on global behavior of solutions to an inverse problem for nonlinear parabolic equations we find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations we also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity

fuzzy goal programming approach for structural optimization a fuzzy nonlinear goal programming approach is presented for solving multiobjective optimization problems involving vague and imprecise information several computational models including simple additive weighted additive and preemptive priority models are given for the numerical solution of the problem the methodolo gies are illustrated with the help of two structural optimization problems involving multiple goals the solution of the first example is obtained using a graphical procedure whereas the second example is solved using nonlinear programming techniques linear membership functions are used in the numerical work for simplicity the methodologies presented in this work aid in the preliminary design of structural systems involving imprecise and vague information about the goals and or constraints inear goal programming has been extensively used in solving decision making problems involving linear equa tions and multiple conflicting goals the goals can be rank ordered depending on their importance to the decision maker goal programming attempts to achieve as man 7 of these goals as possible by minimizing deviational variables from the goal levels depending on their relative weights linear goal pro gramming algorithms were developed by charnes et al 1 ig nizio 2 and zanakis and gupta 3 the extension of goal pro gramming to nonlinear optimization problems has also been considered by several authors 3 4 a fuzzy programming ap proach for linear programming problems with several objec tives was suggested by zimmerman 5 subsequently several authors proposed different fuzzy goal programming ap proaches for solving linear goal programming problems in volving imprecise statements and information 6 10 narasimhan6 suggested a method for solving fuzzy linear goal programming problems under the assumption of linear membership functions his method involves solving a set of 2k linear programming problems each containing 3k constraints where k denotes the number of goals in the original problem hannan7 indicated a procedure for formulating a fuzzy goal programming problem as an equivalent single linear program ming problem with 2k goal related constraints the problems associated with the definition of fuzzy priorities were dis cussed in ref 7 a brief review of the history and the state of the art in fuzzy multicriteria programming as of 1982 was presented by ignizio 8 the distinction between fuzzy goal pro gramming and fuzzy multicriteria formulations was given in ref 9 models are presented by hannan7 for the use of fuzzy goal programming with preemptive priorities with archime dian weights and with the maximization of the membership function corresponding to the minimum goal a methodology based on the use of a nested hierarchy of priorities for each goal was presented by rubin and narasimhan 10 the impor tance of multiple objectives in the design of practical engineer

on hardy s inequality for symmetric integral transforms and analogous abstract in this paper we give a generalization of hardy s theorem for a class of symmetric transforms in both classical and quantum calculus as applications we give q analogues of hardy s theorem for the q fourier cosine q bessel fourier and q dunkl transforms

attractors and error estimates for discretizations of incompressible navier stokes equations by embedding sets of nodal values to function spaces variational arguments are applied to the finite difference approximations to the two dimensional incompressible navier stokes equations the error estimates for trajectories and the it time free error estimates with tolerance for attractors are proved the upper semicontinuity of the attractors with respect to the finite difference discretizations are also proved arguments of applying the techniques to the finite element approximations are also given

finite element analysis of a class of problems in finite elastoplasticity based on the thermodynamical theory of materials of type n abstract this paper describes a finite element approximation scheme for computing finite elastoplastic deformations of a hypothetical class of materials referred to as materials of type n kinematical descriptions of motion and strain rate and rotationally invariant stress rates measures appropriate for finite elastoplastic deformations are derived and the thermodynamical theory for materials of type n is briefly reviewed finite element approximations of the governing nonlinear equations are derived along with numerical algorithms for solving the associated discrete nonlinear system these new methods are applied to some representative plane strain problem including problems with unloading and reloading and the problem of upsetting of a partially constrained rectangular billet

number of degrees of freedom of two dimensional turbulence we derive upper bounds for the number of degrees of freedom of two dimensional navier stokes turbulence freely decaying from a smooth initial vorticity field omega x y 0 omega0 this number denoted by n is defined as the minimum dimension such that for n or n arbitrary n dimensional balls in phase space centered on the solution trajectory omega x y t for t 0 contract under the dynamics of the system linearized about omega x y t in other words n is the minimum number of greatest lyapunov exponents whose sum becomes negative it is found that n or c1re when the phase space is endowed with the energy norm and n or c2re 1 ln re 1 3 when the phase space is endowed with the enstrophy norm here c1 and c2 are constant and re is the reynolds number defined in terms of omega0 the system length scale and the viscosity nu the linear or nearly linear dependence of n on re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number this result is in a sharp contrast to the forced case for which well known estimates for the hausdorff dimension dh of the global attractor scale highly superlinearly with nu 1 we argue that the extra dependence of dh on nu 1 is not an intrinsic property of the turbulent dynamics rather it is a removable artifact brought about by the use of a time independent forcing as a model for energy and enstrophy injection that drives the turbulence

