model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

homogenization of non linear scalar conservation laws we study the limit as e 0 of the entropy solutions of the equation partial t u varepsilon rm div x left a left frac x varepsilon u varepsilon right right 0 we prove that the sequence ue two scale converges toward a function u t x y and u is the unique solution of a limit evolution problem the remarkable point is that the limit problem is not a scalar conservation law but rather a kinetic equation in which the macroscopic and microscopic variables are mixed we also prove a strong convergence result in l rm loc 1

homogenization of linear transport equations with oscillatory vector fields the l 2 weak limit as varepsilon downarrow 0 of the solutions of the linear transport equation is studied begin gathered u t bf a left frac x varepsilon right cdot nabla x u 0 u t 0 u 0 left x frac x varepsilon right end gathered where varepsilon 0 x in r 2 t in 0 t for any t 0 bf a is a smooth divergence free vector field on the two dimensional unit torus and has no equilibrium points the initial condition u 0 is assumed to be smooth compactly supported in x and periodic in x varepsilon using a weak l 2 formulation and the idea of rotation number for flows on the two dimensional torus the effective equations for the weak limit are found they depend sensitively on the underlying flow on the two dimensional torus and its ergodicity

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

the effect of turbulence on mixing in prototype reaction diffusion systems the effect of turbulence on mixing in prototype reaction diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large scale mean flow plus a small scale spatiotemporal periodic flow in the limit of fast reaction and slow diffusion it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large scale velocity field this surprising result contrasts strongly with earlier work of the authors that always yields a large scale propagation speed enhanced by small scale turbulence for propagating fronts the mathematical reasons for these differences are pointed out this main theorem rigorously justifies the limit equilibrium approximations utilized in nonpremixed turbulent diffusion flames and condensation evaporation modeling in cloud physics in the fast reaction limit the subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero diffusion limit with a nontrivial effect of turbulence on mixing in the limit the situation with slow reaction and slow diffusion is also studied in the present work here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction diffusion equations physical examples from non premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results c 2000 john wiley sons inc

a vanishing viscosity approach to quasistatic evolution in plasticity with softening we deal with quasistatic evolution problems in plasticity with softening in the framework of small strain associative elastoplasticity the presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate independent problems to the case of a nonconvex energy functional we argue that in this case the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view thus we analyse a different selection criterion for the solutions of the quasistatic evolution problem based on a viscous approximation this leads to a generalized formulation in terms of young measures developed in the first part of the paper in the second part we apply our approach to some concrete examples

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

stochastic superparameterization in quasigeostrophic turbulence in this article we expand and develop the authors recent proposed methodology for efficient stochastic superparameterization algorithms for geophysical turbulence geophysical turbulence is characterized by significant intermittent cascades of energy from the unresolved to the resolved scales resulting in complex patterns of waves jets and vortices conventional superparameterization simulates large scale dynamics on a coarse grid in a physical domain and couples these dynamics to high resolution simulations on periodic domains embedded in the coarse grid stochastic superparameterization replaces the nonlinear deterministic eddy equations on periodic embedded domains by quasilinear stochastic approximations on formally infinite embedded domains the result is a seamless algorithm which never uses a small scale grid and is far cheaper than conventional sp but with significant success in difficult test problems various design choices in the algorithm are investigated in detail here including decoupling the timescale of evolution on the embedded domains from the length of the time step used on the coarse grid and sensitivity to certain assumed properties of the eddies e g the shape of the assumed eddy energy spectrum we present four closures based on stochastic superparameterization which elucidate the properties of the underlying framework a null hypothesis stochastic closure that uncouples the eddies from the mean a stochastic closure with nonlinearly coupled eddies and mean a nonlinear deterministic closure and a stochastic closure based on energy conservation the different algorithms are compared and contrasted on a stringent test suite for quasigeostrophic turbulence involving two layer dynamics on a plane forced by an imposed background shear the success of the algorithms developed here suggests that they may be fruitfully applied to more realistic situations they are expected to be particularly useful in providing accurate and efficient stochastic parameterizations for use in ensemble based state estimation and prediction

