dimensionality reduction for uncertain dynamic systems dimensionality reduction is a beneficial step to alleviate some of the computation burden as well as to improve the accuracy associated with complex system analyses this paper investigates dimensionality reduction techniques for linear time invariant systems subject to general non linear parameter dependencies in the context of this paper dimensionality reduction refers to simultaneous reductions in both model state order and parameter order i e number of uncertain parameters two complementary approaches will be presented one based on the worst case h infinity norm error associated with both model state and parameter order reductions and another which is essentially the inverse problem that considers the largest allowable parameter bounds for a given total h infinity norm error for the dimensionally reduced problem although applicable to larger order systems a simple low order spring mass example is used to demonstrate the usefulness of the techniques developed herein published in 2009 by john wiley sons ltd

are the incompressible 3d navier stokes equations locally ill posed in the natural energy space abstract an important open problem in the theory of the navier stokes equations is the uniqueness of the leray hopf weak solutions with l 2 initial data in this paper we give sufficient conditions for non uniqueness in terms of spectral properties of a natural linear operator associated to scale invariant solutions recently constructed in 8 if the spectral conditions are satisfied non uniqueness and ill posedness can appear for quite benign compactly supported data just at the borderline of applicability of the classical perturbation theory the verification of the spectral conditions seems to be approachable by numerical simulations which involve only smooth functions

on the large time galerkin approximation of the navier stokes equations in this article we are interested in the difficult problem of relating the large time behaviour of the navier stokes equations to the large time behaviour of their galerkin approximation we restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one if the computed approximation of the time dependent equations seems to converge to some limit as t to infty is the same true for the exact problem and are the limits related we give here some sufficient conditions reasonably easy to verify which guarantee a positive answer to this question

a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

on the stationary solutions and inviscid limit for the generalized proudman johnson equation with o 1 forcing abstract we consider the stationary problem for the generalized proudman johnson equation x x x a x x x x x x x f where a r and 0 are parameters and and f satisfy the periodic boundary conditions on we establish the existence of a time dependent solution and the uniqueness of the stationary solution for some large then we investigate the behavior of solution sequences in the limit 0 fixing a and f where bifurcations appear and new solutions emerge in particular we analyze if there is convergence towards stationary solutions for the corresponding inviscid equation

gevrey class regularity and approximate inertial manifolds for the kuramoto sivashinsky equation abstract we present some results on the kuramoto sivashinsky equation which models pattern formation on unstable flame fronts and thin hydrodynamic films we show gevrey class regularity of the solutions and decay of the orbits to small neighborhoods exponentially small of some simple manifolds approximate inertial manifolds

a model for the unstable manifold of the bursting behavior in the 2d navier stokes flow quasi periodic and bursting behaviors of the two dimensional 2d navier stokes flow are analyzed the tools used are the proper orthogonal decomposition pod method and the artificial neural network ann method the pod is used to extract coherent structures and prominent features from pde simulations of a quasi periodic regime and a bursting regime eigenfunctions of the two regimes were related by the symmetries of the 2d navier stokes equations three eigenfunctions that represent the dynamics of the quasi periodic regime and two eigenfunctions associated with the unstable manifold of the bursting regime were derived calculations of the pod eigenfunctions are performed on the fourier amplitudes in a comoving frame inverse fourier transform is applied to represent the pod eigenfunctions in both streamfunction and vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same projection onto the two eigenfunctions associated with the unstable manifold reduces the data to two time series processing these time series through an ann results in a low dimensional model describing the unstable manifold of the bursting regime that can be used to predict the onset of a burst

onset of chaotic kolmogorov flows resulting from interacting oscillatory modes on the basis of rigorous analysis supported by numerical computation a systematic study is presented to locate and examine chaotic kolmogorov flows resulting from the interaction of a basic steady state flow and oscillatory modes referenced to suitably chosen initial conditions of the kolmogorov flow model these oscillatory modes are derived from the equation linearized around the basic steady state flow the numerical experiments provide insight into the transition process from secondary self oscillation flows or secondary steady state flows to chaotic kolmogorov flows

dynamical indicators for the prediction of bursting phenomena in high dimensional systems drawing upon the bursting mechanism in slow fast systems we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates the indicators are associated with functionals defined in terms of optimally time dependent otd modes one such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes in contrast to other choices of subspaces the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful we illustrate the application of these indicators on three examples a prototype low dimensional model a body forced turbulent fluid flow and a unidirectional model of nonlinear water waves we use bayesian statistics to quantify the predictive power of the proposed indicators

