remarks on the helicity of the 3 d incompressible euler equations in this note we obtain a sufficient condition on the regularity of the weak solutions to guarantee the conservation of helicity for the 3 d incompressible euler equations as a corollary we obtain a lower bound of the vorticities for a weak solution of the euler equations

on nonperiodic euler flows with holder regularity in isett regularity in time along the coarse scale flow for the euler equations 2013 the first author proposed a strengthening of onsager s conjecture on the failure of energy conservation for incompressible euler flows with holder regularity not exceeding 1 3 this stronger form of the conjecture implies that anomalous dissipation will fail for a generic euler flow with regularity below the onsager critical space l t infty b 3 infty 1 3 due to low regularity of the energy profile this paper is the first and main paper in a series of two the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with holder exponent less than 1 5 the main result of the present paper shows that any given smooth euler flow can be perturbed in c 1 5 epsilon t x on any pre compact subset of mathbb r times mathbb r 3 to violate energy conservation furthermore the perturbed solution is no smoother than c 1 5 epsilon t x as a corollary of this theorem we show the existence of nonzero c 1 5 epsilon t x solutions to euler with compact space time support generalizing previous work of the first author isett holder continuous euler flows in three dimensions with compact support in time 2012 to the nonperiodic setting

optimal control of dynamic systems application to spline approximations generally classical polynomial splines tend to exhibit unwanted undulations in this work we discuss a technique based on control principles for eliminating these undulations and increasing the smoothness properties of the spline interpolants we give a generalization of the classical polynomial splines and show that this generalization is in fact a family of splines that covers the broad spectrum of polynomial trigonometric and exponential splines a particular element in this family is determined by the appropriate control data it is shown that this technique is easy to implement several numerical and curve fitting examples are given to illustrate the advantages of this technique over the classical approach finally we discuss the convergence properties of the interpolant

on a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling we consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation we study a sample average approximation saa approach that uses nested sampling in which at each stage a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next stage constraints this approach provides an approximate solution to the multistage problem to establish the consistency of the saa approach we first consider a two stage problem and show that in the second stage problem given a scenario the optimal values and solutions of the saa converge to those of the true problem with probability one when the sample sizes go to infinity these convergence results do not hold uniformly over all possible scenarios for the second stage problem we are nevertheless able to prove that the optimal values and solutions of the saa converge to the true ones with probability one when the sample sizes at both stages increase to infinity we also prove exponential convergence of the probability of a large deviation for the optimal value of the saa the true value of an optimal solution of the saa and the probability that any optimal solution to the saa is an optimal solution of the true problem all of these results can be extended to a multistage setting and we explain how to do it our framework and saa results cover a large variety of resource allocation problems for which at each stage after the first one new information becomes available and the allocation can be readjusted under constraints that involve expectations estimated by monte carlo as an illustration we apply this saa method to a staffing problem in a call center in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service qos the staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

integration of brownian vector fields summary using the wiener chaos decomposition we show that strong solutions of non lipschitzian s d e s are given by random markovian kernels the example of sobolev flows is studied in some detail exhibiting interesting phase transitions

a stochastic approach to stability in stochastic programming abstract a random optimization problem p 0 min x 0 0 x is approximated by a sequence of random surrogate problems p n n n with p n min x n n x p a given probability space we investigate the convergence almost surely and in probability of the optimal values and the solution sets the results can be regarded as random versions of well known stability statements of parametric programming semicontinuous convergence almost surely in probability of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

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

translates of exponential box splines and their related spaces exponential box splines eb splines are multivariate compactly supported functions on a regular mesh which are piecewise in a space z spanned by exponential polynomials this space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients the main part of this paper is devoted to algebraic analysis of the space h of all entire functions spanned by the integer translates of an eb spline this investigation relies on a detailed description of z and its discrete analog s the approach taken here is based on the observation that the structure of z is relatively simple when z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process also we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of fourier analysis thus while generalizing the known theory of polynomial box splines the results here offer a simpler approach and a new insight towards this important special case we also identify and study in detail several types of singularities which occur only for complex eb splines the first is when the fourier transform of the eb spline vanishes at some critical points the second is when z cannot be embedded in y and the third is when h is a proper subspace of z we show among others that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh

asymptotic behavior of optimal solutions in stochastic programming asymptotic behavior of optimal solutions xi n of a sequence of stochastic programming problems is studied variational and generalized equations approaches are discussed an expansion of xi n in terms of a parametrized mathematical programming problem depending on a single random vector is given when optimal solutions of the parametrized program are directionally differentiable this expansion leads to a close form expression for the asymptotic distribution of xi n applicability of the involved regularity conditions to nondifferentiable cases and in particular to stochastic programming with recourse is discussed

