 (1988) 93-118. Zbl0663.76054
- C. Foias, O.P. Manley and R. Temam, Approximate inertial manifolds and effective viscosity in turbulent flows. Phys. Fluids A 3 (1991) 898-911. Zbl0732.76001
- C. Foias, O.P. Manley and R. Temam, Iterated approximate inertial manifolds for Navier-Stokes equations in 2-D. J. Math. Anal. Appl.178 (1994) 567-583. Zbl0806.76015
- C. Foias, O.P. Manley, R. Temam and Y.M. Treve, Asymptotic analysis of the Navier-Stokes equations. Phys. D9 (1983) 157-188. Zbl0584.35007
- C. Foias and B. Nicolaenko, On the algebra of the curl operator in the Navier-Stokes equations (in preparation).
- R.H. Kraichnan, Inertial ranges in two-dimensional turbulence. Phys. Fluids10 (19671) 417-1423.
- W. Heisenberg, On the theory of statistical and isotropic turbulence. Proc. Roy. Soc. Lond. Ser. A.195 (1948) 402-406. Zbl0035.25605
- E. Hopf, A mathematical example displaying features of turbulence. Comm. Appl. Math.1 (1948) 303-322. Zbl0031.32901
- R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York (1997). Zbl0871.35001
- T. von Karman, Tooling up mathematics for engineering. Quarterly Appl. Math.1 (1943) 2-6.

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