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Foiaş, Ciprian; Témam, Roger

doi:10.1512/iumj.1980.29.29062

Cited by 19 publications

(5 citation statements)

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“…The right hand side has two dominant terms; one negative associated with the dissipation, and the dominant positive u 2 ∞ -term. Rewriting the F n+1 term in (16) in terms of the κ n defined in ( 8)…”

Section: Problems With Regularity: An Illustrationmentioning

confidence: 99%

“…Analysis on timeevolving fractal domains with ill-defined boundary conditions is not advanced enough to gain rigorous results by concentrating on one fractal 'spot' in the flow. The partial regularity result of Scheffer [15] proving that the potentially singular set in time alone has zero half-dimensional Hausdorff measure has been of great influence (see also [16] for a short proof). Following this, the most significant space-time result has been that of Caffarelli, Kohn & Nirenberg [17] who showed that the potentially singular set in space-time has zero one-dimensional Hausdorff measure.…”

Section: Introductionmentioning

confidence: 99%

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Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence

Gibbon

Doering

2005

Arch. Rational Mech. Anal.

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Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active 'events' during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray's weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into 'good' and 'bad' intervals: on the 'good' intervals solutions are bounded and regular, whereas singularities are still possible within the 'bad' intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these 'bad' intervals, lower bounds on the local energy dissipation rate and other quantities, such as u(·, t) ∞ and ∇u(·, t) ∞ , are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for n ≥ 1 are related to Scheffer's potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given 'bad' interval, the energy has a lower bound that is decaying exponentially in time. ⋆ Original version Nov 03; 1st revised-version 6th June 04 2 Grashof No Gr = f ℓ 3 ν −2 Reynolds No [45] Re = U ℓν −1 Gr 1/2 ≤ c Re as Gr → ∞

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Section: Problems With Regularity: An Illustrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence

Gibbon

Doering

2005

Arch. Rational Mech. Anal.

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“…5 " 9 In particular, note that no improvement, but a possible worsening of conditions (4), ensues from using basis functions not satisfying the Stokes problem. 9 Since, in two dimensions, \ n ccn for n»l, condition (4) yields a sufficient bound on the number of modes needed for at least qualitatively correct solutions to the Navier-Stokes equations. In order for this result to be of practical value an a priori estimate of a quantity such as llvll 2 is required.…”

Section: V(?o=:sc«fc)w(f) O) I=lmentioning

confidence: 99%

“…Note too that these results are applicable in flows more general than B£nard convection. The physical significance of conditions (4b) and (4c) is readily recognized: A comparison of the inertial and dissipative terms in the Navier-Stokes equation yields the length scale at which the effects of dissipation begin to dominate the inertial effects, i.e., v° vv~iA7 2 v~i;X" +1 v (9) or equivalently, speaking very loosely, it suffices that v\ n+1 >M 2 ,…”

Section: V(?o=:sc«fc)w(f) O) I=lmentioning

confidence: 99%

Number of Modes Governing Two-Dimensional Viscous, Incompressible Flows

et al. 1983

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Recent efforts to estimate the number of modes sufficient for the approximate solutions of Navier-Stokes equations in two dimensions are summarized. Several such estimates have been obtained, and their relations to one another are discussed. The physical significance of the results is noted and used to infer the possible nature of similar estimates in three dimensions.PACS numbers: 03.40. Gc, 47.10.+g, An important unsolved issue in the numerical simulation of fluid flows is the a priori estimate of the number of modes or the number of mesh points needed to obtain correct approximate solutions to the Navier-Stokes equation. The importance of this question lies in (a) the proliferation of published numerical results, and (b) the clear evidence that the qualitative behavior of the approximate solutions (to say nothing of their quantitative behavior) may depend critically on the number of modes retained, for instance, in the Galerkin approximation. 1 ' 2 In this Letter we summarize some recent efforts to determine the sufficient number of modes needed for the approximate solutions of Navier-Stokes equations in two dimensions; we discuss the physical significance of the results and show how they may perhaps be extended to three dimensions; finally we discuss briefly the connection with other related efforts.As is well known, there are estimates of the necessary number of modes for isotropic, homogeneous, fully developed turbulence-they are based on the Kolmogorov length in three dimensions, 3 and on the enstrophy dissipation length in two dimensions. 4 However, it is not clear that such estimates are sufficient, nor how to extend them to anisotropic, inhomogeneous flows of fluids. Thus it is useful to explore the asymptotic properties (i.e., for large times) of the Navier-Stokes equations with a view to establishing a bound for the sufficient number of modes determining an adequate approximation to its true solutions.A relevant asymptotic property is given by the following result 5 : Consider the Navier-Stokes equation in a two-dimensional region fi,where v=(v 19 v 2 ) is the fluid velocity, p is the kinematic pressure, v is the viscosity, T(r 9 t) is the externally imposed force per unit mass, and we have assumed the fluid density to be constant, say, unity. We supplement (1) with the boundary condition v = 0on the boundary a£2, or, if Q is a rectangle, with periodic conditions on v. Associated with (1) and the appropriate boundary conditions is the Stokes problem:where q is determined uniquely by (2). The vector-valued eigenfunctions of (2) may be used as a basis for an expansion of v, i.e., v(?,o=:sc«fc)w,(f). o) i=lLet Xj be the eigenvalue belonging to w i? such that 0

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“…In three dimensions, perturbative renormalisation transformations in momentum space have been used to derive an infrared fixed point describing the long distance physics of the Navier-Stokes equation [2,3,4] . Similarly, probabilistic solutions of the Navier-Stokes equation with no forcing term have been provided by self-similar probability laws for the velocity field [5,6] . We shall use properties of the Navier-Stokes equation under scaling transformations in order to study the time evolution of solutions.…”

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confidence: 99%

A Renormalization Group Study of Three-Dimensional Turbulence

Brax

1997

J. Phys. I France

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Abstract:We study the three dimensional Navier-Stokes equation with a random Gaussian force acting on large wavelengths. Our work has been inspired by Polyakov's analysis of steady states of two dimensional turbulence. We investigate the time evolution of the probability law of the velocity potential. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalisation fixed point, we show that its time evolution is obtained by averaging over small scale features of the velocity potential. The probability law of the velocity potential converges to the fixed point in the long time regime. At the fixed point, the scaling dimension of the velocity potential is determined to be − . We give conditions for the existence of such a fixed point of the renormalisation group describing the long time behaviour of the velocity potential. At this fixed point, the energy spectrum of three dimensional turbulence coincides with a Kolmogorov spectrum.

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Cited by 19 publications

References 0 publications

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence

Number of Modes Governing Two-Dimensional Viscous, Incompressible Flows

A Renormalization Group Study of Three-Dimensional Turbulence

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