exact relations in the theory of developed hydrodynamic turbulence exact relations of two types in the statistical theory of fully developed homogeneous isotropic turbulence in an incompressible fluid were found the relations of the first type connect two point and three point objects of the theory which are correlation functions and susceptibilities the second type of relations are the frequency sum rules which express some frequency integrals from fully dressed many point objects like vertices via corresponding bare values our approach is based on the navier stokes equation in quasi lagrangian variables and on the generating functional technique for correlation functions and susceptibilities the derivation of these relations uses no perturbation expansions and no additional assumptions this means that the relations are exact in the framework of the statistical theory of turbulence we showed that a many point scaling gives birth to the global scaling here many point scaling is the assumption that two point three point etc objects of the theory of turbulence are uniform functions in the inertial interval and may be characterized by some scaling exponents under this assumption the only global scale invariant model of fully developed turbulence suggested by kolmogorov dokl akad nauk ssr 32 19 1941 is consistent with the exact relations deduced

on solutions of the initial value problem for the nonlinear schrodinger equations on considere le probleme aux valeurs initiales pour l equation de schrodinger non lineaire de la forme iu t u f u 2 u t x r r n u 0 x x x r n on etudie la regularite des solutions quand les fonctions initiales ne sont pas regulieres

temporal multiscaling in hydrodynamic turbulence on the basis of the navier stokes equations we develop the high reynolds number statistical theory of different time many point spatial correlation functions of velocity differences we find that their time dependence is not scale invariant n order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents we derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents mathrm ensuremath zeta mathrm q of the standard structure functions we propose a representation of the time dependence using the legendre transform formalism of multifractals that automatically reproduces all the newly found bridge relationships

multiscale correlation functions in strong turbulence under the framework of yakhot phys rev e 57 1737 1998 we model intermittent structure functions in fully developed turbulence based on the experimentally supported markovian nature of turbulence cascades friedrich and peinke phys rev lett 78 863 1997 and calculate the multiscaling correlation functions fusion rules l vov and procaccia phys rev lett 76 2898 1996 which were experimentally tested benzi biferale and toschi phys rev lett 80 3244 1998 to be compatible with almost uncorrelated multiplicative process are analytically checked by direct calculations

locality and stability of the cascades of two dimensional turbulence we investigate and clarify the notion of locality as it pertains to the cascades of two dimensional turbulence the mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions first introduced by l vov and procaccia as a point of departure we use a revised version of the system of hypotheses that was proposed by frisch for three dimensional turbulence we show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non perturbative statistical locality we also investigate the stability conditions for both cascades we have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade for the enstrophy cascade statistical stability requires large scale dissipation and a vanishing downscale energy dissipation a careful discussion of the subtle notion of locality is given at the end of the paper

anomalous scaling in homogeneous isotropic turbulence the anomalous scaling exponents n of the longitudinal structure functions sn for homogeneous isotropic turbulence are derived from the navier stokes equations by using field theoretic methods to develop a low energy approximation in which the kolmogorov theory is shown to act effectively as a mean field theory the corrections to the kolmogorov exponents are expressed in terms of the anomalous dimensions of the composite operators which occur in the definition of sn these are calculated from the anomalous scaling of the appropriate class of nonlinear green function using an uv fixed point of the renormalization group which thereby establishes the connection with the dynamics of the turbulence the main result is an algebraic expression for n which contains no adjustable constants it is valid at orders n below g 1 where g is the fixed point coupling constant this expression is used to calculate n for orders in the range n 2 10 and the results are shown to be in good agreement with experimental data key examples being 2 0 7 3 1 and 6 1 8

analytic calculation of the anomalous exponents in turbulence using the fusion rules to flush out a small parameter the main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter in this paper we show that the formerly established fusion rules can be employed to develop a theory in which kolmogorov s statistics of 1941 k41 acts as the zero order or background statistics and the anomalous corrections to the k41 scaling exponents zeta n of the nth order structure functions can be computed analytically the crux of the method consists of renormalizing a four point interaction amplitude on the basis of the fusion rules this amplitude includes a small dimensionless parameter which is shown to be of the order of the anomaly of zeta 2 delta 2 zeta 2 2 3 approximately 0 03 higher order interaction amplitudes are shown to be even smaller the corrections to k41 to o delta 2 result from standard logarithmically divergent ladder diagrams in which the four point interaction acts as a rung the theory allows a calculation of the anomalous exponents zeta n in powers of the small parameter delta 2 the n dependence of the scaling exponents zeta n stems from pure combinatorics of the ladder diagrams in this paper we calculate the exponents zeta n up to o delta32 previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter delta 2 the actual evaluation of the small parameter delta 2 from first principles requires additional developments that are outside the scope of this paper