identification of nonlinear membership function for failure of structural material and its application to structural optimization abstract the purpose of this paper is the identification and application of fuzzy membership function for material failure the concept of evolutionary damage in continuum damage mechanics is related with the decision of material failure in fuzzy set theory the failure phenomenon has a fuzzy nature and the internal state variable which indicates the degree of damage in continuum damage mechanics can be thought as a fuzzy membership function for failure in the structural design of machines such as automobiles airplanes and ships the results of the numerical analysis based on continuum damage mechanics which are verified by the uniaxial tensile test are introduced to obtain the evolution of the damage variable from which the fuzzy membership function for failure can be derived an exponential type fuzzy membership function is proposed to represent the fuzziness of material failure from the results of the numerical analysis in this study an elastoviscoplastic isotropic damage model of an isotropic hardening material is introduced and the three bar truss problem is revisited with the proposed fuzzy membership function which is applied to structural optimization

smallest scale estimates for the navier stokes equations for incompressible fluids we consider solutions of the navier stokes equations for incompressible fluids in two and three space dimensions we obtain improved estimates in the limit of vanishing viscosity for the fourier coefficients the coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity this defines the minimum scale the size of the smallest feature in the flow

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

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

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

multi objective optimization of fuzzy structural systems it is recognized that there exists a vast amount of fuzzy information in both the objective and constraint functions of the optimum design of structures since most practical structural design problems involve several often conflicting objectives to be considered a multi objective fuzzy programming method is outlined in this work the fuzzy constraints define a fuzzy feasible domain in the design space and each of the fuzzy objective functions defines the optimum solution by a fuzzy set of points a method of solving a fuzzy multi objective structural optimization problem using ordinary single objective programming techniques is presented the computational approach is illustrated with two numerical examples

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

alternating cluster estimation a new tool for clustering and function approximation many clustering models define good clusters as extrema of objective functions optimization of these models is often done using an alternating optimization ao algorithm driven by necessary conditions for local extrema we abandon the objective function model in favor of a generalized model called alternating cluster estimation ace ace uses an alternating iteration architecture but membership and prototype functions are selected directly by the user virtually every clustering model can be realized as an instance of ace out of a large variety of possible instances of non ao models we present two examples 1 an algorithm with a dynamically changing prototype function that extracts representative data and 2 a computationally efficient algorithm with hyperconic membership functions that allows easy extraction of membership functions we illustrate these non ao instances on three problems a simple clustering of plane data where we show that creating an unmatched ace algorithm overcomes some problems of fuzzy c means fcm ao and possibilistic c means pcm ao b functional approximation by clustering on a simple artificial data set and c functional approximation on a 12 input 1 output real world data set ace models work pretty well in all three cases

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

attractor local dimensionality nonlinear energy transfers and finite time instabilities in unstable dynamical systems with applications to two dimensional fluid flows we examine the geometry of the finite dimensional attractor associated with fluid flows described by navier stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components modes of the flow specifically we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order reduction and describe the stochastic attractor in the reduced order phase space in terms of the associated probability measure we introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite time lyapunov exponents subsequently we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms in particular we illustrate the connection of the low dimensionality of the attractor with the circulation of energy i from the mean flow to the unstable modes due to their linearly unstable character ii from the unstable modes to the stable ones due to a nonlinear energy transfer mechanism and iii from the stable modes back to the mean due to the linearly stable character of these modes

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

optimal design of laminated composite plates in a fuzzy environment the present study focuses on a fuzzy optimization approach to the optimal design of simply supported laminated composite plates the goal is minimum weight design of the composite plates the design constraints are the magnitude of center displacement and lamina failure criteria a procedure of two level optimization in a fuzzy environment is presented and this procedure is applied to the design of laminated composite plates to consider the influences of fiber direction and thickness of each lamina separately it is shown that the membership function in the case of failure can be deduced from continuum damage mechanics the membership function for failure criteria is the function of the internal state variable for degree of damage in the continuum damage mechanics the modified weibull membership function is chosen to be used as a simple case of damage progression the finite element method is used to calculate the displacement of the plate and the stresses of each lamina the fuzzy optimization formulation as a max min problem is transformed into a classical one this classical optimization problem is solved using the augmented lagrange multiplier method with the quasi newton method two examples are solved to demonstrate the procedure from the results it is shown that optimal design of laminated composite plates is more useful in a fuzzy environment and the modified weibull membership function characterizes the fuzziness of failure nomenclature bi upper limit of the inequality constraint d fuzzy decision f x objective function f x fuzzy goal

