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

Abstract

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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.

How to cite

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Foias, 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

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  1. P. Constantin and C. Foias, Navier-Stokes Equations, Univ. Chicago Press, Chicago, IL (1988).  Zbl0687.35071
  2. N. Dunford and J.T. Schwartz, Book Linear Operators, Wiley, New York (1958) Part II.  
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. C. Foias and B. Nicolaenko, On the algebra of the curl operator in the Navier-Stokes equations (in preparation).  
  9. R.H. Kraichnan, Inertial ranges in two-dimensional turbulence. Phys. Fluids10 (19671) 417-1423.  
  10. W. Heisenberg, On the theory of statistical and isotropic turbulence. Proc. Roy. Soc. Lond. Ser. A.195 (1948) 402-406.  Zbl0035.25605
  11. E. Hopf, A mathematical example displaying features of turbulence. Comm. Appl. Math.1 (1948) 303-322.  Zbl0031.32901
  12. R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York (1997).  Zbl0871.35001
  13. T. von Karman, Tooling up mathematics for engineering. Quarterly Appl. Math.1 (1943) 2-6.  

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