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

computing the scaling exponents in fluid turbulence from first principles the formal setup abstract we propose a scheme for the calculation from the navier stokes equations of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence the scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the euler equation it constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n as a consequence the scaling exponents are determined by solvability conditions and not from power counting it is argued that in order to achieve such a formulation one must recognize that the many point space time correlation functions are not scale invariant in their time arguments the assumption of full scale invariance leads unavoidably to kolmogorov exponents it is argued that the determination of all the scaling exponents n requires equations for infinitely many renormalized objects one can however proceed in controlled successive approximations by successive truncations of the infinite hierarchy of equations clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory to this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the nth order correlation functions and green s function in light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations this paper presents the conceptual and technical details of the scheme the analysis of the high order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with v belinicher

universal direct cascade in two dimensional turbulence we consider the correlation functions of vorticity ensuremath omega in the region of the direct cascade in a steady two dimensional turbulence the nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates and the main problem is to take those logarithms into account our procedure starts directly with the euler equation rewritten in the comoving reference frame we express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain it enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval we find the indices characterizing the logarithmic behavior of different correlation functions for example the two point simultaneous functions are as follows mathrm ensuremath omega mathit n mathbf r 1 mathrm ensuremath omega mathit n mathbf r 2 ensuremath sim mathit p 2 ln l mathbf r 1 mathbf r 2 ensuremath vert 2 mathit n 3 where l is the pumping scale we demonstrate that the form of those correlation functions is universal i e independent of the pumping the only pumping related value which enters the expressions is the enstrophy production rate mathit p 2 the contributions related to pumping rates mathit p mathit n of the higher order integrals of motion are demonstrated to be small in comparison with the ones induced by mathit p 2 we establish also the time dependence of the correlation functions the correlation time ensuremath tau in the comoving reference frame is the same for the vorticity and strain and is scale dependent ensuremath tau ensuremath propto mathrm ln 2 3 l r we reformulate our procedure in the diagrammatic language to reinforce the conclusions

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

inhomogeneous dirichlet boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques abstract we consider the inhomogeneous dirichlet boundary value problem for the cubic nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques

nonlinear chains inside walls abstract the conservative dynamics of a 1d chain of units coupled with fpu type nonlinear interactions is considered stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope this results in a zig zag pattern in the case of periodic boundary conditions and in kink or anti kink solutions in the case of the free boundary conditions imposing constraints on the chain by way of two confining infinitely high walls has repercussions for the stability of these stationary states here such stationary states commensurable with the available space between the two walls are examined in detail and their respective stability properties are determined analytically by invoking the transfer matrix method strikingly stationary anti kink solutions and periodic zig zag states being unstable in the absence of confining walls become stable when confining walls are introduced furthermore simulations reveal that chains with randomly generated initial conditions can seek out these patterns thus localising energy and persist for considerable time

on the dual cascade in two dimensional turbulence we study the dual cascade scenario for two dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range kmin kmax with kmin 0 such that the respective energy and enstrophy injection rates and satisfy k 2 k 2 the classical kraichnan leith batchelor paradigm based on the simultaneous conservation of energy and enstrophy and the scaleselectivity of the molecular viscosity requires that the domain be unbounded in both directions for two dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the acrosschannel direction a direct enstrophy cascade is not possible in the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers constant spectral slopes must satisfy 5 and 8 where is the asymptotic slope of the range of wavenumbers lower higher than the forcing wavenumber the influence of a large scale dissipation on the realizability of a dual cascade is analyzed we discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain

local energy flux and the refined similarity hypothesis in this paper we demonstrate the locality of energy transport for incompressible euler equations both in space and in scale the key to the proof is the proper definition of a local subscale flux t r which is supposed to be a measure of energy transfer to length scales l at the space point r kraichnan suggested that for such a quantity the refined similarity hypothesis will hold which kolmogorov originally assumed to hold instead for volume averaged dissipation we derive a local energy balance relation for the large scale motions which yields a natural definition of such a subscale flux for this definition a precise form of the refined similarity hypothesis is rigorously proved as a big o bound the established estimate is t r o l3h 1 in terms of the local holder exponenth at the point r which is also the estimate assumed in the parisi frisch multifractal model our method not only establishes locality of energy transfer but it also clarifies the physical reason that convection effects which naively violate locality do not contribute to the subscale flux in fact we show that as a consequence of incompressibility such effects enter into the local energy balance only as the divergence of a spatial current therefore they describe motion of energy in space and cancel in the integration over volume we also discuss theorems of onsager eyink and constantinet al on energy conservation for euler dynamics particularly to explain their relation with the parisi frisch model the constantinet al proof may be interpreted as giving a bound on the total flux t d d r t r of the form t r o lz3h 1 wherez3 is the third order scaling exponent or besov index in agreement with the multifractal model finally we discuss how the local estimates are related to the results of caffarelli kohn nirenberg on partial regularity for solutions of navier stokes equations they provide some heuristic support to a scenario proposed recently by pumir and siggia for singularities in the solutions of navier stokes with small enough viscosity