symmetry breaking mixing instability and low frequency variability in a minimal lorenz like system starting from the classical saltzman two dimensional convection equations we derive via a severe spectral truncation a minimal 10 ode system which includes the thermal effect of viscous dissipation neglecting this process leads to a dynamical system which includes a decoupled generalized lorenz system the consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables with the ensuing modifications to the structural properties of the attractor and of the spectral features when the relevant nondimensional number eckert number ec is different from zero an additional time scale of o text ec ensuremath 1 is introduced in the system as shown with standard multiscale analysis and made clear by several numerical evidences moreover the system is ergodic and hyperbolic the slow variables feature long term memory with 1 f 3 2 power spectra and the fast variables feature amplitude modulation increasing the strength of the thermal viscous feedback has a stabilizing effect as both the metric entropy and the kaplan yorke attractor dimension decrease monotonically with ec the analyzed system features very rich dynamics it overcomes some of the limitations of the lorenz system and might have prototypical value in relevant processes in complex systems dynamics such as the interaction between slow and fast variables the presence of long term memory and the associated extreme value statistics this analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic in fact this leads to spurious invariances that affect essential dynamical properties ergodicity hyperbolicity and that cause the model losing ability in describing intrinsically multiscale processes

the role of bkm type theorems in 3d euler navier stokes and cahn hilliard navier stokes analysis abstract the beale kato majda theorem contains a single criterion that controls the behaviour of solutions of the 3 d incompressible euler equations versions of this theorem are discussed in terms of the regularity issues surrounding the 3 d incompressible euler and navier stokes equations together with a phase field model for the statistical mechanics of binary mixtures called the 3 d cahn hilliard navier stokes chns equations a theorem of bkm type is established for the chns equations for the full parameter range moreover for this latter set it is shown that there exists a reynolds number and a bound on the energy dissipation rate that remarkably reproduces the r e 3 4 upper bound on the inverse kolmogorov length normally associated with the navier stokes equations alone an alternative length scale is introduced and discussed together with a set of pseudo spectral computations on a 12 8 3 grid

a determining form for the two dimensional navier stokes equations the fourier modes case the determining modes for the two dimensional incompressible navier stokes equations nse are shown to satisfy an ordinary differential equation ode of the form dv dt f v in the banach space x of all bounded continuous functions of the variable s r with values in certain finite dimensional linear space this new evolution ode named determining form induces an infinite dimensional dynamical system in the space x which is noteworthy for two reasons one is that f is globally lipschitz from x into itself the other is that the long term dynamics of the determining form contains that of the nse the traveling wave solutions of the determining form i e those of the form v t s v0 t s correspond exactly to initial data v0 that are projections of solutions of the global attractor of the nse onto the determining modes the determining form is also shown to be dissipative an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the grashof num

extensivity of two dimensional turbulence this study is concerned with how the attractor dimension of the two dimensional navier stokes equations depends on characteristic length scales including the system integral length scale the forcing length scale and the dissipation length scale upper bounds on the attractor dimension derived by constantin foias and temam are analysed it is shown that the optimal attractor dimension estimate grows linearly with the domain area suggestive of extensive chaos for a sufficiently large domain if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed for sufficiently small domain area a slightly super extensive estimate becomes optimal in the extensive regime the attractor dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined on physical grounds in terms of the average rate of shear this dissipation length scale which is not necessarily the scale at which the energy or enstrophy dissipation takes place can be identified with the dimension correlation length scale the square of which is interpreted according to the concept of extensive chaos as the area of a subsystem with one degree of freedom furthermore these length scales can be identified with a minimum length scale of the flow which is rigorously deduced from the concept of determining nodes

