random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

attractors for the semilinear reaction diffusion equation with distribution derivatives in unbounded domains abstract in this paper we prove the existence of global attractors for a nonlinear reaction diffusion equation with a nonlinearity having a polynomial growth of arbitrary order p 1 p 2 and with distribution derivatives in the inhomogeneous term the global attractors are obtained in l 2 r n and l p r n respectively a new a priori estimate method has been used since the solutions of the equation have no higher regularity and the semigroup associated with the solutions is not continuous in l p r n the results are new and appear to be optimal

theorems about the attractor for incompressible non newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average this paper discusses the incompressible non newtonian fluid with rapidly oscillating external forces g x t g x t t possessing the average g 0 x t as 0 where 0 0 then we establish that the hausdorff distance between the uniform attractors a v and a 0 v in space v is also less than o as 0 finally we show a a v for each 0 0

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

optimal maps for the multidimensional monge kantorovich problem let 1 n be borel probability measures on d denote by 1 n the set of all n tuples t t1 tn such that ti d d i 1 n are borel measurable and satisfy 1 i v for all borel v d the multidimensional monge kantorovich problem investigated in this paper consists of finding s s1 sn 1 n minimizing over the set 1 n we study the case where the i s have finite second moments and vanish on d 1 rectifiable sets we prove existence and uniqueness of optimal maps s when we impose that s1 x x and give an explicit form of the maps si the result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality as a consequence we obtain uniqueness and characterization of an optimal measure for the multidimensional kantorovich problem 1998 john wiley sons inc

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

limiting behavior of non autonomous stochastic reaction diffusion equations on thin domains abstract this paper deals with the limiting behavior of stochastic reaction diffusion equations driven by multiplicative noise and deterministic non autonomous terms defined on thin domains we first prove the existence uniqueness and periodicity of pullback tempered random attractors for the equations in an n 1 dimensional narrow domain and then establish the upper semicontinuity of these attractors when a family of n 1 dimensional thin domains collapses onto an n dimensional domain

finite fractal dimensions of random attractors for stochastic fitzhugh nagumo system with multiplicative white noise abstract in this paper we consider the asymptotic behavior of solutions for stochastic non autonomous fitzhugh nagumo system with multiplicative white noise first we prove the existence of random attractor of the random dynamical system generated by the solutions of considered system then we present some conditions for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable banach space and apply these conditions to prove the finiteness of fractal dimension of random attractor

adaptive control of linearizable discrete time systems adaptive control is applied to a siso discrete time nonlinear system the control objective is the tracking of the trajectory a generalized classical adaptive scheme is proposed two results are obtained imposing different restrictions on the resulting closed loop

local histogram based segmentation using the wasserstein distance we propose and analyze a nonparametric region based active contour model for segmenting cluttered scenes the proposed model is unsupervised and assumes pixel intensity is independently identically distributed our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region based image term that uses histograms of pixel intensity to distinguish different regions more specifically the region data encourages segmentation so that local histograms within each region are approximately homogeneous an advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem we use wasserstein distance with exponent 1 to determine the dissimilarity between two histograms the wasserstein distance is a metric and is able to faithfully measure the distance between two histograms compared to many pointwise distances moreover it is insensitive to oscillations and therefore our model is robust to noise a fast global minimization method based on chan et al in siam j appl math 66 5 1632 1648 2006 bresson et al in j math imaging vis 28 2 151 167 2007 is employed to solve the proposed model the advantages of using this method are two fold first the computational time is less than that of the method by gradient descent of the associated euler lagrange equation chan et al in proc of ssvm pp 697 708 2007 second it is able to find a global minimizer finally we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

dynamics for a stochastic reaction diffusion equation with additive noise abstract in this paper we present a new scheme to study the dynamics of a stochastic reaction diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p 2 firstly we use this scheme to establish some new estimates higher order integrability of the difference of the solutions near the initial time instead of using the usual estimates about higher regularities and higher order integrability of solutions secondly we verify that the attraction in the usual l 2 q l 2 q d pullback random attractor indeed can be l 2 norm for any 0 and the solutions of the equation are continuous in h 0 1 q with respect to initial data thirdly we obtain the existence of pullback random attractor in h 0 1 q as an application of the continuity even for the corresponding deterministic cases the results and methods introduced in this paper are new

partial regularity of suitable weak solutions to the system of the incompressible non newtonian fluids abstract in this paper we investigate the partial regularity of the generalized solutions which are called the suitable weak solutions to the modified navier stokes equations which describes the dynamics of the incompressible monopolar non newtonian fluids it is proved that the singular points are concentrated on a closed set whose 5 2p for p 5 2 dimensional hausdorff measure is zero and the solution is a regular one for p 5 2

h1 random attractors for stochastic reaction diffusion equations with additive noise abstract in this paper a new theorem obtained by li and guo y li b guo random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations j differential equations 245 7 2008 1775 1800 is employed to some reaction diffusion equations with additive noise we prove the unique existence of d random attractors in h 0 1 for the corresponding random dynamical system we also obtain the existence of a unique random equilibrium for the generated random dynamical system under some restrictive assumption on the coefficients

