limit sets and strengths of convergence for sequences in the duals of thread like lie groups we consider a properly converging sequence of non characters in the dual space of a thread like group g n n geq 3 and investigate the limit set and the strength with which the sequence converges to each of its limits we show that if k is a properly convergent sequence of non characters in widehat g n then there is a trade off between the number of limits which are not characters their degrees and the strength of convergence i to each of these limits theorem 3 2 this enables us to describe various possibilities for maximal limit sets consisting entirely of non characters theorem 4 6 in sect 5 we show that if k is a properly converging sequence of non characters in widehat g n and if the limit set contains a character then the intersection of the set of characters which is homeomorphic to mathbb r 2 with the limit set has at most two components in the case of two components each is a half plane in theorem 7 7 we show that if a sequence has a character as a cluster point then by passing to a properly convergent subsequence and then a further subsequence it is possible to find a real null sequence c k with c k ne 0 such that for a in the pedersen ideal of c g n lim k rightarrow infty c k rm tr pi k a exists not identically zero and is given by a sum of integrals over mathbb r 2

discrete time lyapunov stability of large matrices a hybrid method for analyzing the discrete time lyapunov stability of large matrices is proposed the method combines lyapunov theory with krylov subspace techniques several numerical tests illustrate the behavior of the proposed method

on the behaviour of sequences in the dual of a nilpotent lie group let a be a separable c algebra and let c he a subset of the dual space 4 of a denote by a the cone of the positive elements in a we say that x e a is an element with bounded trace on c if the function n tr it x is bounded on c put bt xea x with bounded trace on 7 and let btc be the two sided selfadjoint ideal in a generated by bt let us set hc for the hull of btc in 4 i e

on ɛ spectra and stability radii techniques of krylov subspace iterations play an important role in computing e spectra of large matrices to obtain results about the reliability of this kind of approximations we propose to compare the position of the e spectrum of a with those of its diagonal submatrices we give theoretical results which are valid for any block decomposition in four blocks a11 a12 a21 a22 we then illustrate our results by numerical experiments the same kind of problem arises when we compute the stability radius of a large matrix in that context we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices

solvable lie algebras of dimension six six dimensional solvable lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes this completes mubarakzyanov s classification of the real six dimensional solvable lie algebras

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

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

on the construction of families of approximate inertial manifolds abstract the concept of approximate inertial manifold is related to the study of the longtime behaviour of partial differential equations in this paper we present a general method in order to construct sequences of such manifolds providing better and better approximations to the attractor le concept de variete inertielle approximative intervient dans l etude du comportement pour les grands temps des equations aux derivees partielles nous presentons dans cet article une methode generale pour construire des suites de telles varietes qui approximent de mieux en mieux l attracteur

rigorous estimates of small scales in turbulent flows we derive rigorous bounds on the length scale of determining local averages volume elements for the 3 d navier stokes equations these length scale estimates are related to kolmogorov s notion of a dissipation length scale in turbulent flows

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

bounded trace c algebras and integrable actions let g be a second countable group a be a separable c algebra with bounded trace and a strongly continuous action of g on a suppose that the action of g on open image in new window induced by is free and the g orbits are locally closed we show that the crossed product a g has bounded trace if and only if g acts integrably in the sense of rieffel and an huef on open image in new window in the course of this we show that the extent of non properness of an integrable action gives rise to a lower bound for the size of the finite upper multiplicities of the irreducible representations of the crossed product

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

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

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

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray 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 model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray 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 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

on the stability of large matrices the distance rstab a of a stable matrix a to the set of unstable matrices and the norm of the exponential of matrices constitute two important topics in stability theory we treat in this note the case of large matrices the method proposed partitions the matrix into two blocks a small block in which the stability is studied and a large block whose field of values is located in the complex plane using the information on the blocks and some results on perturbation theory we give sufficient conditions for the stability of the original matrix a lower bound of rstab a and an upper bound on the norm of the exponential of a we illustrate these theoretical bounds on a practical test problem

a block arnoldi chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices in this paper we describe a block version of arnoldi s method for computing a few eigenvalues with largest or smallest real parts the method is accelerated via chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum a parallel implementation of this method is investigated on the eight processor alliant fx 80 numerical results and comparisons with simultaneous iteration on some harwell boeing matrices are reported