low dimensional dynamo modelling and symmetry breaking bifurcations motivated by the successful karlsruhe dynamo experiment a relatively low dimensional dynamo model is proposed it is based on a strong truncation of the magnetohydrodynamic mhd equations with an external forcing of the roberts type and the requirement that the model system satisfies the symmetries of the full mhd system so that the first symmetry breaking bifurcations can be captured the backbone of the roberts dynamo is formed by the roberts flow a helical mean magnetic field and another part of the magnetic field coupled to these two by triadic mode interactions a minimum truncation model mtm containing only these energetically dominating primary mode triads is fully equivalent to the widely used first order smoothing approximation however it is shown that this approach works only in the limit of small wave numbers of the excited magnetic field or small magnetic reynolds numbers rm 1 to obtain dynamo action under more general conditions secondary mode triads must be taken into account altogether a set of six primary and secondary mode types is found to be necessary for an optimum truncation model otm corresponding to a system of 152 ordinary differential equations in a second step the otm is used to study symmetry breaking bifurcations on its route to chaos with the reynolds number or strength of the driving force as the control parameter a decisive role in this scenario is played by a symmetry of the form of z2 s 1 resulting from the z2 reflection symmetry of the magnetic field in the mhd equations in conjunction with a circle symmetry s 1 of the roberts flow under its influence in a secondary hopf bifurcation from a circle of steady

common periodic behavior in larger and larger truncations of the navier stokes equations the periodic behavior ofn mode truncations of the navier stokes equations on a two dimensional torus is studied forn 44 60 80 and 98 significant common features are found particularly for not too high reynolds numbers in all models periodicity ends giving rise though at quite different parameter values to quasiperiodicity

time dependent periodic navier stokes flows on a two dimensional torus the dynamical behaviour of an incompressible viscous fluid flow on a two dimensional torus externally excited by a spatially periodic force is investigated the flow field described by navier stokes equations is found to possess a sequence of time periodic solutions which bifurcate from a single steady state solution i e hopf bifurcations this result is based on a combination of analysis and computations and each provides corroborative evidence to the findings of the other

coexistence of chaotic and non chaotic states in the two dimensional gauss navier stokes dynamics recently gallavotti proposed an equivalence conjecture in hydrodynamics which states that forced damped fluids can be equally well represented by means of the navier stokes equations ns and by means of time reversible modifications of ns called gauss navier stokes equations gns this equivalence conjecture received numerical support in several recent papers concerning two dimensional fluid mechanics the corresponding results rely on the fact that the ns and gns systems only have one attracting set performing similar two dimensional simulations we find that there are conditions to be met by the gns system for this to be the case in particular increasing the reynolds number while keeping fixed the number of fourier modes leads to the coexistence of different attractors this makes difficult a test of the equivalence conjecture but constitutes a spurious effect due to the insufficient spectral resolution with sufficiently fine spectral resolution the steady states are unique and the equivalence conjecture can be conveniently established

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

bifurcation computations on an approximate inertial manifold for the 2d navier stokes equations abstract we compare the minimum number of modes needed to capture certain bifurcations of the 2d navier stokes equations by a certain nonlinear galerkin method and by the traditional galerkin method the boundary conditions are periodic and the forcing is in a single mode two different choices of the forcing mode are tested we introduce and implement a homotopy which allows continuation between approximations of different size

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

symmetries and dynamics for 2 d navier stokes flow abstract simulations of forced 2 d navier stokes equations are analyzed the forcing is spatially periodic and temporally steady a karhunen loeve analysis is used to identify the structures in phase space that generate the pde behavior their relationship to the invariant subspaces generated by the symmetry group is discussed it is shown that certain modes that are in the stable eigenspace of the kolmogorov flow solution play an essential role for the dynamics of the attractor for the 2 d navier stokes equations below a reynolds number of about 30 in this regime all stable solutions are identified and thier relation to the symmetry structure is elucidated a new type of gluing bifurcation generated by the symmetry is found and analyzed a mechanism for the generation of bursting behavior is suggested

on the use of pod based roms to analyze bifurcations in some dissipative systems abstract this paper deals with the use of pod based reduced order models to construct bifurcation diagrams which requires calculating steady and time dependent attractors in complex bifurcation problems involving dissipative systems the method proposed in the paper relies on the observation that pod manifolds resulting from snapshots calculated in time dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters the reason for this property is explained for a general class of dissipative systems which includes many problems of scientific industrial interest the consequence is that appropriate pod manifolds can be calculated in a quite computationally efficient way the method is illustrated considering both a simple bifurcation problem for a fisher like equation and a fairly complex bifurcation problem for the complex ginzburg landau equation

preserving symmetries in the proper orthogonal decomposition the proper orthogonal decomposition pod also called karhunen loeve expansion has been recently used in turbulence to derive optimally fast converging bases of spatial functions leading to efficient finite truncations whether a finite number of these modes can be used in numerical simulations to derive an accurate finite set of ordinary differential equations over a certain range of bifurcation parameter values still remains an open question it is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite dimensional system in most cases this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric pod basis the kuramoto sivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view however the conclusion is general and can be applied to other equations such as the navier stokes equations the complex g