convergence theory for nonconvex stochastic programming with an application to mixed logit monte carlo methods have extensively been used and studied in the area of stochastic programming their convergence properties typically consider global minimizers or first order critical points of the sample average approximation saa problems and minimizers of the true problem and show that the former converge to the latter for increasing sample size however the assumption of global minimization essentially restricts the scope of these results to convex problems we review and extend these results in two directions we allow for local saa minimizers of possibly nonconvex problems and prove under suitable conditions almost sure convergence of local second order solutions of the saa problem to second order critical points of the true problem we also apply this new theory to the estimation of mixed logit models for discrete choice analysis new useful convergence properties are derived in this context both for the constrained and unconstrained cases and associated estimates of the simulation bias and variance are proposed

stochastic least action principle for the incompressible navier stokes equation abstract we formulate a stochastic least action principle for solutions of the incompressible navier stokes equation which formally reduces to hamilton s principle for the incompressible euler solutions in the case of zero viscosity we use this principle to give a new derivation of a stochastic kelvin theorem for the navier stokes equation recently established by constantin and iyer which shows that this stochastic conservation law arises from particle relabelling symmetry of the action we discuss issues of irreversibility energy dissipation and the inviscid limit of navier stokes solutions in the framework of the stochastic variational principle in particular we discuss the connection of the stochastic kelvin theorem with our previous martingale hypothesis for fluid circulations in turbulent solutions of the incompressible euler equations

approximation with the output of linear control systems abstract in this paper we give a new proof that for controllable and observable linear systems every l 2 0 t function can be approximated in the l 2 0 t sense with an output function generated by an l 2 0 t input function we also give a new characterization of how continuous functions on 0 t are uniformly approximated by an output generated by a continuous input function the relative degree of the transfer function of the system determines those functions that can be approximated we further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions

a note on generalized flows the problem of constructing flows associated with first order ordinary differential equations odes with spatially non lipschitz right hand side is considered due to the lack of uniqueness of the solutions standard flows cannot be defined in this case for these situations it is natural to introduce generalized flows which are random fields constructed by assigning a probability measure on the set of maps associated with the solutions of the odes some properties of the generalized flows are discussed here in particular in terms of transport via simple one dimensional examples these simple examples display a wide variety of behaviors and indicate that a general theory of generalized flows is likely to be inaccessible because they typically lack desirable properties such as stability with respect to perturbations or markovianity in time

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

weak solutions with decreasing energy of incompressible euler equations weak solution of the euler equations is an l2 vector field u x t satisfying certain integral relations which express incompressibility and the momentum balance our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations as both viscosity and compressibility tend to zero thus we believe that weak solutions describe turbulent flows with very high reynolds numbers

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

numerical simulations of possible finite time singularities in the incompressible euler equations comparison of numerical methods abstract the numerical simulation of the 3d incompressible euler equations is analyzed with respect to different integration methods the numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods based on this comparison a kida pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given this estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective

numerical solutions of the isotropic 3 wave kinetic equation we show that the isotropic 3 wave kinetic equation is equivalent to the mean field rate equations for an aggregation fragmentation problem with an unusual fragmentation mechanism this analogy is used to write the theory of 3 wave turbulence almost entirely in terms of a single scaling parameter a new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending lee s method for solving aggregation equations the new algorithm is validated against some analytical calculations of the kolmogorov zakharov kz constant for some families of model interaction coefficients the algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature firstly it is shown that for finite capacity cascades the dissipation of energy becomes independent of the cut off frequency as this cut off is taken to infinity this is an explicit indication of the presence of a dissipative anomaly secondly a preliminary numerical study is presented of the so called bottleneck effect in a wave turbulence context it is found that the structure of the bottleneck depends non trivially on the interaction coefficient finally some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so called mixed solutions of the kinetic equation which exhibit aspects of both kz and equilibrium equipartition spectra

dissipative anomalies in singular euler flows abstract we discuss onsager s conjecture that non vanishing energy dissipation in high reynolds number turbulence is associated to singular distributional solutions of the incompressible euler equations we carefully explain the physical and mathematical meaning of the conjecture and also review relevant theoretical experimental and numerical work emphasizing some of the dramatic successes of onsager s point of view finally we present several new ideas and results on lagrangian dynamics of circulations and vortex lines that we believe will be important for future progress

flows coalescence and noise we are interested in stationary fluid random evolutions with independent increments under some mild assumptions we show they are solutions of a stochastic differential equation sde there are situations where these evolutions are not described by flows of diffeomorphisms but by coalescing flows or by flows of probability kernels in an intermediate phase for which there exist a coalescing flow and a flow of kernels solution of the sde a classification is given all solutions of the sde can be obtained by filtering a coalescing motion with respect to a subnoise containing the gaussian part of its noise thus the coalescing motion cannot be described by a white noise

local approximation by certain spaces of exponential polynomials approximation order of exponential box splines and related interpolation problems local approximation order to smooth complex valued functions by a finite dimensional space 7b h spanned by certain products of exponentials by polynomials is investigated the results obtained together with a suitable quasi interpolation scheme are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline the analysis of a typical 7b h is based here on the identification of its dual with a certain space 7b p of multivariate polynomials this point of view allows us to solve a class of multivariate interpolation problems by the polynomials from 7b p

