Edriss S. Titi
Weizmann Institute of Science
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arXiv: Chaotic Dynamics | 2002
Ciprian Foias; Darryl D. Holm; Edriss S. Titi
We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, ℓ∈, as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/ℓ∈)3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-α equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-α model to the NSE by proving a convergence theorem, that as the length scale α1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.
Communications in Mathematical Physics | 1994
Peter Constantin; Weinan E; Edriss S. Titi
We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Eulers equation.
Physics Letters A | 1988
Ciprian Foias; M. S. Jolly; Ioannis G. Kevrekidis; George R. Sell; Edriss S. Titi
Abstract A modified Galerkin (the “Euler-Galerkin”) algorithmj for the computational of inertial manifolds is described and applied to reaction diffusion and the Kuramoto-Sivashinsky (KS) equation. In the context of the (KS) equation, a low-dimensional Euler-Galerkin approximation ( n = 3) is distinctly superior to the traditional Galerkin of the same dimension, and comparable to a traditional Galerkin of a much higher dimension ( n = 16).
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | 2005
Alexey Cheskidov; Darryl D. Holm; Eric Olson; Edriss S. Titi
In this paper we introduce and study a new model for three–dimensional turbulence, the Leray–α model. This model is inspired by the Lagrangian averaged Navier–Stokes–α model of turbulence (also known Navier–Stokes–α model or the viscous Camassa–Holm equations). As in the case of the Lagrangian averaged Navier–Stokes–α model, the Leray–α model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray–α model of the order of (L/ld)12/7, where L is the size of the domain and ld is the dissipation length–scale. This upper bound is much smaller than what one would expect for three–dimensional models, i.e. (L/ld)3. This remarkable result suggests that the Leray–α model has a great potential to become a good sub–grid–scale large–eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual k−5/3 Kolmogorov power law the inertial range has a steeper power–law spectrum for wavenumbers larger than 1/α. Finally, we propose a Prandtl–like boundary–layer model, induced by the Leray–α model, and show a very good agreement of this model with empirical data for turbulent boundary layers.
Journal of Dynamics and Differential Equations | 1989
Ciprian Foias; George R. Sell; Edriss S. Titi
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 has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.
Physics of Fluids | 1999
Shiyi Chen; Ciprian Foias; Darryl D. Holm; Eric Olson; Edriss S. Titi; Shannon Wynne
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 compare the results with empirical data for turbulent flows in channels and pipes. The data suggest that the constant α version of the Camassa–Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increas...
international symposium on physical design | 1999
Shiyi Chen; Ciprian Foias; Darryl D. Holm; Eric Olson; Edriss S. Titi; Shannon Wynne
Abstract In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett 81 (1998) 5338. S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, A connection between the Camassa–Holm equations and turbulent flows in channels and pipes, Phys. Fluids, in press. In particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large Reynolds numbers. There are many facts connecting the Camassa–Holm equations to turbulent fluid flows. The dimension of the attractor agrees with the heuristic argument based on the Kolmogorov statistical theory of turbulence. The statistical properties of the energy spectrum agree in numerical simulation with the Kolmogorov power law. Furthermore, comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the Camassa–Holm equations.
Journal of Mathematical Analysis and Applications | 1990
Edriss S. Titi
Abstract Recently, the theory of Inertial Manifolds has shown that the long time behavior (the dynamics) of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system. Although we are still unable to prove existence of Inertial Manifolds to the Navier-Stokes equations, we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two-dimensional case, and certain bounded invariant sets in the three-dimensional case. This approximate manifold and others allow us to introduce modified Galerkin approximations.
Archive for Rational Mechanics and Analysis | 2011
Chongsheng Cao; Edriss S. Titi
In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, that is, the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier–Stokes equations in the whole space, as well as for the case of periodic boundary conditions.
Nonlinearity | 1991
Ciprian Foias; Edriss S. Titi
The authors present a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations. In order to illustrate this connection they consider the 1D Kuramoto-Sivashinsky equation as a instructive paradigm. They remark that the results presented here apply to many other equations such as the 1D complex Ginzburg-Landau equation, the Chafee-Infante equation, etc.