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

the energy operator and new scaling relations for the incompressible navier stokes equations the incompressible navier stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena the nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling here the incompressible navier stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales of motion in particular the self adjoint linear reynolds orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear navier stokes operator the eigensystem of the reynolds orr operator is shown to yield a dynamical range of scales that can be characterized by an energy containing range an inertial range and a dissipation range it i

on approximate inertial manifolds to the navier stokes equations 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

low dimensional attractors arise from forcing at small scales abstract standard estimates of the dimension of the attractor of the 2d navier stokes equations are given in terms of a dimensionless grashof number that measures the total amount of energy injected into the flow however this result takes no account of whether the forcing is concentrated at large or small scales by a simple modification of the usual argument this paper provides a bound that is linear in a modified grashof number that reflects the spatial structure of the forcing in particular as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases

c1approximations of inertial manifolds for dissipative nonlinear equations abstract in this paper we study a class of nonlinear dissipative partial differential equations that have inertial manifolds this means that the long time behavior is equivalent to a certain finite system of ordinary differential equations we investigate ways in which these finite systems can be approximated in thec1sense geometrically this may be interpreted as constructing manifolds in phase space that arec1close to the inertial manifold of the partial differential equation under such approximations the invariant hyperbolic sets of the global attractor persist

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

analytic study of shell models of turbulence abstract in this paper we study analytically the viscous sabra shell model of the energy turbulent cascade we prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom specifically a finite dimensional global attractor and globally invariant inertial manifolds moreover we establish the existence of an exponentially decaying energy dissipation range for sufficiently smooth forcing

uniform in time error estimates for the postprocessing galerkin method applied to a data assimilation algorithm we apply the postprocessing galerkin method to a recently introduced continuous data assimilation downscaling algorithm for obtaining a numerical approximation of the solution of the two dimensional navier stokes equations corresponding to given measurements from a coarse spatial mesh under suitable conditions on the relaxation nudging parameter the resolution of the coarse spatial mesh and the resolution of the numerical scheme we obtain uniform in time estimates for the error between the numerical approximation given by the postprocessing galerkin method and the reference solution corresponding to the measurements our results are valid for a large class of interpolant operators including low fourier modes and local averages over finite volume elements notably we use here the two dimensional navier stokes equations as a paradigm but our results apply equally to other evolution equations such as the boussinesq system of benard convection and other oceanic and atmospheric circulation models

a posteriori regularity of the three dimensional navier stokes equations from numerical computations in this paper we consider the role that numerical computations in particular galerkin approximations can play in problems modeled by the three dimensional 3d navier stokes equations for which no rigorous proof of the existence of unique solutions is currently available we prove a robustness theorem for strong solutions from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem we then consider galerkin approximations and show that if a strong solution exists the galerkin approximations will converge to it thus if one is prepared to assume that the navier stokes equations are regular one can justify this particular numerical method rigorously combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time we thus introduce the possibility of rigorous com

the emergence of large scale coherent structure under small scale random bombardments we provide mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments with small forcing and appropriate scaling assumptions the analysis shows that the large scale structure emerging out of the small scale random forcing is not the one predicted by equilibrium statistical mechanics but the error is very small which explains earlier successful prediction of the large scale structure based on equilibrium statistical mechanics 2005 wiley periodicals inc

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

instability for the navier stokes equations on the 2 dimensional torus and a lower bound for the hausdorff dimension of their global attractors we prove instability of stationary solutions of the navier stokes equations on the domain 0 2 0 2 with periodic boundary condition for a class of external forces for large reynolds number moreover we give a lower bound for the hausdorff dimension of the global attractors

determining finite volume elements for the 2d navier stokes equations we consider the 2d navier stokes equations on a square with periodic boundary conditions dividing the square into n equal subsquares we show that if the asymptotic behavior of the average of solutions on these subsquares finite volume elements is known then the large time behavior of the solution itself is completely determined provided n is large enough we also establish a rigorous upper bound for n needed to determine the solutions to the navier stokes equation in terms of the physical parameters of the problem

nonlinear instability for the boussinesq equations with diabatic forcing abstract in this paper we prove the linear instability for the boussinesq equations experiencing a diabatic forcing to do so we first prove the local well posedness of the governing set of equations an example of an unstable solution to the linearized set of equations is then constructed finally we show that the unstable solution to the linearized problem is also unstable in the nonlinear sense

