on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

error estimates for the discretization of the velocity tracking problem in this paper we are continuing our work casas and chrysafinos siam j numer anal 50 5 2281 2306 2012 concerning a priori error estimates for the velocity tracking of two dimensional evolutionary navier stokes flows the controls are of distributed type and subject to point wise control constraints the discretization scheme of the state and adjoint equations is based on a discontinuous time stepping scheme in time combined with conforming finite elements in space for the velocity and pressure provided that the time and space discretization parameters tau and h h respectively satisfy tau le ch 2 ch2 error estimates of order mathcal o h 2 o h2 and mathcal o h frac 3 2 frac 2 p o h32 2p with p 3 p 3 depending on the regularity of the target and the initial velocity are proved for the difference between the locally optimal controls and their discrete approximations when the controls are discretized by the variational discretization approach and by using piecewise linear functions in space respectively both results are based on new duality arguments for the evolutionary navier stokes equations

dynamics of stochastic three dimensional navier stokes voigt equations on unbounded domains abstract the aim of this paper is to study the long time behavior of the following stochastic 3d navier stokes voigt equation u t u 2 u t u u p g x e h d d t in an arbitrary bounded or unbounded domain satisfying the poincare inequality by famous j ball s energy equation method we obtain a unique random attractor a e for the random dynamical system generated by the equation moreover we prove that the random attractor a e tends to the global attractor a 0 of the deterministic equation in the sense of hausdorff semi distance as e 0

weak and strong solutions for the incompressible navier stokes equations with damping in this paper we show that the cauchy problem of the navier stokes equations with damping u 1u 0 has global weak solutions for any 1 global strong solution for any 7 2 and that the strong solution is unique for any 7 2 5

a study of boundedness in probabilistic normed spaces abstract it was shown in lafuerza guillen rodriguez lallena and sempi 1999 8 that uniform boundedness in a serstnev pn space v named boundedness in the present setting of a subset a v with respect to the strong topology is equivalent to the fact that the probabilistic radius r a of a is an element of d here we extend the equivalence just mentioned to a larger class of pn spaces namely those pn spaces that are topological vector spaces briefly tv spaces but are not serstnev pn spaces we present a characterization of those pn spaces whether they are tv spaces or not in which the equivalence holds then a characterization of the archimedeanity of triangle functions of type t l is given this work is a partial solution to a problem of comparing the concepts of distributional boundedness d bounded in short and that of boundedness in the sense of associated strong topology

random attractors for stochastic 2d navier stokes equations in some unbounded domains abstract we show that the stochastic flow generated by the 2 dimensional stochastic navier stokes equations with rough noise on a poincare like domain has a unique random attractor one of the technical problems associated with the rough noise is overcomed by the use of the corresponding cameron martin or reproducing kernel hilbert space our results complement the result by brze niak and li 2006 10 who showed that the corresponding flow is asymptotically compact and also generalize caraballo et al 2006 12 who proved existence of a unique attractor for the time dependent deterministic navier stokes equations

a study of boundedness in probabilistic normed spaces probabilistic normed spaces have been redefined by alsina schweizer and sklar we give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

on almost surely bounded semigroups of random linear operators menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry following menger s idea the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations the main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators which is not involved in the classical case subsequently some random operators equations are studied in particular we discuss the schrodinger type random equation which considerably generalizes the corresponding result in the sense of skorohod