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

a semi spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates abstract we describe a new diabatic primitive equation model for studying regional and basin scale ocean circulation processes the model features coordinate transformations that efficiently incorporate moderately irregular basin geometries and large variations in bottom topography and permits the inclusion of both thermal and wind forcing a novel semi spectral solution procedure in which the vertical structure of the model variables is represented as a finite sum of user specifiable structure functions e g chebyshev polynomials provides faster than algebraic convergence of the vertical approximation scheme model performance is assessed on a variety of test problems drawn from coastal and large scale oceanography including unforced linear wave propagation in both regular and irregular geometries non linear flow over rough bottom topography and eddy mean flow interaction in a wind driven midlatitude ocean basin computational efficiency of the model is found to be comparable to other existing primitive equation ocean models despite the utilization of the higher order spectral methods

taylor series associated with a differential difference operator on the real line we extend the classical theory of taylor series to a first order differential difference operator on the real line which includes as a particular case the dunkl operator associated with the reflection group z2 on r more precisely we establish first a generalized taylor formula with integral remainder and then specify sufficient conditions for a function on r to be expanded as a generalized taylor series moreover we provide a criterion of analyticity for functions on r involving the differential difference operator

determining modes for the surface quasi geostrophic equation abstract we introduce a determining wavenumber for the surface quasi geostrophic sqg equation defined for each individual trajectory and then study its dependence on the force while in the subcritical and critical cases this wavenumber has a uniform upper bound it may blow up when the equation is supercritical a bound on the determining wavenumber provides determining modes and measures the number of degrees of freedom of the flow or resolution needed to describe a solution to the sqg equation

a rigorous treatment of experimental observations for the two dimensional navier stokes equations within a fully rigorous framework this paper shows that a finite number of point observations distributed in both space and time serve to determine the fully developed flow of a two dimensional fluid throughout the whole flow domain with the measurements taken at one fixed time the results imply that if the flow is resolved on a sufficiently small scale then its dynamics are entirely determined applied to experimental observations taken randomly from a small neighbourhood of one point in space but at staggered times they imply a version of taken s time delay embedding theorem

determining modes nodes and volume elements for stationary solutions of the navier stokes problem past a three dimensional body in this paper we show that every solution of the three dimensional exterior navier stokes boundary value problem corresponding to a given non zero constant velocity at infinity flow past a body and belonging to a very general functional class open image in new window can be determined by a finite number of parameters our results extend the analogous classical results by foias temam 6 7 and by jones titi 14 for the interior problem this extension is by no means trivial in that all fundamental tools used in the case of the interior problem such as compactness of the sobolev embeddings poincare s inequality and the special basis constituted by eigenfunctions of the stokes operator are no longer available for the exterior problem an important consequence of our results is that any solution in open image in new window is uniquely determined by the knowledge of the associated velocity field only near the boundary just how near it has to be depends only on the reynolds number and on the body

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

long time dynamics of forced critical sqg we prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi geostrophic equation sqg and prove that it has finite fractal box counting dimension in order to do so we give a new proof of global regularity for critical sqg the main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional laplacian which is used to bootstrap the regularity directly from l to c without the use of de giorgi techniques we prove that for large time the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force which is assumed to belong merely to l h1 using the fact that the solution is bounded independently of the initial data after a transient time in spaces conferring enough regularity we prove the existence of a compact absorbing set for the dynamics in h1 obtain the compactness of the linearization and the continuous differentiability of the solution map we then prove exponential decay of high yet finite dimensional volume elements in h1 along solution trajectories and use this property to bound the dimension of the global attractor

distinguishing smooth functions by a finite number of point values and a version of the takens embedding theorem abstract we prove a general result showing that a finite dimensional collection of smooth functions whose differences cannot vanish to infinite order can be distinguished by their values at a finite collection of points this theorem is then applied to the global attractors of various dissipative parabolic partial differential equations in particular for the one dimensional complex ginzburg landau equation and for the kuramoto sivashinsky equation we show that a finite number of measurements at a very small number of points two and four respectively serve to distinguish between different elements of the attractor this gives an infinite dimensional version of the takens time delay embedding theorem