sharp lower bounds for the dimension of the global attractor of the sabra shell model of turbulence in this work we derive lower bounds for the hausdorff and fractal dimensions of the global attractor of the sabra shell model of turbulence in different regimes of parameters we show that for a particular choice of the forcing term and for sufficiently small viscosity term the sabra shell model has a global attractor of large hausdorff and fractal dimensions proportional to log 1 for all values of the governing parameter e except for e 1 the obtained lower bounds are sharp matching the upper bounds for the dimension of the global attractor obtained in our previous work moreover the complexity of the dynamics of the shell model increases as the viscosity tends to zero and we describe a precise scenario of successive bifurcations for different parameters regimes in the three dimensional regime of parameters this scenario changes when the parameter e becomes sufficiently close to 0 or to 1 we also show that in the two dimensional regime of parameters for a certain non zero forcing term the long term dynamics of the model becomes trivial for every value of the viscosity

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

global solutions of the navier stokes equations in thin three dimensional domains this paper is concerned with the spatially periodic navier stokes equations in a thin three dimensional domain it is shown that the classical solution exists globally when the initial velocityu0and the external forcefare chosen in the set significantly larger than those presented in earlier works what is more it is shown that these global solutions which seem large in a function space are small in nature

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

fredholm theory for pairs of closed subspaces of a banach space we study the fredholm theory for pairs of closed subspaces of a banach space developed by kato we define the strictly singular and the strictly cosingular pairs of subspaces and we show that some of the results of operator theory can be extended to this context however there are some notable differences on the one hand the perturbation classes problem has a positive answer in this context the upper and lower semi fredholm pairs are stable under strictly singular and strictly cosingular perturbations respectively and this stability characterizes the strictly singular and the strictly cosingular pairs note that it has been proved recently that the perturbation classes problem for continuous semi fredholm operators has a negative answer on the other hand unlike in the case of operators the fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs we also show the stability under composition of the compact the strictly singular and the strictly cosingular pairs of subspaces

long time behavior of navier stokes flow on a two dimensional torus excited by an external sinusoidal force in this paper we study the navier stokes flow on the two dimensional toruss1 s1 excited by the external force k2 sinky 0 and find the long time behavior for the flow starting from some states wheres1 0 2 mod 2 especially for the casek 2 it follows from an analysis and computation that the navier stokes flow with the initial state cos mx ny or sin mx ny will likely evolve through at most one step bifurcation to either a steady state solution or a time dependent periodic solution for any reynolds number and integersm andn

the perturbation classes problem in fredholm theory abstract we give a negative answer to the perturbation classes problem the perturbation class of the upper semi fredholm operators contains properly the strictly singular operators and the perturbation class of the lower semi fredholm operators contains properly the strictly cosingular operators

estimates on enstrophy palinstrophy and invariant measures for 2 d turbulence we construct semi integral curves which bound the projection of the global attractor of the 2 d navier stokes equations in the plane spanned by enstrophy and palinstrophy of particular interest are certain regions of the plane where palinstrophy dominates enstrophy previous work shows that if solutions on the global attractor spend a significant amount of time in such a region then there is a cascade of enstrophy to smaller length scales one of the main features of 2 d turbulence theory the semi integral curves divide the plane into regions having limited ranges for the direction of the flow this allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy we also derive a sharp universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero

some numerical aspects of computing inertial manifolds various aspects of the problem of computing inertial manifolds for dissipative partial differential equations are investigated in particular methods are proposed which generalize previous ones and improve upon some of their limitations the basic ideas and techniques are exemplified mainly for the kuramoto sivashinsky equation this equation is chosen because it is fairly simple but the dynamics are sufficiently complicated and it is a classical case for which a considerable amount of computational experience has been well documented

on the non linear stability of the 1 1 1 abc flow abstract abc flows which can be considered as prototypes for the study of the onset of three dimensional spatio temporal turbulence are known both analytically and numerically to be linearly unstable we analyze the nonlinear evolution of the abc flow a 1 with a b c 1 and with characteristic wavenumber k0 1 in the interval of reynolds number 13 r 50 we solve numerically the forced navier stokes equations with periodic boundary conditions for up to 9 9 104 eddy turnover times bifurcations towards progressively more complex flows obtain with a relaminarization window loss of symmetries and chaotic oscillations probably revealing an underlying heteroclinic structure in the chaotic regime only three steady solutions emerge besides a1 they consist of a perturbed abc flow a 2 with a b c plus cyclic permutations at 23 r 50 an unstructured temporal chaos is observed with the flow still dominated by the largest scales