spectral stochastic two scale convergence method for parabolic pdes following the theory of two scale convergence method introduced by nguetseng siam j math anal 1989 20 608 623 and further developed by allaire siam j math anal 1992 23 1482 1518 we introduce the chaos two scale method as a spectral stochastic tool to tackle parabolic partial differential equations where the material properties are stochastic processes e t x of the form t x t e x e oscillating in both space and time variables with different speeds periodicity with respect to the fast or local variables is assumed and stationary gaussian material properties processes are considered copyright 2010 john wiley sons ltd

taylor dispersion in curved channels the taylor dispersion problem is to determine the long time behavior of the concentration density of a passive scalar contaminant diffusing in an incompressible channel flow given an initial injection of contaminant which is slowly varying order epsilon in the longitudinal direction taylor showed that the cross channel mean of the concentration experiences an enhanced diffusion relative to the mean flow this is due to mechanical rather than molecular events mercer and roberts have treated this problem by a formal application of infinite dimensional center manifold theory and extended the analysis to the case of a curved channel which is slowly varying in the longitudinal direction it is shown that after the introduction of appropriate multiple scales their center manifold technique can be used to carry out the taylor dispersion calculations for a channel whose cross section has large order 1 variation in the longitudinal directions also more general underlying flows can be allowed than was heretofore possible a heuristic argument is given to show the existence of some channel shape and flow field for which effective diffusion is reduced rather than enhanced a similar advection diffusion problem but with no channel boundary has been rigorously analyzed by homogenization theory and discussed in a recent paper of majda andmore mclaughlin the two problems are briefly compared less

some characterizations of operators satisfying a browder s theorem abstract we characterize the bounded linear operators t defined on banach spaces satisfying a browder s theorem or a weyl s theorem by means of the discontinuity of some maps defined on certain subsets of c several other characterizations are given in terms of localized svep as well as by means of the quasi nilpotent part the hyper kernel or the analytic core of i t

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

operators which do not have the single valued extension property abstract in this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator t l x on a banach space x and some quantities associated with t which play an important role in fredholm theory in particular we shall consider some conditions for which t does not have the single valued extension property at a point o c

corps de nombres peu ramifies et formes automorphes autoduales let s be a finite set of primes p in s and q s a maximal algebraic extension of q unramified outside s and infinity assume that s 2 we show that the natural maps gal q p bar q p gal q s q are injective much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of gl 2n a q with prescribed properties at all places that we study via the twisted trace formula of j arthur the techniques we develop shed also some lights on the orthogonal symplectic alternative for selfdual representations of gl 2n

a variational formulation of rate independent phase transformations using an extremum principle we propose a rate independent mesoscopic model for the hysteretic evolution of phase transformations in shape memory alloys the model uses the deformation and phase indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws using the associated functionals admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality

from discrete visco elasticity to continuum rate independent plasticity rigorous results we show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco elastic crystal lattice with quenched disorder the constitutive structure changes as a result of two concurrent limiting procedures the vanishing viscosity limit and the discrete to continuum limit in the course of these limits a non convex elastic problem transforms into a convex elastic problem while the quadratic rate dependent dissipation of visco elastic lattice transforms into a singular rate independent dissipation of an ideally plastic solid in order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape memory alloys the approach however is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models

weak turbulence plasma induced by two scale homogenization abstract this paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the vlasov poisson system it is shown that the homogenization limit in the sense of two scale limit of the distribution function satisfies the linear vlasov poisson equations moreover the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part the equilibrium distribution function is shown to be the solution of the nonlocal quasilinear velocity space diffusion equation we also investigate the landau damping from the point of view of homogenization through the two scale limit

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

numerical analysis of the m p p model of turbulence abstract two different numerical schemes are proposed for the study of the behavior of a viscous incompressible fluid in a turbulent regime using the m p p model they differ in their treatment of both velocity and turbulent kinetic energy at the time discretization stage the first a gear scheme is fully implicit while the second is of the alternate direction type in both the lagrangian coordinate is calculated by applying an explicit lax wendroff scheme various numerical experiments showed good agreement between the two schemes

single valued extension property at the points of the approximate point spectrum abstract a localized version of the single valued extension property is studied at the points which are not limit points of the approximate point spectrum as well as of the surjectivity spectrum in particular we shall characterize the single valued extension property at a point o c in the case that oi t is of kato type from this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

