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

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 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

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

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

ergodic complex structures on hyperkahler manifolds let m be a compact complex manifold the corresponding teichmuller space teich is the space of all complex structures on m up to the action of the group rm diff 0 m of isotopies the mapping class group gamma rm diff m rm diff 0 m acts on teich in a natural way an ergodic complex structure is a complex structure with a gamma orbit dense in teich let m be a complex torus of complex dimension ge 2 or a hyperkahler manifold with b 2 3 we prove that m is ergodic unless m has maximal picard rank there are countably many such m this is used to show that all hyperkahler manifolds are kobayashi non hyperbolic

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

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

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

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 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

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

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

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

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

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

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

robust solutions of uncertain quadratic and conic quadratic problems we consider a conic quadratic and in particular a quadratically constrained optimization problem with uncertain data known only to reside in some uncertainty set cal u the robust counterpart of such a problem leads usually to an np hard semidefinite problem this is the case for example when cal u is given as the intersection of ellipsoids or as an n dimensional box for these cases we build a single explicit semidefinite program which approximates the np hard robust counterpart and we derive an estimate on the quality of the approximation which is essentially independent of the dimensions of the underlying conic quadratic problem

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

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

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

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

finiteness of klein actions and real structures on compact hyperkahler manifolds one central problem in real algebraic geometry is to classify the real structures of a given complex manifold we address this problem for compact hyperkahler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence we actually prove more generally that there are only finitely many up to conjugacy faithful finite group actions by holomorphic or anti holomorphic automorphisms the so called klein actions in other words the automorphism group and the klein automorphism group of a compact hyperkahler manifold contain only finitely many conjugacy classes of finite subgroups we furthermore answer a question of oguiso by showing that the automorphism group of a compact hyperkahler manifold is finitely presented

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

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

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

mapping class group and a global torelli theorem for hyperkahler manifolds a mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies we compute a mapping class group of a hyperkahler manifold m showing that it is commensurable to an arithmetic lattice in so 3 b2 3 a teichmuller space of m is a space of complex structures on m up to isotopies we define a birational teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds we show that the period map gives the isomorphism between connected components of the birational teichmuller space and the corresponding period space so b2 3 3 so 2 so b2 3 1 we use this result to obtain a torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group when m is a hilbert scheme of n points on a k3 surface with n 1 a prime power our torelli theorem implies the usual hodge theoretic birational torelli theorem for other examples of hyperkahler manifolds the hodge theoretic torelli theorem is known to be false

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

families of lagrangian fibrations on hyperkahler manifolds abstract a holomorphic lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with lagrangian fibers it is known due to huybrechts that a given compact manifold admits only finitely many holomorphic symplectic structures up to deformation we prove that a given compact simple hyperkahler manifold with b 2 7 admits only finitely many deformation types of holomorphic lagrangian fibrations we also prove that all known hyperkahler manifolds are never kobayashi hyperbolic

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 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

relaxed robust second order cone programming in this paper we propose an approximate optimization model for the robust second order cone programming problem with a single ellipsoid uncertainty set for which the computational complexity is not known yet we prove that this approximate robust model can be equivalently reformulated as a finite convex optimization problem

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

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

the kahler cone of a compact hyperkahler manifold this paper is a sequel to 11 we study a number of questions only touched upon in 11 in more detail in particular what is the relation between two birational compact hyperkahler manifolds what is the shape of the cone of all kahler classes on such a manifold how can the birational kahler cone be described most of the results are motivated by either the well established two dimensional theory i e the theory of k3 surfaces or the theory of calabi yau threefolds and string theory

explicit reformulations for robust optimization problems with general uncertainty sets we consider a rather general class of mathematical programming problems with data uncertainty where the uncertainty set is represented by a system of convex inequalities we prove that the robust counterparts of this class of problems can be reformulated equivalently as finite and explicit optimization problems moreover we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions our results provide a unified treatment of many situations that have been investigated in the literature and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before the analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems the analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set

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

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

robust convex quadratically constrained programs in this paper we study robust convex quadratically constrained programs a subset of the class of robust convex programs introduced by ben tal and nemirovski 4 in contrast to 4 where it is shown that such robust problems can be formulated as semidefinite programs our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second order cone programs socp we propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit socp and present examples where these classes of uncertainty sets are natural

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