conditional statistics in scalar turbulence theory versus experiment we consider turbulent advection of a scalar field t b r passive or active and focus on the statistics of gradient fields conditioned on scalar differences delta t r across a scale r in particular we focus on two conditional averages langle nabla 2 t big delta t r rangle and langle nabla t 2 big delta t r rangle we find exact relations between these averages and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function p delta t r of delta t r these results offer a new way to analyze experimental data that is presented in this paper the main question that we ask is whether the conditional average langle nabla 2 t big delta t r rangle is linear in delta t we show that there exists a dimensionless parameter which governs the deviation from linearity the data analysis indicates that this parameter is very small for passive scalar advection and is generally a decreasing function of the rayleigh number for the convection data

two dimensional turbulence in the inverse cascade range numerical and physical experiments on forced two dimensional navier stokes equations show that transverse velocity differences are described by normal kolmogorov scaling proportional r 2n 3 and obey gaussian statistics since nontrivial scaling is a sign of the strong nonlinearity of the problem these two results seem to contradict each other a theory explaining these observations is presented in this paper the derived self consistent expression for the pressure gradient contributions leads to the conclusion that small scale transverse velocity differences are governed by a linear langevin like equation stirred by a nonlocal universal solution dependent gaussian random force this explains the experimentally observed gaussian statistics of transverse velocity differences and their kolmogorov scaling the solution for the pdf of longitudinal velocity differences is based on the numerical smallness of the energy flux in two dimensional turbulence the theory makes a few quantitative predictions that can be tested experimentally

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

the solitons of zabusky and kruskal revisited perspective in terms of the periodic spectral transform abstract we consider the numerical soliton experiments of zabusky and kruskal 1 and address their results in light of the spectral structure of the periodic korteweg devries kdv equation we derive and exploit a simple relationship between the infinite line transfer matrix and the monodromy matrix of periodic theory this establishes the connection at all stages of our analysis between the infinite line soliton and the soliton limit of periodic theory we develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic fourier analysis finally we introduce the concept of a reference level which is defined to be the level upon which the solitons propagate these results together with judicious numerical experiments allow us to reassess the work of zabusky and kruskal from the standpoint of periodic theory for the kdv equation we find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper we also find that even in the presence of periodic boundary conditions the solitons behave like infinite line solitons in that they undergo phase shifts upon collision with each other our estimate of the fpu recurrence time which includes the effect of phase shifting is within one percent of their observed value

anomalous scaling in the anisotropic sectors of the kraichnan model of passive scalar advection kraichnan s model of passive scalar advection in which the driving gaussian velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields we show here that the solutions of the kraichnan equation for the n order correlation functions foliate into sectors that are classified by the irreducible representations of the so d symmetry group we find a discrete spectrum of universal anomalous exponents with a different exponent characterizing the scaling behavior in every sector generically the correlation functions and structure functions appear as sums over all these contributions with nonuniversal amplitudes that are determined by the anisotropic boundary conditions the isotropic sector is always characterized by the smallest exponent and therefore for sufficiently small scales local isotropy is always restored the calculation of the anomalous exponents is done in two complementary ways in the first they are obtained from the analysis of the correlation functions of gradient fields the theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result in the second method we compute the exponents from the zero modes of the kraichnan equation for the correlation functions of the scalar field itself in this case the renormalization scale is the outer scale the two approaches lead to the same scaling exponents for the same statistical objects illuminating the relative role of the outer and inner scales as renormalization scales in addition we derive exact fusion rules which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions

on the elimination of the sweeping interactions from theories of hydrodynamic turbulence in this paper we revisit the claim that the eulerian and quasi lagrangian same time correlation tensors are equal this statement allows us to transform the results of an msr quasi lagrangian statistical theory of hydrodynamic turbulence back to the eulerian representation we define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity it is shown that both the elimination of the sweeping interactions and the derivation of the 4 5 law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity the quasi lagrangian transformation on the other hand requires an even stronger homogeneity assumption which is many time rather than one time but still weaker than many time global homogeneity we argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi lagrangian transformation