computation of invariant tori by the method of characteristics in this paper we present a technique for the numerical approximation of a branch of invariant tori of finite dimensional ordinary differential equations systems our approach is a discrete version of the graph transform technique used in analytical work by fenichel indiana univ math j 21 1971 pp 193 226 in contrast to our previous work l dieci j lorenz and r d russell siam j sci statist comput 12 1991 pp 607 647 the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori but determines and updates such a coordinate system during a continuation process we give general convergence results for the method and present its algorithmic description we also show how the method performs on two physically important nonlinear problems a system of two coupled oscillators and the forced van der pol oscillator in the latter case we discuss some modifications needed to approximate an invariant curve for the poincare map

young measure solutions for a viscoelastically damped wave equation with nonmonotone stress strain relation we study the viscoelastically damped wave equation with a nonmonotone stress strain relation this system describes the dynamics of phase transitions which is closely related to the creation of microstructures in order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states two questions are addressed in this work how do oscillations propagate in space and time what can be said about the long time behavior an appropriate tool to deal with oscillations are young measures they describe the local distribution or one point statistics of a sequence of fast fluctuating functions we demonstrate that highly oscillatory initial states generate in a unique fashion an evolution in the space of young measures and we derive the determining equations further on we prove a generalized dissipation identity for young measure solutions as a consequence it is shown that every low energy solution converges to a young measure equilibrium as t this is a generalization of g friesecke s j b mcleod s fm96 convergence result for classical solutions to the case of young measure solutions

homogenization of two dimensional linear flows with integral invariance we study the homogenization of two dimensional linear transport equations u t av xv ep cdot nabla xv u 0 where av is a nonvanishing vector field with integral invariance on the torus t 2 when the underlying flow on t 2 is ergodic we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate this result unifies and illuminates the previously known results in the special cases of incompressible flows and shear flows when the flow on t 2 is nonergodic the homogenized limit is an average over t 1 of solutions of linear transport equations with constant coefficients the convergence here is in the weak sense of w 1 infty loc rra 1 and the sharp convergence rate is oe one of the main ingredients in our analysis is a classical theorem due to kolmogorov regarding flows with integral invariance on t 2 to which we present here an elementary and constructive proof

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

a variational view of partial brittle damage evolution under time dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state elaborating on prior work 14 we establish the existence of a relaxed variational evolution where at each time the two states of the material combine to form a fine mixture optimal from the standpoint of the applied load at that time yet preserving the irreversibility of the damaging process

threshold based quasi static brittle damage evolution we introduce models for static and quasi static damage in elastic materials based on a strain threshold and then investigate the relationship between these threshold models and the energy based models introduced in francfort and marigo eur j mech a solids 12 149 189 1993 and francfort and garroni ration mech anal 182 1 125 152 2006 a somewhat surprising result is that while classical solutions for the energy models are also threshold solutions this is shown not to be the case for nonclassical solutions that is solutions with microstructure a new and arguably more physical definition of solutions with microstructure for the energy based model is then given in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors rather than by the tensors themselves we prove existence for this energy based problem and show that these solutions are also threshold solutions a by product of this analysis is that all local minimizers in both the classical setting and for the new microstructure definition are also global minimizers

parametrized invariant manifolds a recipe for multiscale modeling given a fine scale physical theory characterized by an evolutionary system of equations and a set of quantities defined from the variables of the fine theory that serve as a coarse representation of the fine scale phenomena a method for developing an evolutionary set of equations for the coarse quantities is presented this set may be considered as governing equations for a coarse model corresponding to the fine one since its solutions can be formally shown to generate exact approximate solutions to the fine theory in a well defined sense the method for generating the coarse representation is illustrated on model problems

quasistatic evolution problems for linearly elastic perfectly plastic materials the problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate independent processes existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation this approach provides a new approximation result for the solutions of the quasistatic evolution problem which are shown to be absolutely continuous in time four equivalent formulations of the problem in rate form are derived a strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain

probability density functions of decaying passive scalars in periodic domains an application of sinai yakhot theory employing the formalism introduced by phys rev lett 63 1962 1989 we study the probability density functions pdf s of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields the particular regime we focus on is one where the normalized scalar pdf s attain a self similar profile in finite time i e the so called strange or statistical eigenmode regime in accordance with the work of sinai and yakhot the central regions of the pdf s are power laws however the details of the pdf profiles are dependent on the physical parameters in the problem interestingly for small peclet numbers the pdf s resemble stretched or pure exponential functions whereas in the limit of large peclet numbers there emerges a universal gaussian form for the pdf numerical simulations are used to verify these predictions