the least action principle and the related concept of generalized flows for incompressible perfect fluids the link between the euler equations of perfect incompressible flows and the least action principle has been known for a long time i solu tions can be considered as geodesic curves along the manifold of volume pre serving mappings here the shortest path problem is investigated given two different volume preserving mappings at two different times find for the inter mediate times an incompressible flow map that minimizes the kinetic energy or more generally the action in its classical formulation this problem has been solved 7 only when the two different mappings are sufficiently close in some very strong sense in this paper a new framework is introduced where gen eralized flows are defined in the spirit of l c young as probability measures on the set of all possible trajectories in the physical space then the minimiza tion problem is generalized as the continuous linear programming problem that is much easier to handle the existence problem is completely solved in this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms the least action principle 255 the case of the d dimensional torus it is also shown that under natural re strictions a classical solution to the euler equations is the unique optimal flow in the generalized framework finally a link is established with the concept of measure valued solutions to the euler equations 6 and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic inria bp 105 78153 le chesnay cedex france this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms

weak solutions renormalized solutions and enstrophy defects in 2d turbulence enstrophy half the integral of the square of vorticity plays a role in 2d turbulence theory analogous to the role of kinetic energy in the kolmogorov theory of 3d turbulence it is therefore interesting to obtain a description of the way enstrophy is dissipated at high reynolds numbers in this article we explore the notions of viscous and transport enstrophy defect which model the spatial structure of the dissipation of enstrophy these notions were introduced by g eyink in an attempt to reconcile the kraichnan batchelor theory of 2d turbulence with current knowledge of the properties of weak solutions of the equations of incompressible and ideal fluid motion three natural questions arise from eyink s theory i existence of the enstrophy defects ii conditions for the equality of transport and viscous enstrophy defects iii conditions for the vanishing of the enstrophy defects in 10 eyink proved a number of results related to these questions and formulated a conjecture on how to answer these problems in a physically meaningful context in the present article we improve and extend some of eyink s results and present a counterexample to his conjecture

finding interpolating curves minimizing l infty acceleration in the euclidean space via optimal control theory we study the problem of finding an interpolating curve passing through prescribed points in the euclidean space the interpolating curve minimizes the pointwise maximum length i e l infty norm of its acceleration we reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory we characterize solutions associated with singular and nonsingular controls some of the results we obtain are new even for the scalar interpolating function case we reduce the infinite dimensional interpolation problem to an ensuing finite dimensional one and derive closed form expressions for interpolating curves consequently we devise efficient numerical techniques and illustrate them with examples

lattice gases large deviations and the incompressible navier stokes equations we study the incompressible limit for a class of stochastic particle systems on the cubic lattice zd d 3 for initial distributions corresponding to arbitrary macroscopic l2 initial data the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible navier stokes equations furthermore explicit exponential rates for the convergence large deviations are obtained the probability to violate the divergence free condition decays at rate at least exp e d l while the probability to violate the momentum conservation equation decays at rate exp fe d 2 with an explicit rate function given by an hi1 norm

budget dependent convergence rate of stochastic approximation convergence rate results are derived for a stochastic optimization problem where a performance measure is minimized with respect to a vector parameter t assuming that a gradient estimator is available and that both the bias and the variance of the estimator are known functions of the budget devoted to its computation the gradient estimator is employed in conjunction with a stochastic approximation sa algorithm our interest is to figure out how to allocate the total available computational budget to the successive sa iterations the effort is devoted to solving the asymptotic version of this problem by finding the convergence rate of sa toward the optimizer first as a function of the number of iterations and then as a function of the total computational effort as a result the optimal rate of increase of the computational budget per iteration can be found explicit expressions for the case where the computational budget devoted to an iteration is a polynomial in the iteration number and where the bias and variance of the gradient estimator are polynomials of the computational budget are derived applications include the optimization of steady state simulation models with likelihood ratio perturbation analysis or finite difference gradient estimators optimization of infinite horizon models with discounting optimization of functions of several expectations and so on several examples are discussed our results readily generalize to general root finding problems

three dimensional wavelet based multifractal method the need for revisiting the multifractal description of turbulence dissipation data we generalize the wavelet transform modulus maxima wtmm method to multifractal analysis of 3d random fields this method is calibrated on synthetic 3d monofractal fractional brownian fields and on 3d multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration then we apply the 3d wtmm method to the dissipation field issued from 3d isotropic turbulence simulations we comment on the need to revisit previous box counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

overconvergence properties of quintic interpolatory splines let q be a quintic spline with equi spaced knots on a b interpolating a given function y at the knots the parameters which determine q are used to construct a piecewise defined polynomial p of degree six it is shown that p can be used to give at any point of a b better orders of approximation to y and its derivatives than those obtained from q it is also shown that the superconvergence properties of the derivatives of q at specific points of a b are all simple consequences of the properties of p