multidimensional inhomogeneous mixed initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous mixed initial boundary value problem for the nonlinear multidimensional schrodinger equation formulated on upper right quarter plane we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions our approach to get well posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument to get a linear theory for the multidimensional model we propose general method based on laplace approach and theory cauchy type integral equations to get smooth solutions in l we modify a method based on the factorization for the free schrodinger evolution group the advantage of our approach is that it can also be applied to non integrable equations and arbitrary boundary conditions this approach is new and it is not standard we believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis

multiscale correlations and conditional averages in numerical turbulence the equations of motion for the nth order velocity differences raise the interest in correlation functions containing both large and small scales simultaneously we consider the scaling of such objects and also their conditional average representation with emphasis on the question of whether they behave differently in the inertial or the viscous subranges the turbulent flow data are obtained by navier stokes solutions on a 603 grid with periodic boundary conditions and re 70 our results complement previous high re data analysis based on measured data a l fairhall v s l vov and i procaccia europhys lett 43 277 1998 whose preference were the larger scales and the analysis of both experimental and synthetic turbulence data by r benzi and co workers phys rev lett 80 3244 1998 phys fluids 11 2215 1999 the inertial range fusion rule is confirmed and insight is obtained for the conditional averages the local dissipation rate conditioned on the velocity fluctuations

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

towards a nonperturbative theory of hydrodynamic turbulence fusion rules exact bridge relations and anomalous viscous scaling functions in this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence of paramount importance for this theory are the fusion rules that describe the asymptotic properties of n point correlation functions when some of the coordinates tend toward one other we first derive here on the basis of two fundamental assumptions a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval using this set of fusion rules we consider the standard hierarchy of equations relating the n mathrm th order correlations originating from the viscous term in the navier stokes equations to n 1 mathrm th order originating from the nonlinear term and demonstrate that for fully unfused correlations the viscous term is negligible consequently the hierarchic chain of equations is decoupled in the sense that the correlations of n 1 mathrm th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions using the same hierarchy of equations when some separations go to zero we derive on the basis of the navier stokes equations a second set of fusion rules for correlations with differences in the viscous range the latter includes gradient fields we demonstrate that every n mathrm th order correlation function of velocity differences mathcal f n r 1 r 2 dots exhibits its own crossover length ensuremath eta n to dissipative behavior as a function of say r 1 this length depends on n and on the remaining separations r 2 r 3 dots when all these separations are of the same order r this length scales as ensuremath eta n r ensuremath sim ensuremath eta frac r l x n with x n frac ensuremath zeta n ensuremath ensuremath zeta n 1 ensuremath zeta 3 ensuremath ensuremath zeta 2 2 ensuremath ensuremath zeta 2 with ensuremath zeta n being the scaling exponent of the n mathrm th order structure function we derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents ensuremath zeta n of the n mathrm th order structure functions one of these relations is the well known bridge relation for the scaling exponent of dissipation fluctuations ensuremath mu 2 ensuremath ensuremath zeta 6

multilocality and fusion rules on the generalized structure functions in two dimensional and three dimensional navier stokes turbulence using the fusion rules hypothesis for three dimensional and two dimensional navier stokes turbulence we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences we call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator multilocality the resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry our main result is that the fusion rules hypothesis implies both locality and multilocality in both the ir and uv limits for the downscale energy cascade of three dimensional navier stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two dimensional navier stokes turbulence we stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two point correlation and response functions

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

inhomogeneous neumann boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques

inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity on the upper half plane abstract we consider the inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity in the upper half plane we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the problem by using the classical energy method and factorization techniques

from global scaling a la kolmogorov to local multifractal scaling in fully developed turbulence in the first part of the paper a modern presentation of scaling ideas is made it includes a reformulation of kolmogorov s 1941 theory bypassing the universality problem pointed out by landau and a presentation of the multifractal theory with emphasis on scaling rather than on cascades in the second part various historical aspects are discussed the importance of kolmogorov s rigorous derivation of the 4 5 l law for the third order structure function in his last 1941 turbulence paper is stressed this paper also contains evidence that he was aware of universality not being essential to the 1941 theory an inequality is established relating the exponents 2 p of the structure functions of order 2 p and the maximum velocity excursion it follows that models such as the obukhov kolmogorov 1962 log normal model in which 2 p does not increase monotonically are inconsistent with the basic physics of incompressible flow this result is independent of novikov s 1971 inequality in particular the proof presented here does not rely on the questionable relation proposed by obukhov and kolmogorov between instantaneous velocity increments and local averages of the dissipation