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

spectral analysis and computation for homogenization of advection diffusion processes in steady flows advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of d involving the molecular diffusivity and a spectral measure of a self adjoint operator that depends on the characteristics of a randomly perturbed periodic flow in discrete formulations of each approach we express explicitly in terms of standard or generalized eigenvalues and eigenvectors of hermitian matrices we develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach we use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorou

periodic homogenization for inertial particles we study the problem of homogenization for inertial particles moving in a periodic velocity field and subject to molecular diffusion we show that under appropriate assumptions on the velocity field the large scale long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone to achieve this we use a formal multiple scale expansion in the scale parameter this expansion relies on the hypo ellipticity of the underlying diffusion an expression for the diffusivity tensor is found and various of its properties studied in particular an expansion in terms of the non dimensional particle relaxation time the stokes number is shown to co incide with the known result for passive non inertial tracers in the singular limit 0 this requires the solution of a singular perturbation problem achieved by means of a formal multiple scales expansion in incompressible and potential fields are studied as well as fields which are neither and theoretical findings are supported by numerical simulations

property w and perturbations ii abstract a bounded linear operator t l x defined on a banach space x satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate spectrum wa t coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a bounded operator t acting on a banach space under perturbations by finite rank operators by nilpotent operator and quasi nilpotent operators commuting with t

invariant manifolds in mechanics i the general construction of coarse theories from fine theories la theorie dite fine fournit potentiellement des informations tres detaillees sur le comportement du materiau et generalement plus d informations que necessaire tandis que la theorie dite grossiere donne des informations brutes plus accessibles qui peuvent etre suffisantes dans des circonstances donnees presentation d une methode permettant de deduire les theories grossieres des theories fines

calculating effective diffusivities in the limit of vanishing molecular diffusion in this paper we study the problem of the numerical calculation by monte carlo methods of the effective diffusivity for a particle moving in a periodic divergent free velocity field in the limit of vanishing molecular diffusion in this limit traditional numerical methods typically fail since they do not represent accurately the geometry of the underlying deterministic dynamics we propose a stochastic splitting method that takes into account the volume preserving property of the equations of motion in the absence of noise and when inertial effects can be neglected an extension of the method is then proposed for the cases where the noise has a non trivial time correlation structure and when inertial effects cannot be neglected the method of modified equations is used to explain failings of euler based methods the new stochastic geometric integrators are shown to outperform standard euler based integrators various asymptotic limits of physical interest are investigated by means of numerical experiments using the new integrators

some notes on periodic beltrami fields in cartesian geometry some mathematical facts about beltrami fields which are time independent solutions of the three dimensional incompressible euler equations with nontrivial helicity are assembled the linearization about an arbitrary fixed beltrami field is studied in a hamiltonian framework a factorization of this linearization is introduced and used to characterize null spaces for beltrami fields with ergodic streamlines an interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed

rate independent dynamics and kramers type phase transitions in nonlocal fokker planck equations with dynamical control the hysteretic behavior of many particle systems with non convex free energy can be modeled by nonlocal fokker planck equations that involve two small parameters and are driven by a time dependent constraint in this paper we consider the fast reaction regime related to kramers type phase transitions and show that the dynamics in the small parameter limit can be described by a rate independent evolution equation with hysteresis for the proof we first derive mass dissipation estimates by means of muckenhoupt constants formulate conditional stability estimates and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities

convection of microstructure and related problems we study how two flows that vary spatially in two widely separated scales evolve under the dynamics of euler s equations we use a multiple scale approach which we motivate by studying some simpler model problems