upper multiplicity and bounded trace ideals inc algebras upper and lower multiplicitiesmu andml for an irreducible representation of ac algebraa relative to a net in are shown to generalize the multiplicity numbers obtained by previous authors in trace formulae for group c algebras this leads in the presence of an auxiliary finiteness condition to anuppersemi continuity result in 0 for trace functions on limsuptr a mu tr a a a where the summation is taken over the cluster points of a characterization is given for the conditionmu k wherekis a positive integer from which it follows that ac algebra has all upper multiplicities finite if and only if it has bounded trace more generally the largest bounded trace idealjof ac algebraais given by mu

discrete data assimilation in the lorenz and 2d navier stokes equations consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing in this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time this paper studies discrete data assimilation for the lorenz equations and the incompressible two dimensional navier stokes equations in both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

the camassa holm equations and turbulence 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

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

observers for compressible navier stokes equation we consider a multidimensional model of a compressible fluid in a bounded domain we want to estimate the density and velocity of the fluid based on the observations for only velocity we build an

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

data assimilation algorithm for 3d benard convection in porous media employing only temperature measurements abstract in this paper we propose a continuous data assimilation downscaling algorithm for the benard convection in porous media using only discrete spatial mesh measurements of the temperature in this algorithm we incorporate the observables as a feedback nudging term in the evolution equation of the temperature we show that under an appropriate choice of the nudging parameter and the size of the mesh and under the assumption that the observed data is error free the solution of the proposed algorithm converges at an exponential rate asymptotically in time to the unique exact unknown reference solution of the original system associated with the observed finite dimensional projection of temperature data moreover we note that in the case where the observational measurements are not error free one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements

strength of convergence in the orbit space of a transformation group abstract let g x be a second countable transformation group with g acting freely on x it is shown that measure theoretic accumulation of the action and topological strength of convergence in the orbit space x g provide equivalent ways of quantifying the extent of nonproperness of the action these notions are linked via the representation theory of the transformation group c algebra c 0 x g

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

strength of convergence in duals of c algebras and nilpotent lie groups abstract by using trace formulae the recent concept of upper multiplicity for an irreducible representation of a c algebra is linked to the earlier notion of strength of convergence in the dual of a nilpotent lie group g in particular it is shown that if has finite upper multiplicity then this integer is the greatest strength with which a sequence in can converge to upper multiplicities are calculated for all irreducible representations of the groups in the threadlike generalization of the heisenberg group the values are computed by combining new c theoretic results with detailed analysis of the convergence of coadjoint orbits and they show that every positive integer occurs for this class of groups

efficient form of the lans α turbulence model in a primitive equation ocean model the lagrangian averaged navier stokes alpha email protected model is a turbulence parameterization that has been shown to capture some of the most important features of high resolution ocean modeling at lower resolution the email protected model improves turbulence statistics with an additional nonlinear term and a smoothed advecting velocity in this work we investigate two smoothing techniques helmholtz inversions and filters and their effect on the email protected model s results and efficiency we show that convolution filters are an effective smoothing method that filters and helmholtz inversions produce similar trends statistics like higher resolution email protected simulations as the smoothing parameter is increased and that the filter is computationally more efficient filters must be constructed such that a pressure velocity numerical instability is not excited we show analytically that certain ranges of filter weights are unstable and confirm this with numerical experiments our stability criterion also guarantees that the kinetic energy is well defined and that the filtered velocity is smoother that the original velocity simulations of email protected using the largest filter width nine in the pop primitive equation ocean model resemble doubled resolution simulations of standard pop in statistics such as kinetic energy eddy kinetic energy and potential temperature fields the computational cost of adding email protected with this filter is only 27 as compared to a factor of 8 10 for a doubling of resolution

the c algebras of the heisenberg group and of thread like lie groups we describe the c algebras of the heisenberg group h n n 1 and the thread like lie groups g n n 3 in terms of c algebras of operator fields