on solutions of the initial value problem for the nonlinear schrodinger equations in one space dimension t 0 xe has a regularizing property that the solutions become smooth instantly for t 0 even if their initial data are not smooth for example in l n our aim of this paper is to show that the nls equation 1 1 1 2 has a similar regularizing effect more precisely all solutions of finite energy to 1 1 1 2 are smooth for tq 0 provided the initial functions in hi n or in

a new approach to computing the scaling exponents in fluid turbulence from first principles in this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication v s l vov and i procaccia physica a 1998 in press chao dyn 9707015 the scaling exponents in this set of equations cannot be found from power counting in this short paper we discuss in detail low order non trivial closures of this infinite set of equations and prove that these closures lead to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l

recovering isotropic statistics in turbulence simulations the kolmogorov 4 5th law one of the main benchmarks in direct numerical simulations of three dimensional turbulence is the kolmogorov prediction for third order structure functions with homogeneous and isotropic statistics in the infinite reynolds number limit previous direct numerical simulations dns techniques to obtain isotropic statistics have relied on time averaging structure functions in a few directions over many eddy turnover times using forcing schemes carefully constructed to generate isotropic data motivated by recent theoretical work which removes isotropy requirements by spherically averaging the structure functions over all directions we will present results which supplement long time averaging by angle averaging over up to 73 directions from a single flow snapshot the directions are among those natural to a square computational grid and are weighted to approximate the spherical average we use this angle averaging procedure to compare the statistically steady flows generated by two different forcing schemes in a periodic box our results show that despite the apparent differences in the two flows their isotropic components as measured by the kolmogorov laws are essentially identical this procedure may be used to investigate the isotropic part of the small scale statistics of any quantity of interest the averaging process is inexpensive and for the kolmogorov 4 5 law reasonable results can be obtained from a single snapshot of data this implies consistency with the recently derived local versions of the kolmogorov laws which do not require long time averages

smoothing effect for nonlinear schrodinger equations in exterior domains abstract we consider the following nonlinear schrodinger equations in exterior domains i t u 1 2 u u 2 u t x r x d u 0 x x x d u t x 0 or u t6v 0 t x r x d where d x rn x r d r 0 and v denotes the outward normal unit vector at x d in this paper we prove the radially symmetric solutions of have a smoothing property

functional and graphical methods for classical statistical dynamics i a formulation of the martin siggia rose method a formulation of the martin siggia rose msr method for describing the statistical dynamics of classical systems is presented the present formulation is very similar in structure to the original msr operator formalism and very different from the alternative functional integral formalism of janssen de dominicis peliti and others the need for imposing certain boundary conditions in the msr formalism as pointed out by deker is clarified the basic results of this paper include a construction of the msr formalism in a way that demonstrates its internal consistency a definition of a functional whose functional derivatives give all the correlation functions and response functions of an ensemble of mechanical systems a graphical expression for the correlation functions and response functions a graphical expression for the legendre transform of the functional and of the resulting vertex functions and a graphical derivation of the appropriate dyson equation the present formulation is applicable to systems with highly non gaussian statistics including systems of classical particles described in terms of the particle density in single particle phase space in this paper we consider only the case of ensembles of systems whose coordinates are continuous and whose time evolution is described by deterministic first order differential equations that are local in time the method is easily extended to systems whose dynamics is governed by stochastic differential equations and to spin systems

robustness of the inverse cascade in two dimensional turbulence we study quasisteady inverse cascades in unbounded and bounded two dimensional turbulence driven by time independent injection and dissipated by molecular viscosity it is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy range energy spectrum to be steeper than k ensuremath 5 ruling out a direct cascade unless 1 ensuremath r ensuremath ll 1 a direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales 1 ensuremath r ensuremath ll 1 these facts underlie the robustness of the kolmogorov kraichnan k ensuremath 5 3 inverse cascade which is readily observable in numerical simulations without an accompanying direct enstrophy cascade we numerically demonstrate an instance where the k ensuremath 5 3 inverse cascading range is realizable with 79 of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region as equilibrium is approached the respective logarithmic slopes ensuremath alpha and ensuremath beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy ensuremath alpha ensuremath beta ensuremath approx 8 these results are consistent with recent theoretical predictions

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

computing the scaling exponents in fluid turbulence from first principles demonstration of multiscaling we develop a consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydro dynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication by v s l vov and i procaccia the scaling exponents in this set of equations cannot be found from power counting in this paper we present in detail the lowest non trivial closure of this infinite set of equations and prove that this closure leads to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l we demonstrate that the solvability condition of our equations leads to non kolmogorov values of the scaling exponents n finally we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence

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