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

variations on weyl s theorem in this note we study the property w a variant of weyl s theorem introduced by rakocevic by means of the localized single valued extension property svep we establish for a bounded linear operator defined on a banach space several sufficient and necessary conditions for which property w holds we also relate this property with weyl s theorem and with another variant of it a weyl s theorem we show that weyl s theorem a weyl s theorem and property w for t respectively t coincide whenever t respectively t satisfies svep as a consequence of these results we obtain that several classes of commonly considered operators have property w

diffusing passive tracers in random incompressible flows statistical topography aspects the paper studies statistical characteristics of the passive tracer concentrations and of its spatial gradient in random incompressible velocity fields from the viewpoint of statistical topography the statistics of interest include mean values probability distributions as well as various functionals characterizing topographic features of tracers the functional approach is used we consider the influence of the mean flow the linear shear flow and the molecular diffusion coefficient on the statistics of the tracer most of our analysis is carried out in the framework of the delta correlated in time approximation and conditions for its applicability are established but we also consider the diffusion approximation scheme for finite correlation radius the latter is applied to a diffusing passive tracer that undergoes sedimentation in a random velocity field

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

enhanced diffusivity and intercell transition layers in 2 d models of passive advection a class of two dimensional isotropic divergence free vector fields is introduced and the effective diffusivity of the corresponding advection diffusion equations is studied these examples are very idealized flows but they can be solved exactly in the limit pe 1 scaling laws d d0 pe are obtained where d0 molecular diffusion pe peclet number with exponents in the range 0 1 and examples of stream functions with logarithmic singularities for which d d0pe the exponent is related by a simple formula to the shape of the stream function along cell boundaries suggesting that similar scaling laws should hold for more general 2 d closed cell flows

fracture paths from front kinetics relaxation and rate independence crack fronts play a fundamental role in engineering models for fracture they are the location of both crack growth and the energy dissipation due to growth however there has not been a rigorous mathematical definition of crack front nor rigorous mathematical analysis predicting fracture paths using these fronts as the location of growth and dissipation here we give a natural weak definition of crack front and front speed and consider models of crack growth in which the energy dissipation is a function of the front speed that is the dissipation rate at time t is of the form int f t psi v x t rm d mathcal h n 2 x where f t is the front at time t and v is the front speed we show how this dissipation can be used within existing models of quasi static fracture as well as in the new dissipation functionals of mielke ortiz an example of a constrained problem for which there is existence is shown but in general if there are no constraints or other energy penalties this dissipation must be relaxed we prove a general relaxation formula that gives the surprising result that the effective dissipation is always rate independent

diffusion coefficients as function of kubo number in random fields particle standard diffusion d k in 2d homogeneous isotropic stationary random velocity fields is studied numerically as a function of the dimensionless kubo number k defined as the ratio between the correlation time of the velocity field and its sweeping time the three cases corresponding to the energy spectra proportional in the inertial range respectively to k 3 k 5 3 and k 2 are considered and two different power law regimes are found for large but finite k

gradient systems with wiggly energies and related averaging problems gradient systems with wiggly energies of the form and a d were proposed by abeyaratne chu james 2 to study the kinetics of martensitic phase transitions their model may be recast in the framework of the theory of averaging as a dynamical system on d d with the slow variable x d and fast variable d however this problem lies completely outside the classical theory of averaging since the vertical flow on d is not ergodic for sets of positive measure and we must interpret averages to mean weak limits

coarse variables of autonomous ode systems and their evolution abstract given an autonomous system of ordinary differential equations ode we consider developing practical models for the deterministic slow coarse behavior of the ode system two types of coarse variables are considered the first type consists of running finite time averages of phase functions approaches to construct the coarse evolution equation for this type are discussed and implemented on a forced lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium we explore two strategies in one we compute locally invariant manifolds of the fast dynamics parameterized by the slow variables in the other the choice of our coarse variables automatically guarantees them to be slow in a precise sense this allows their evolution to be phrased in terms of averaging utilizing limit measures probability distributions of the fast flow coarse evolution equations are constructed based on these approaches and tested against coarse response of the microscopic models the second type of coarse variables are defined as non trivial scalar state functions that are required by design to evolve autonomously to the extent possible with the goal of being candidate state functions for unambiguously initializable coarse dynamics the question motivates a mathematical restatement in terms of a first order pde a computational approximation is developed and tested on the lorenz system and the hald hamiltonian system

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis