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2 Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368; also Department of Mathematics, Indiana University, Bloomington, Indiana 47405, and 1997–1998 Ulam Scholar at Los Alamos National Laboratory, Center for Nonlinear Sciences, MS B258, Los Alamos, New Mexico 87545. E-mail: foias@indiana.edu and foias@math.tamu.edu 3… (More)

We review the properties of the nonlinearly dispersive Navier-Stokesalpha (NS−α) model of incompressible fluid turbulence – also called the viscous Camassa-Holm equations in the literature. We first re-derive the NS−α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier-Stokes equations. Then we show that this… (More)

Shiyi Chen,1 Ciprian Foias,1,2 Darryl D. Holm,1 Eric Olson,1,2 Edriss S. Titi,3,4,5 and Shannon Wynne3,5 1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 2Department of Mathematics, Indiana University, Bloomington, Indiana 47405 3Department of Mathematics, University of California, Irvine,… (More)

In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and… (More)

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 manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold… (More)

- Ciprian Foias, OSCAR P. MANLEY, Ricardo M. S. Rosa
- 2001

A starting point for the conventional theory of turbulence (A. N. Kolmogorov (1941a,b, 1942)) is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes (L. Onsager (1945)). Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade… (More)

- Ciprian Foias, Luan T. Hoang, +4 authors Basil Nikolaenko
- 2008

We study the asymptotic behavior of the statistical solutions to the Navier–Stokes equations using the normalization map [9]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also… (More)

We show that for the periodic 2D Navier-Stokes equations (NSE) the set of initial data for which the solution exists for all negative times and has exponential growth is rather rich. We study this set and show that it is dense in the phase space of the NSE. This yields to a positive answer to a question in [BT]. After an appropriate resealing the large… (More)

Let C denote the category of Hilbert modules which are similar to contractive Hilbert modules. It is proved that if H0, H ∈ C and if H1 is similar to an isometric Hilbert module, then the sequence 0 → H0 → H → H1 → 0 splits. Thus the isometric Hilbert modules are projective in C. It follows that ExtC (K, H) = 0, whenever n > 1, for H, K ∈ C. In addition, it… (More)

- Ciprian Foias, OSCAR P. MANLEY, Ricardo M. S. Rosa, Roger Temam
- 2001

The phenomenological theory of turbulence in three dimensions postulates that at large Reynolds numbers there exists an interval of wavenumbers within which the direct effects of the molecular viscosity are negligible. Within that interval, the so-called inertial range, an eddy characterized by a wavenumber given in that range decays principally by breaking… (More)