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

on strong solutions of the multi layer quasi geostrophic equations of the ocean abstract in this article we study the multi layer quasi geostrophic equations of the ocean the existence of strong solutions is proved we also prove the existence of a maximal attractor in l 2 and we derive estimates of its hausdorff and fractal dimensions in terms of the data our estimates rely on a new formulation that we introduce for the multi layer quasi geostrophic equation of the ocean which replaces the nonhomogeneous boundary conditions and the nonlocal constraint on the stream function by a simple homogeneous dirichlet boundary condition this work improves the results given in c bernier existence of attractor for the quasi geostrophic approximation of the navier stokes equations and estimate of its dimension adv math sci appl 4 2 1994 465 489

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

a variant of cowling price s theorem for the dunkl transform on r abstract in this paper we prove a new version of cowling price s theorem for the dunkl transform on r

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

a note on the numerical analysis of material damage based on the theory of materials of type n abstract a generalization of the theory of materials of type n to continuum damage mechanics is presented then algorithms developed earlier by the authors for plasticity problems are extended and applied to the analysis of progressive damage of materials under large elastoplastic deformation results obtained by applying these methods to representative problems of metal forming are presented

the finiteness of determining degrees of freedom for the quasi geostrophic multi layer ocean model in this paper we consider a multi layer quasi geostrophic model of the ocean dynamics and we prove that the long time behaviour of its solutions can be described by a finite number of determining parameters under some additional condition we also show that the dynamics of the bottom layer is completely determined by parameters connected with upper layers only it means that the information about the bottom layer is not essential for a description of the long time behaviour of the system under consideration

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

mean convergence of fourier dunkl series abstract in the context of the dunkl transform a complete orthogonal system arises in a very natural way this paper studies the weighted norm convergence of the fourier series expansion associated to this system we establish conditions on the weights in terms of the a p classes of muckenhoupt which ensure the convergence necessary conditions are also proved which for a wide class of weights coincide with the sufficient conditions

exponential stability of the multi layer quasi geostrophic ocean model with delays abstract in this paper the multi layer quasi geostrophic ocean model with delays is studied first a few criteria are derived for asymptotic stability of the stationary solutions then to quantify the impact of delay on the overall system evolution an estimate is derived for comparing the quasi geostrophic ocean states with or without delays

exponential decay of the power spectrum and finite dimensionality for solutions of the three dimensional primitive equations in this article we estimate the number of modes volumes and nodes sufficient to describe well the solution of the three dimensional primitive equations the physical meaning of these estimates is also discussed we also study the exponential decay of the spatial power spectrum for the three dimensional primitive equations

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

determining nodes for semilinear parabolic equations we are concerned with the determination of the asymptotic behavior of strong solutions to the initial boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set more precisely if the asymptotic behavior of the strong solution is known on a suitable finite set which is called determining nodes then the asymptotic behavior of the strong solution itself is entirely determined we prove the above property by the energy method

on degrees of freedom of certain conservative turbulence models for the navier stokes equations abstract we study the degrees of freedom of several conservative computational turbulence models that are derived via a non dissipative regularizations of the navier stokes equations for the navier stokes the leray and the navier stokes equations we prove that the longtime behavior of their respective solutions is completely determined by a finite set of grid values and by a finite set of fourier modes for each turbulence model the number of determining nodes and of determining modes is estimated in terms of flow parameters such as viscosity smoothing length forcing and domain size these estimates are global as they do not depend on an individual solution

further properties of steady state solutions to the navier stokes problem past a three dimensional obstacle we show that the weak formulation of the steady state navier stokes problem in the exterior of a three dimensional compact set closure of a bounded domain corresponding to a nonzero velocity at infinity and subjected to a given body force is equivalent to a nonlinear equation in appropriate banach spaces we thus show that the relevant nonlinear operator enjoys a number of fundamental properties that allow us to derive many significant results for the original problem in particular we prove that the manifold constituted by the pairs u with the nondimensional speed at infinity reynolds number and u weak solution corresponding to and to a given body force f is for generic f a c one dimensional manifold and that for almost any the number of solutions is finite we also show that for any given f in the appropriate function space and any given 0 the corresponding solutions can be controlled by their specification only in a suitable neighborhood i of the boundary the size