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

gevrey regularity for the attractor of a partially dissipative model of benard convection in a porous medium abstract convective flow though a porous medium can be modeled by darcy s law a linear weakly damped momentum equation coupled with an advection diffusion equation for the energy the solution semigroup for this system is not smoothing and the solution of the momentum equation does not gain regularity with respect to its initial value in finite time however it is known that the semigroup is asymptotically smoothing so that the system possesses a finite dimensional global attractor as well as exponential attractors we show that the global attractor is contained in a special gevrey class of regularity and in particular is real analytic the key idea is the use of a fourier splitting method to approximate every orbit asymptotically in time by a gevrey regular function

lower multiplicity for irreducible representations of nilpotent locally compact groups let g be a nilpotent locally compact group the lower multiplicity ml is defined for every irreducible representation of g which does not form an open point in the dual space of g it is shown that ml 1 if either g is connected or is finite dimensional conversely for g a nilpotent group with small invariant neighbourhoods ml implies that is finite dimensional

the c algebra of ax b like groups abstract we describe the c algebras of a x b like groups in terms of algebras of operator fields defined over their dual spaces

on the hilbert schmidt semi norms ofl 1 of a nilpotent lie group in 2 appears the convolution product o f f with f on the non abelian group g combined with the abelian fourier transform on r a consequence of this mixture of non abelian with abelian operations is the fact that the integrand f f o exp need not be nonnegative this prohibits the extension of 2 to general ll functions it makes it also difficult to give estimations of ih f la s in terms o f i 7 it would be much better if one could replace 2 by

completely bounded mappings and simplicial complex structure in the primitive ideal space of a c algebra we consider the natural contraction from the central haagerup tensor product of a c algebra a with itself to the space of completely bounded maps cb a on a and investigate those a where there exists an inverse map with finite norm l a we show that a stabilised version l a sup n l m n a depends only on the primitive ideal space prim a the dependence is via simplicial complex structures defined from primal intersections on finite sets of primitive ideals that contain a glimm ideal of a moreover l a l a k h with k h the compact operators which requires us to develop the theory in the context of c algebras that are not necessarily unital

on the controllability of distributed systems to control a system is to make it behave hopefully according to our wishes in a way compatible with safety and ethics at the least possible cost the systems considered here are distributed i e governed modeled by partial differential equations pdes of evolution our wish is to drive the system in a given time by an adequate choice of the controls from a given initial state to a final given state which is the target if this can be achieved respectively if we can reach any neighborhood of the target the system with the controls at our disposal is exactly respectively approximately controllable a very general and fuzzy idea is that the more a system is unstable chaotic turbulent the simplest or the cheapest it is to achieve exact or approximate controllability when the pdes are the navier stokes equations it leads to conjectures which are presented and explained recent results reported in this expository paper essentially prove the conjectures in two space dimensions in three space dimensions a large number of new questions arise some new results support without proving the conjectures such as generic controllability and cases of decrease of cost of control when the instability increases short comments are made on models arising in climatology thermoelasticity non newtonian fluids and molecular chemistry the introduction of the paper and the first part of all sections are not technical many open questions are mentioned in the text

a class of numerical algorithms for large time integration the nonlinear galerkin methods the nonlinear galerkin methods are numerical schemes well adapted to the long term integration of evolution partial differential equations they consist of looking for approximate solutions lying in nonlinear manifolds that are close in some sense to the attractor the aim of this paper is to extend these schemes to a large class of manifolds convergence results are derived for the schemes introduced

data assimilation for the navier stokes α equations abstract the well posedness of the data assimilation problem for the navier stokes equations on a bounded three dimensional domain is investigated the data assimilation procedures under consideration are the adjoint method of variational data assimilation 4d var and the method of continuous data assimilation concerning the adjoint method the existence of optimal initial conditions with respect to an observation dependent cost functional is proven the optimizers are characterized by a first order necessary condition involving the adjoint linearized navier stokes equations and conditions for the uniqueness of the initial conditions are given well posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided

strength of convergence in non free transformation groups let g x be a transformation group where the group g does not necessarily act freely on the space x we investigate the extent to which the action of g may fail to be proper stability subgroups are used to define new notions of strength of convergence in the orbit space and of measure accumulation along orbits by using the representation theory of the associated crossed product c algebra we show that these notions are equivalent under certain conditions