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

h2 compact attractor for a non newtonian system in two dimensional unbounded domains abstract in this paper the authors study the long time behavior of a non newtonian system in two dimensional unbounded domains they prove the existence of h2 compact attractor for the system by showing the corresponding semigroup is asymptotically compact

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping abstract in this paper we proved local existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping in one dimensional bounded intervals the main difficulty is due to the strong nonlinearity of the system and initial vacuum

a note on direct adaptive control of systems with bounded disturbances abstract a direct adaptive controller for discrete time minimum phase systems with bounded disturbances is presented it is shown that asymptotically the tracking error is smaller than the upperbound of the regression form disturbance egardt 1978 parameter increment convergence to zero and uniform boundedness of all signals are established

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

random attractors for damped non autonomous wave equations with memory and white noise abstract we first prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped non autonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent then we study the upper semicontinuity of random attractors when the coefficient of random term approaches zero

a note on the asymptotic smoothing effect of solutions to a non newtonian system in 2 d unbounded domains abstract in this note the authors consider the regularity of the global attractor for a non newtonian system in two dimensional unbounded domains they prove that the l 2 compact attractor and the h 2 compact attractor of the system are the same and thus establish the asymptotic smoothing effect of solutions to this system

a fixed point approach to barycenters in wasserstein space abstract let p 2 a c be the set of borel probabilities on r d with finite second moment and absolutely continuous with respect to lebesgue measure we consider the problem of finding the barycenter or frechet mean of a finite set of probabilities 1 k p 2 a c with respect to the l 2 wasserstein metric for this task we introduce an operator on p 2 a c related to the optimal transport maps pushing forward any p 2 a c to 1 k under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter the procedure allows effective computation of barycenters in any location scatter family including the gaussian case in such cases the barycenter must belong to the family thus it is characterized by its mean and covariance matrix while its mean is just the weighted mean of the means of the probabilities the covariance matrix is characterized in terms of their covariance matrices 1 k through a nonlinear matrix equation the performance of the iterative procedure in this case is illustrated through numerical simulations which show fast convergence towards the barycenter

a finite dimensional attractor of the moore greitzer pde model the moore greitzer pde model with viscosity is presented and the equations are rewritten as an evolution equation on a hilbert space it is proven that the model has a unique global solution which is smooth in space and time furthermore it is proven that there exists a global attractor i e a compact set which attracts all bounded sets finally it is shown that the attractor has a finite fractal dimension

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

on the largest lyapunov exponent for products of gaussian matrices the paper provides a new integral formula for the largest lyapunov exponent of gaussian matrices which is valid in the real complex and quaternion valued cases this formula is applied to derive asymptotic expressions for the largest lyapunov exponent when the size of the matrix is large and compare the lyapunov exponents in models with a spike and no spikes

slic superpixels compared to state of the art superpixel methods computer vision applications have come to rely increasingly on superpixels in recent years but it is not always clear what constitutes a good superpixel algorithm in an effort to understand the benefits and drawbacks of existing methods we empirically compare five state of the art superpixel algorithms for their ability to adhere to image boundaries speed memory efficiency and their impact on segmentation performance we then introduce a new superpixel algorithm simple linear iterative clustering slic which adapts a k means clustering approach to efficiently generate superpixels despite its simplicity slic adheres to boundaries as well as or better than previous methods at the same time it is faster and more memory efficient improves segmentation performance and is straightforward to extend to supervoxel generation

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

pullback attractors for a non autonomous incompressible non newtonian fluid this paper studies the pullback asymptotic behavior of solutions for a non autonomous incompressible non newtonian fluid in two dimensional 2d bounded domains we first prove the existence of pullback attractors av in space v has h2 regularity see notation in section 2 and ah in space h has l2 regularity for the cocycle corresponding to the solutions of the fluid then we verify the regularity of the pullback attractors by showing av ah which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data

random exponential attractor for cocycle and application to non autonomous stochastic lattice systems with multiplicative white noise abstract we first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable banach space and weighted spaces of infinite sequences then we apply our abstract result to study the existence of random exponential attractors for non autonomous first order dissipative lattice dynamical systems with multiplicative white noise

existence regularity and decay rate of solutions of non newtonian flow the existence and the regularity of young measure valued solutions to non newtonian flows are considered furthermore the uniqueness of solutions and their asymptotic behavior are given