local lyapunov exponents of the quasi geostrophic ocean dynamics the predictability of the quasi geostrophic ocean model is considered in this paper this is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density this model is driven by the wind stress on the surface it includes effects of lateral and bottom friction of the earth rotation and the nonlinear interaction between adjacent layers internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system to estimate the divergence rate of trajectories we compute lyapunov exponents of this system along with lyapunov exponents their generalization local lyapunov exponents are computed as a measure of divergence rate on finite time scales these exponents provide us with the information about principal mechanisms of local instability they can be useful in the forecasting of the forecast skill in meteorology and oceanology numerous studies are performed in order to understand the influence of physical form of the basin number of layers and numerical grid resolution parameters on the predictability of the system to study the geographical structure of the perturbation the distribution of initial error with the fastest norm growth is distinguished this distribution shows us the initial data structure the most sensible to errors and points out the regions where the most accuracy should be required the structure of this function depends much on the time scale of prediction

a two level finite element discretization of the streamfunction formulation of the stationary quasi geostrophic equations of the ocean in this paper we proposed a two level finite element discretization of the nonlinear stationary quasi geostrophic equations which model the wind driven large scale ocean circulation optimal error estimates for the two level finite element discretization were derived numerical experiments for the two level algorithm with the argyris finite element were also carried out the numerical results verified the theoretical error estimates and showed that for the appropriate scaling between the coarse and fine mesh sizes the two level algorithm significantly decreases the computational time of the standard one level algorithm

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

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

a comparison of two numerical methods for integrating a quasi geostrophic multilayer model of ocean circulations to study the feasibility of using the finite elements method for ocean modelling a quasi geostrophic eddy resolving general circulation model has been driven using either a classical finite differences technique or a finite elements technique this feasibility has been clearly demonstrated and a comparison of the models performances shows that for simulations of realistic basins the finite elements model cpu cost is only four times the finite difference model cost

nonlinear membership functions in multiobjective fuzzy optimization of mechanical and structural systems an application of fuzzy mathematical programming techniques to multiple objective design problems is presented two examples dealing with the multiobjective design of mechanical and structural systems are considered the concept of a pareto optimal and fuzzy pareto optimal solution is discussed and it is shown that the resulting formulation yields pareto optimal solutions the fundamental assumption in fuzzy mathematical programming applications involving the use of linear membership functions is critically examined several nonlinear shapes for the membership functions of the fuzzy sets are proposed consistent with varying perceptions of the designer and are analyzed to determine their impact on the overall design process these shapes correspond to what we define as the coefficient of membership satiation it is seen that optimum designs for both examples are strongly influenced by the sign of the membership satiation coefficient

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

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

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

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

determining nodes for partly dissipative reaction diffusion systems abstract in this paper we study the long time dynamical properties of the solutions of partly dissipative reaction diffusion systems of the form u t d u f x u g x v 0 x r n t 0 v t v h x u 0 the main result is that under appropriate assumptions the large time behavior of a solution is completely determined in certain sense if the large time behavior of the solution is known on a densely chosen yet finite set of points which are called determining nodes a concept initially introduced by foias and temam for the navier stokes equations math comput 43 1984 117

optimum design of structures in a fuzzy environment the basic concepts of the theory of fuzzy sets and its applications are briefly reviewed a method is then presented for solving fuzzy optimization problems using ordinary nonlinear programming techniques along with a numerical example possible applications of the approach described here are mentioned including application to the analysis of structural failures and development of expert systems for structural design 9 references

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

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

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

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

convergence of galerkin solutions and continuous dependence on data in spectrally hyperviscous models of 3d turbulent flow we obtain results on the convergence of galerkin solutions and continuous dependence on data for the spectrally hyperviscous navier stokes equations let un denote the galerkin approximates to the solution u and let wn u un then our main result uses the decomposition wn pnwn qnwn where for fixed n pn is the projection onto the first n eigenspaces of a and qn i pn for assumptions on n that compare well with those in related previous results the convergence of qnwn t h as n depends linearly on key parameters and on negative powers of n thus reflective of kolmogorov theory predictions that in high wavenumber modes viscous i e linear effects dominate meanwhile pnwn t h satisfies a more standard exponential estimate with positive but fractional dependence on n modifications of these estimates demonstrate continuous dependence on data

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