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

on the 3d kelvin voigt brinkman forchheimer equations in some unbounded domains abstract considered here is the first initial boundary value problem for the kelvin voigt brinkman forchheimer equations in three dimensional domains satisfying the poincare inequality the existence of a weak solution to the problem is proved by using the faedo galerkin method then we show the existence of a unique minimal pullback d attractor for the process associated to the problem with respect to a large class of non autonomous forcing terms finally the existence uniqueness and stability of a stationary solution is studied when the external force is time independent and small

the algebraic structure of finitely generated l0 f k modules and the helly theorem in random normed modules abstract let k be the scalar field of real numbers or complex numbers and l 0 f k the algebra of equivalence classes of k valued random variables defined on a probability space f p in this paper we first characterize the algebraic structure of finitely generated l 0 f k modules and then combining the recently developed separation theorem in random locally convex modules we prove the helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property

countable products of probabilistic normed spaces abstract countable products of probabilistic normed spaces are introduced and studied in particular a comparison is made with the analogous constructions for probabilistic metric spaces

probabilistic norms and convergence of random variables abstract we prove that the probabilistic norms of suitable probabilistic normed spaces induce convergence in probability l p convergence and almost sure convergence

monotone circuits for matching require linear depth it is proven that monotone circuits computing the perfect matching function on n vertex graphs require o n depth this implies an exponential gap between the depth of monotone and nonmonotone circuits

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

on the uniqueness of strong solution to the incompressible navier stokes equations with damping abstract in this paper we show that the cauchy problem of the incompressible navier stokes equations with damping u 1 u 0 has global strong solution for any 3 and the strong solution is unique when 3 5 this improves earlier results

on the regularity and convergence of solutions to the 3d navier stokes voigt equations abstract in this paper we consider the 3d navier stokes voigt equations with periodic boundary conditions we first prove the higher order global regularity including both sobolev and gevrey regularity of solutions to the navier stokes voigt equations then we show the convergence of solutions of the 3d navier stokes voigt equations to the corresponding strong solution of the limit 3d navier stokes equations on the interval of existence of the latter as the parameter tends to zero

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

lower bounds for lovasz schrijver systems and beyond follow from multiparty communication complexity we prove that an omega log 4 n lower bound for the three party number on the forehead nof communication complexity of the set disjointness function implies an n omega 1 size lower bound for treelike lovasz schrijver systems that refute unsatisfiable formulas in conjunctive normal form cnfs more generally we prove that an n omega 1 lower bound for the k 1 party nof communication complexity of set disjointness implies a 2 n omega 1 size lower bound for all treelike proof systems whose formulas are degree k polynomial inequalities

gevrey regularity for a class of dissipative equations with applications to decay abstract in this paper following the techniques of foias and temam we establish gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional laplacian the initial data is taken to be in besov type spaces defined via caloric extension we apply our result to the navier stokes equations the surface quasi geostrophic equations the kuramoto sivashinsky equation and the barotropic quasi geostrophic equation consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the navier stokes equations in the 3d case we extend the class of initial data where such decay holds while in 2d we provide a new class for such decay similar decay result and uniform analyticity band on the attractor is also proven for the sub critical 2d surface quasi geostrophic equation

the topological structure of a certain menger space in this paper we will reconsider the topological structure of menger probabilistic normed spaces briefly pn spaces under the t norm m we will prove that this topology is compatible with the topology induced by a countable and separating family of semi norms and hence the well known theorems of classical functional analysis such as the principle of uniform boundedness open mapping and closed graph theorems are valid in this context also we will meanwhile obtain a method by which one may construct easily a large class of pn spaces finally using this method we see that a certain subspace of bounded linear operators between pn spaces i e the class of strongly bounded linear operators has a natural pn structure

navier stokes equations with delays on unbounded domains abstract some results on the existence and uniqueness of solutions to navier stokes equations when the domain is unbounded and the external force contains some hereditary characteristics are proved for both the evolutionary and the stationary cases exponential stability of the stationary solution is also established in dimension two

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

the bishop phelps theorem in complete random normed modules endowed with the e λ topology abstract in this paper we adopt a new approach so that we can prove that the bishop phelps theorem in complete random normed modules still holds under the e topology which solves an open problem posed in t x guo y j yang ekeland s variational principle for an l 0 valued function on a complete random metric space j math anal appl 389 2012 1 14