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

modelling the dynamics of nonlinear partial differential equations using neural networks the dynamics of two nonlinear partial differential equations pdes known as the kuramoto sivashinsky k s equation and the two dimensional navier stokes n s equations are analyzed using karhunen loeve k l decomposition and artificial neural networks ann for the k s equation numerical simulations using a pseudospectral galerkin method is presented at a bifurcation parameter 17 75 where a dynamical behavior represented by a heteroclinic connection is obtained we apply k l decomposition on the numerical simulation data with the task of reducing the data into a set of data coefficients then we use ann to model and predict the data coefficients at a future time it is found that training the neural networks with only the first data coefficient is enough to capture the underlying dynamics and to predict for the other remaining data coefficients as for the two dimensional n s equation a quasiperiodic behavior represented in phase space by a torus is analyzed at re 14 0 applying the symmetry observed in the two dimensional n s equations on the quasiperiodic behavior eight different tori were obtained we show that by exploiting the symmetries of the equation and using k l decomposition in conjunction with neural networks a smart neural model is obtained

a seven mode truncation of the plane incompressible navier stokes equations a model obtained by a seven mode truncation of the navier stokes equations for a two dimensional incompressible fluid on a torus is studied this model extending a previously studied five mode one exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis i e coexistence of attractors a stochastic behavior is found for high values of the reynolds number when no stable fixed points closed orbits or tori are present

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

fixed point limit behavior of n mode truncated navier stokes equations as n increases the fixed point behavior ofn mode truncations of the navier stokes equations on a two dimensional torus is investigated asn increases fromn 44 on the behavior does not undergo any qualitative change furthermore the bifurcations occur at critical parameter values which clearly tend to stabilize asn approaches 100

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

some specific mathematical constraints on 2d turbulence abstract we derive upper and lower bounds for ensemble averages of energy enstrophy and palinstrophy for the 2d periodic navier stokes equations this is carried out both in the general case and in the case where the energy power law for fully developed turbulence holds in the turbulent case the bounds are sharp up to a logarithm and provide a new lower bound on the landau lifschitz degrees of freedom we also prove two properties of the inertial term under the turbulence assumption one is that as the grashof number is increased the ensemble average of this term approaches the force the other is that an estimate of it via the ladyzhenskaya inequality is sharp on a considerable portion of the global attractor

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

chaotic principle some applications to developed turbulence some models for developed turbulence are considered they are shown to obey a large fluctuations theorem and one of them also obeys a response reciprocity relation of onsager type this illustrates and extends ideas and techniques developed in earlier works mainly for nonequilibrium problems in statistical mechanics

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

dynamical ensembles equivalence in fluid mechanics abstract an attempt to put together various theoretical mathematical or experimental results recently developed in apparently unrelated subjects namely ruelle s 1980 approach to turbulence the body of nose hoover type of molecular dynamics experiments evans and morriss 1990 evans et al 1993 bonetto et al 1996 mathematical results on lyapunov exponents the pairing rule dressler 1988 and dettman and morriss 1995 and experimental results on them sarman et al 1992 bohr et al 1996 theoretical as well as mathematical results on fluctuations frisch and parisi 1984 multifractality and gallavoti and cohen 1995 chaotic hypothesis the key idea that we try to clarify is that of dynamical ensembles as a generalization of the classical equilibrium ensembles arguing that they should be identified with the srb distributions and that they share several properties with the classical ensembles most of the results invoked here did not deal directly with the navier stokes equations and yet they seem to have a lot to do with them as we shall argue here the discussion will focus on the navier stokes and dissipative euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis the ideas developed to a great extent from the efforts put in interpreting the experimental results in bonetto et al 1996

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

finite reductions for dissipative systems and viscous fluid dynamic models on t2 abstract we reconsider the reduction method introduced for hamiltonian systems by amann conley and zehnder we propose an extension of these techniques to evolutive pde systems of dissipative type and prove that under suitable regularity conditions a finite number of spectral modes controls exactly the time evolution of the complete problem the problem of finite reduction for a two dimensional modified navier stokes equations is considered and an estimate of the dimension of the reduced space is given valid for any time t 0 comparison is made with the asymptotic finite dimension that has been obtained for the true navier stokes equations

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time