limiting dynamics for stochastic wave equations abstract in this paper relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered by introducing almost surely d contracting property for random dynamical systems we obtain a global random attractor of the stochastic wave equation u t t u t u f u w endowed with dirichlet boundary condition for any 0 1 the upper semicontinuity of this global random attractor and the global attractor of the heat equation z t z f z 0 with dirichlet boundary condition as goes to zero is investigated furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation

uniform attractor for a two dimensional nonautonomous incompressible non newtonian fluid abstract we discuss the long time behavior of solutions for a two dimensional 2d nonautonomous incompressible non newtonian fluid in 2d bounded domains when the external force g 0 x t is translation compact tr c in l loc 2 r h we obtain the existence and reveal the structure of the uniform attractor for the processes associated to the fluid by constructing skew product flow on the extended phase space when g 0 l b 2 is properly small we establish that there exists a unique bounded asymptotically stable solution to the fluid

existence of global stochastic flow and attractors for navier stokes equations for 2 d stochastic navier stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors the equations considered do not admit a pathwise method of solution

random attractor for the ladyzhenskaya model with additive noise this paper studies the dynamical behavior of the ladyzhenskaya model with additive noise with some conditions we prove that the generated random dynamical system has a compact random attractor which is a random compact set absorbing any bounded nonrandom subset of the phase space

regularity of trajectory attractor and upper semicontinuity of global attractor for a 2d non newtonian fluid abstract there are two results within this paper the one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non newtonian fluid on 2d bounded domains for which the solution to each initial value could be non unique the other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from m to r l l where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m as m that is let a and a m be the global attractors of the fluid corresponding to and m respectively we establish that for any neighborhood o a of a the global attractor a m enters o a if m is large enough

turbulent transport and the random occurrence of coherent events abstract we suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events which correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor homoclinic excursions are trajectories in the noncompact phase space that are attracted to special orbits which connect saddle points in the finite region of phase space to infinity and represent organized structures in the flow field this picture also suggests that one can compute fluxes using a relatively low dimensional description of the flow a method for extracting the organized structures from a time series is given and provides a local analogue of the notion of lyapunov exponents

l2 compact uniform attractors for a nonautonomous incompressible non newtonian fluid with locally uniformly integrable external forces in distribution space we consider the long time behavior of solutions for a two dimensional nonautonomous incompressible non newtonian fluid with external forces in distribution space when the external force g0 x t is locally uniformly integrable see definition 3 1 in lloc2 r w we prove the existence of l2 compact uniform attractor in space h and reveal its structure for the families of processes corresponding to the fluid moreover if g0 lb2 r w is properly small we establish the unique existence of bounded asymptotically stable solutions and give two interesting corollaries

asymptotic behavior of two dimensional stochastic magneto hydrodynamics equations with additive noises this paper is devoted to the investigation of the asymptotic behavior of solutions of the stochastic magneto hydrodynamics equations driven by random exterior forced terms both in the velocity and in the magnetic field the nonlinear term is supposed to be time dependent and satisfies certain dissipative conditions the existence of a random attractor for this random dynamical system is obtained

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

the norm of products of free random variables let xi denote free identically distributed random variables this paper investigates how the norm of products pi n x 1 x 2 cdots x n behaves as n approaches infinity in addition for positive xi it studies the asymptotic behavior of the norm of y n x 1 circ x 2 circ cdots circ x n where circ denotes the symmetric product of two positive operators a circ b a 1 2 ba 1 2 it is proved that if exi 1 then left vert y n right vert is between c 1 sqrt n and c2n for certain constant c1 and c2 for left vert pi n right vert it is proved that the limit of n 1 log left vert pi n right vert exists and equals log sqrt e left x i ast x i right finally if is a cyclic representation of the algebra generated by xi and if is a cyclic vector then n 1 log left vert pi left pi n right xi right vert log sqrt e left x i ast x i right for all n these results are significantly different from analogous results for commuting random variables

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

a cell dynamical system model of chemical turbulence a cellular automaton like caricature of chemical turbulence on an infinite one dimensional lattice is studied the model exhibits apparently turbulent space time patterns to make this statement precise the following problems or points are discussed 1 the infinite system size limit of such cell dynamical systems and its observability is defined 2 it is proved that the invariant state in the large system size limit of the turbulent phase exhibits spatial patterns governed by a gibbs random field 3 potential characteristics of turbulent space time patterns are critically surveyed and a working definition of weak turbulence is proposed 4 it is proved that the invariant state of the turbulent phase is actually weak turbulent furthermore we conjecture that the turbulent phase of our model is an example of a k system that is not bernoulli