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

linear operators in finite dimensional probabilistic normed spaces in this paper we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

decay characterization of solutions to navier stokes voigt equations in terms of the initial datum abstract the navier stokes voigt equations are a regularization of the navier stokes equations that share some of its asymptotic and statistical properties and have been used in direct numerical simulations of the latter in this article we characterize the decay rate of solutions to the navier stokes voigt equations in terms of the decay character of the initial datum and study the long time behavior of its solutions by comparing them to solutions to the linear part

discontinuous galerkin approximations for an optimal control problem of three dimensional navier stokes voigt equations we analyze a fully discrete scheme based on the discontinuous in time galerkin approach which is combined with conforming finite element subspaces in space for the distributed optimal control problem of the three dimensional navier stokes voigt equations with a quadratic objective functional and box control constraints the space time error estimates of order o sqrt tau h where tau and h are respectively the time and space discretization parameters are proved for the difference between the locally optimal controls and their discrete approximations

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

gevrey regularity for the supercritical quasi geostrophic equation abstract in this paper following the techniques of foias and temam we establish suitable gevrey class regularity of solutions to the supercritical quasi geostrophic equations in the whole space with initial data in critical sobolev spaces moreover the gevrey class that we obtain is near optimal and as a corollary we obtain temporal decay rates of higher order sobolev norms of the solutions unlike the navier stokes or the subcritical quasi geostrophic equations the low dissipation poses a difficulty in establishing gevrey regularity a new commutator estimate in gevrey classes involving the dyadic littlewood paley operators is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

on the normability of generalized serstnev pn spaces probabilistic normed spaces have been redefined by c alsina b schweizer and a sklar but even today whether the generalized serstnev pn space is normable or not is still an open question in this paper through applying kolmogorov s theorem we will give several sufficient conditions under which many generalized serstnev pn spaces are normable as the application of our results we will give two examples which are normable generalized serstnev pn space but not serstnev space

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

the velocity tracking problem for navier stokes flows with boundary control we present some systematic approaches to the mathematical formulation and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in a bounded two dimensional domain with boundary control we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and full space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

semidiscretization and error estimates for distributed control of the instationary navier stokes equations optimal control problems governed by the two dimensional instationary navier stokes equations and their spatial discretizations with finite elements are investigated a concept of semi discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions furthermore an optimal error estimate in terms of the spatial discretization parameter is given

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

error estimates for the numerical approximation of a distributed control problem for the steady state navier stokes equations we obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary navier stokes equations with pointwise control constraints we show that the l 2 norm of the error for the control is of order h 2 if the control set is not discretized while it is of order h if it is discretized by piecewise constant functions these error estimates are obtained for local solutions of the control problem which are nonsingular in the sense that the linearized navier stokes equations around these solutions define some isomorphisms and which satisfy a second order sufficient optimality condition we establish a second order necessary optimality condition the gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems

upper bound of decay rate for solutions to the navier stokes voigt equations in r 3 in this paper we first show the global existence uniqueness and regularity of weak solutions for the navier stokes voigt equations in r 3 then we combine the fourier splitting method of schonbek and the gronwall inequality to prove that the solutions have the following decay rates m u x t 2 m 1 u x t 2 c 1 t 3 2 m for large t when u 0 h m r 3 l 1 r 3 and m 0 1

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

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

lacunary ideal convergence of multiple sequences in probabilistic normed spaces an ideal i is a family of subsets of positive integers n n which is closed under finite unions and subsets of its elements the aim of this paper is to study the notion of lacunary i convergence of double sequences in probabilistic normed spaces as a variant of the notion of ideal convergence also lacunary i limit points and lacunary i cluster points have been defined and the relation between them has been established furthermore lacunary cauchy and lacunary i cauchy lacunary i cauchy lacunary i convergent double sequences are introduced and studied in probabilistic normed spaces finally we provided example which shows that our method of convergence in probabilistic normed space is more general