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

strength of convergence in the orbit space of a groupoid abstract let g be a second countable locally compact hausdorff groupoid with a haar system and let x n be a sequence in the unit space g 0 of g we show that the notions of strength of convergence of x n in the orbit space g 0 g and measure theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid c algebra

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

on the number of determining nodes for the 2d navier stokes equations abstract it is known that the solutions of the 2d navier stokes equations in bounded domains are determined by a finite discrete set of nodal values that is if the large time behavior of the solutions to the navier stokes equations is known on an appropriate finite discrete set then the large time behavior of the solution itself is totally determined here an upper bound is rigorously established for the number of nodes needed to determine the solutions of the navier stokes equations in two dimensions with periodic boundary conditions

the solvable lie group n_ 6 28 an example of an almost c_0 mathcal k c algebra abstract motivated by the description of the c algebra of the affine automorphism group n 6 28 of the siegel upper half plane of degree 2 as an algebra of operator fields defined over the unitary dual n 6 28 of the group we introduce a family of c algebras which we call almost c 0 k and we show that the c algebra of the group n 6 28 belongs to this class

accuracy and stability of filters for dissipative pdes data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system filters refer to a class of data assimilation algorithms designed to update the estimation of the state in an on line fashion as data is acquired sequentially for linear problems subject to gaussian noise filtering can be performed exactly using the kalman filter for nonlinear systems filtering can be approximated in a systematic way by particle filters however in high dimensions these particle filtering methods can break down hence for the large nonlinear systems arising in applications such as oceanography and weather forecasting various ad hoc filters are used mostly based on making gaussian approximations the purpose of this work is to study the accuracy and stability properties of these ad hoc filters we work in the context of the 2d incompressible navier stokes equation although the ideas readily generalize to a range of dissipative partial differential equations pdes by working in this infinite dimensional setting we provide an analysis which is useful for the understanding of high dimensional filtering and is robust to mesh refinement we describe theoretical results showing that in the small observational noise limit the filters can be tuned to perform accurately in tracking the signal itself filter accuracy provided the system is observed in a sufficiently large low dimensional space roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics the tuning corresponds to what is known as variance inflation in the applied literature numerical results are given which illustrate the theory the positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal

global in time stability and accuracy of imex fem data assimilation schemes for navier stokes equations abstract we study numerical schemes for incompressible navier stokes equations using imex temporal discretizations finite element spatial discretizations and equipped with continuous data assimilation a technique recently developed by azouani et al 2014 we analyze stability and accuracy of the proposed methods and are able to prove well posedness long time stability and long time accuracy estimates under restrictions of the time step size and data assimilation parameter we give results for several numerical tests that illustrate the theory and show that for good results the choice of discretization parameter and element choices can be critical

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

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

analysis and approximation of the velocity tracking problem for navier stokes flows with distributed control we consider the mathematical formulation analysis and the numerical solution of a time dependent optimal control problem associated with the tracking of the velocity of a navier stokes flow in a bounded two dimensional domain through the adjustment of a distributed control the existence of optimal solutions is proved and the first order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls semidiscrete in time and fully discrete space time approximations are defined and their convergence to the exact optimal solutions is shown a gradient method for the solution of the fully discrete equations is examined as are its convergence properties finally the results of some illustrative computational experiments are presented

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

filter accuracy for the lorenz 96 model fixed versus adaptive observation operators in the context of filtering chaotic dynamical systems it is well known that partial observations if sufficiently informative can be used to control the inherent uncertainty due to chaos the purpose of this paper is to investigate both theoretically and numerically conditions on the observations of chaotic systems under which they can be accurately filtered in particular we highlight the advantage of adaptive observation operators over fixed ones the lorenz 96 model is used to exemplify our findings we consider discrete time and continuous time observations in our theoretical developments we prove that for fixed observation operator the 3dvar filter can recover the system state within a neighbourhood determined by the size of the observational noise it is required that a sufficiently large proportion of the state vector is observed and an explicit form for such sufficient fixed observation operator is given numerical experiments where the data is incorporated by use of the 3dvar and extended kalman filters suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction adaptive observation operators are then studied numerically we show that for carefully chosen adaptive observation operators the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator indeed we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive lyapunov exponents of the underlying system