Geometry of inertial manifolds probed via a Lyapunov projection method

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a simple way to determine also experimentally inertial manifolds and their geometric characteristics

On the interaction of small and large eddies in two dimensional turbulent flows

Some results concerning the interaction of small and large eddies to two dimensional turbulent flows are presented. It is shown that the amplitude of small structures decays exponentially to a small value, and from this is inferred a simplified interaction law of small and large eddies. Beside their intrinsic interest for the understanding of the physics of turbulence, these results lead to new numerical schemes to be studied in a separate work

Variational bounds on the energy dissipation rate in body-forced shear flow

A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations.Comment: 15 pages, 7 figure

Confluent operator algebras and the closability property

Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence.Comment: Preliminary versio

Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number \bG, whose character depends on the ratio of the forcing to the viscosity $\nu$, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number \Rey, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias \cite{DF} to the two-dimensional Navier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating quantities of physical relevance, particularly long-time averages \left, in terms of the Reynolds number \Rey = U\ell/\nu, where U^{2}= L^{-2}\left and $\ell$ is the forcing scale. In particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor dimension converts to a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}, while the estimate for the inverse Kraichnan length is (a_{\ell}^{2}\Rey)^{1/2}, where $a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency : it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for the special issue on mathematical fluid mechanic

Optimal solutions to matrix-valued Nehari problems and related limit theorems

In a 1990 paper Helton and Young showed that under certain conditions the optimal solution of the Nehari problem corresponding to a finite rank Hankel operator with scalar entries can be efficiently approximated by certain functions defined in terms of finite dimensional restrictions of the Hankel operator. In this paper it is shown that these approximants appear as optimal solutions to restricted Nehari problems. The latter problems can be solved using relaxed commutant lifting theory. This observation is used to extent the Helton and Young approximation result to a matrix-valued setting. As in the Helton and Young paper the rate of convergence depends on the choice of the initial space in the approximation scheme.Comment: 22 page

On the Clark-alpha model of turbulence: global regularity and long--time dynamics

In this paper we study a well-known three--dimensional turbulence model, the filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width alpha ($\alpha$). We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to $(L/l_d)^3$, where $L$ is the integral spatial scale and $l_d$ is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark-$\aa$ model has the usual $k^{-5/3}$ Kolmogorov power law for wave numbers $k\aa \ll 1$ and $k^{-3}$ decay power law for $k\aa \gg 1.$ This is evidence that the Clark$-\alpha$ model parameterizes efficiently the large wave numbers within the inertial range, $k\aa \gg 1$, so that they contain much less translational kinetic energy than their counterparts in the Navier-Stokes equations.Comment: 11 pages, no figures, submitted to J of Turbulenc

On the behavior of the Lorenz equation backward in time

AbstractThe sets of solutions to the Lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined. The set associated with the middle eigenvalue is shown to project surjectively onto a plane, thereby providing a lower estimate for its dimension. Specific bounds are also found for a cone containing this set