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

hausdorff dimension of attractors for the two dimensional navier stokes equations with boundary conditions we consider a viscous incompressible fluid enclosed in a region of ir2 and subject to boundary conditions we obtain explicit bounds depending only on the data for the entropy kolmogorov sinai invariant and the hausdorff dimension of attracting sets

self tuning controllers for nonlinear systems abstract two new self tuning control stc strategies for nonlinear control problems are proposed these strategies are applicable to a broad class of nonlinear single input single output systems which can include arbitrary nonlinear functions of the output and the old inputs as well as the products of these functions and any power of the most recent input simulation results for a stirred tank chemical reactor demonstrate that the new nonlinear stc strategy provides significant improvement over conventional stcs based on linear models

uniform attractors for nonautonomous incompressible non newtonian fluid with locally uniform integrable external forces this paper discusses the long time behavior of solutions for a two dimensional 2d nonautonomous incompressible non newtonian fluid in 2d bounded domains when the external force g0 x t is locally uniform integrable see definition 1 1 in lloc2 r h the authors obtain the existence and structure of uniform attractors in space v for the family of processes associated with the fluid moreover when g0 lb2 is properly small they provide some interesting corollaries

trajectory attractor and global attractor for a two dimensional incompressible non newtonian fluid the authors construct the trajectory attractor and global attractor for an autonomous two dimensional non newtonian fluid

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

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

attractors for partly dissipative reaction diffusion systems in rn in this paper we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2 n 2 n l l 2000 academic press

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

the convergence of non newtonian fluids to navier stokes equations abstract in this paper the convergence of solutions for incompressible dipolar viscous non newtonian fluids is investigated we obtain the conclusion that the solutions of non newtonian fluids converge to the solutions of navier stokes equations in the sense of l 2 norm resp h 1 norm as the viscosities tend to zero and the initial data belong to h 1 resp h 2 moreover we obtain l norm convergence of solutions if the initial data belong to h 2

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

characteristic exponents for a viscous fluid subjected to time dependent forces we consider a viscous incompressible fluid enclosed in a bounded region of 2 or 3 and subjected to time dependent forces using bound state estimates for the schrodinger operator we obtain rigorous bounds for the characteristic exponents entropy kolmogorov sinai invariant and hausdorff dimension of attracting sets our methods are of potential use for more general time evolutions described by nonlinear partial differential equations

on characteristic exponents in turbulence ruelle has found upper bounds to the magnitude and to the number of non negative characteristic exponents for the navier stokes flow of an incompressible fluid in a domain the latter is particularly important because it yields an upper bound to the hausdorff dimension of attracting sets however ruelle s bound on the number has three deficiences i it relies on some unproved conjectures about certain constants ii it is valid only in dimensions 3 and not 2 iii it is valid only in the limit in this paper these deficiences are remedied and in addition the final constants in the inequality are improved

pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on two dimensional unbounded domains this paper studies the pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on r l l for some l 0 we first use the technique of truncation functions together with the decomposition of spatial domain to prove the existence of a pullback attractor in then we discuss the upper semicontinuity of the pullback attractors when the spatial domains vary from m to where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m 1 m let a a r and am am r be the pullback attractors corresponding to and m respectively we establish for each r that for any neighborhood o a of a am enters o a if m is large enough

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s

limiting behavior of non autonomous stochastic reaction diffusion equations on unbounded thin domains this article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains we first prove the existence of tempered pullback random attractors for the equations defined on n 1 dimensional unbounded thin domains then we show the upper semicontinuity of these attractors when the n 1 dimensional unbounded thin domains collapse onto the n dimensional space rn here the tail estimates are utilized to deal with the noncompactness of sobolev embeddings on unbounded domains

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

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

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

a finite dimensional attractor for three dimensional flow of incompressible fluids abstract the asymptotic behavior of the flow for a system of the navier stokes type is investigated in the considered model the viscous part of the stress tensor is generally a nonlinear function of the symmetric part of the velocity gradient provided that the function describing this dependence satisfies the polynomial p 1 growth condition a unique weak solution exists if either p 2 n 2 and u 0 h or p 1 2 n n 2 and u 0 w 1 2 c n h in the first case the existence of a global attractor in h is proved in order to indicate the finite dimensional behavior of the flow at infinity the fractal dimension of the new invariant set composed from all short trajectories with initial value in the attractor is estimated in the l 2 0 h topology having uniqueness only for more regular data in the second case many trajectories can start from the initial value u 0 h this does not allow one to define a semigroup on the space h therefore the set of short trajectories x s closed in l 2 0 h is introduced along with a semigroup working on this set the existence of a global attractor with a finite fractal dimension is then demonstrated

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

a variational model for histogram transfer of color images in this paper we propose a variational formulation for histogram transfer of two or more color images we study an energy functional composed by three terms one tends to approach the cumulative histograms of the transformed images the other two tend to maintain the colors and geometry of the original images by minimizing this energy we obtain an algorithm that balances equalization and the conservation of features of the original images as a result they evolve while approaching an intermediate histogram between them this intermediate histogram does not need to be specified in advance but it is a natural result of the model finally we provide experiments showing that the proposed method compares well with the state of the art

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159