almost sure well posedness of the cubic nonlinear schrodinger equation below l 2 mathbb t we consider the cauchy problem for the one dimensional periodic cubic non linear schrodinger equation nls with initial data below l 2 in particular we exhibit nonlinear smoothing when the initial data are randomized then we prove local well posedness of nls almost surely for the initial data in the support of the canonical gaussian measures on h s t for each s 1 and global well posedness for each s 1 12

a variational formula for the free energy of an interacting many particle system we consider n bosons in a box in r d with volume n rho under the influence of a mutually repellent pair potential the particle density p is an element of 0 infinity is kept fixed our main result is the identification of the limiting free energy f beta p at positive temperature 1 beta in terms of an explicit variational formula for any fixed rho if beta is sufficiently small and for any fixed beta if rho is sufficiently small the thermodynamic equilibrium is described by the symmetrized trace of e beta h n where h n denotes the corresponding hamilton operator the well known feynman kac formula reformulates this trace in terms of n interacting brownian bridges due to the symmetrization the bridges are organized in an ensemble of cycles of various lengths the novelty of our approach is a description in terms of a marked poisson point process whose marks are the cycles this allows for an asymptotic analysis of the system via a large deviations analysis of the stationary empirical field the resulting variational formula ranges over random shift invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process in our proof of the lower bound for the free energy we drop all interaction involving infinitely long cycles and their possible presence is signalled by a loss of mass of the finitely long cycles in the variational formula in the proof of the upper bound we only keep the mass on the finitely long cycles we expect that the precise relationship between these two bounds lies at the heart of bose einstein condensation and intend to analyze it further in future

derivation of the gross pitaevskii hierarchy for the dynamics of bose einstein condensate consider a system of n bosons on the three dimensional unit torus interacting via a pair potential n2v n xi xj where x x1 xn denotes the positions of the particles suppose that the initial data n 0 satisfies the condition where hn is the hamiltonian of the bose system this condition is satisfied if n 0 wn n 0 where wn is an approximate ground state to hn and n 0 is regular let n t denote the solution to the schrodinger equation with hamiltonian hn gross and pitaevskii proposed to model the dynamics of such a system by a nonlinear schrodinger equation the gross pitaevskii gp equation the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices k ut ut solves the gp hierarchy we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy our analysis requires that the n boson dynamics be described by a modified hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance our proof can be extended to a modified hamiltonian that only forbids at least n particles from coming close together for any fixed n 2006 wiley periodicals inc

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

on well posedness for general hierarchy equations of gross pitaevskii and hartree type gross pitaevskii and hartree hierarchies are infinite systems of coupled pdes emerging naturally from the mean field theory of bose gases their solutions are known to be related to initial value problems in particular the gross pitaevskii and hartree equations due to their physical and mathematical relevance the issues of well posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques in this article we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain liouville equations and secondly solving the uniqueness and existence questions for the latter as an outcome we formulate a hierarchy equation starting from any initial value problem which is u 1 invariant and prove a general principle which can be stated formally as follows in particular several new well posedness results as well as a counterexample to uniqueness for the gross pitaevskii hierarchy equation are proved the novelty in our work lies in the aforementioned duality and the use of liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

the quintic nls as the mean field limit of a boson gas with three body interactions we investigate the dynamics of a boson gas with three body interactions in dimensions d 1 2 we prove that in the limit of infinite particle number the bbgky hierarchy of k particle marginals converges to a limiting gross pitaevskii gp hierarchy for which we prove existence and uniqueness of solutions factorized solutions of the gp hierarchy are shown to be determined by solutions of a quintic nonlinear schrodinger equation our proof is based on and extends methods of erdos schlein yau klainerman machedon and kirkpatrick schlein staffilani

random data cauchy problem for the navier stokes equations on t3 in this paper we study the random data initial value problem of the navier stokes equations in l2 t3 by using the randomization approach recently developed by n burq and n tzvetkov we prove that for almost all where is the sample space of a probability space a p for the randomized initial data f h l 2 t3 the navier stokes equations have a global solution u c 0 l 2 t3 or more precisely u et f h c 0 h 12 t3

on the uniqueness of solutions to the periodic 3d gross pitaevskii hierarchy abstract in this paper we present a uniqueness result for solutions to the gross pitaevskii hierarchy on the three dimensional torus under the assumption of an a priori spacetime bound we show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear schrodinger equation in this way we obtain a periodic analogue of the uniqueness result on r 3 previously proved by klainerman and machedon 75 except that in the periodic setting we need to assume additional regularity in particular we need to work in the sobolev class h for 1 by constructing a specific counterexample we show that on t 3 the existing techniques from the work of klainerman and machedon approach do not apply in the endpoint case 1 this is in contrast to the known results in the non periodic setting where these techniques are known to hold for all 1 in our analysis we give a detailed study of the crucial spacetime estimate associated to the free evolution operator in this step of the proof our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice this method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius r depending on the number theoretic properties of the normal vector to the plane we are hence able to obtain a sharp range of admissible sobolev exponents for which the spacetime estimate holds

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

on the uniqueness of solutions to the gross pitaevskii hierarchy the purpose of this note is to give a new proof of uniqueness of the gross pitaevskii hierarchy first established in 1 in a different space based on space time estimates similar in spirit to those of 2

mean field and classical limit of many body schrodinger dynamics for bosons we present a new proof of the convergence of the n particle schrodinger dynamics for bosons towards the dynamics generated by the hartree equation in the mean field limit for a restricted class of two body interactions we obtain convergence estimates uniform in hbar up to an exponentially small remainder for hbar 0 the classical dynamics in the mean field limit is given by the vlasov equation

rigorous derivation of the cubic nls in dimension one we derive rigorously the one dimensional cubic nonlinear schrodinger equation from a many body quantum dynamics the interaction potential is rescaled through a weak coupling limit together with a short range one we start from a factorized initial state and prove propagation of chaos with the usual two step procedure in the former step convergence of the solution of the bbgky hierarchy associated to the many body quantum system to a solution of the bbgky hierarchy obtained from the cubic nls by factorization is proven in the latter we show the uniqueness for the solution of the infinite bbgky hierarchy

on the cauchy problem for gross pitaevskii hierarchies the purpose of this paper is to investigate the cauchy problem for the gross pitaevskii infinite linear hierarchy of equations on rn n 1 we prove local existence and uniqueness of solutions in certain sobolev type spaces h of sequences of marginal density operators with n 2 in particular we give a clear discussion of all cases n 2 which covers the local well posedness problem for the gross pitaevskii hierarchy in this situation

on the propagation of weakly nonlinear random dispersive waves we study several basic dispersive models with random periodic initial data such that the different fourier modes are independent random variables motivated by the vast physics literature on related topics we then study how much the fourier modes of the solution at later times remain decorrelated our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

a central limit theorem in many body quantum dynamics we study the many body quantum evolution of bosonic systems in the mean field limit the dynamics is known to be well approximated by the hartree equation so far the available results have the form of a law of large numbers in this paper we go one step further and we show that the fluctuations around the hartree evolution satisfy a central limit theorem interestingly the variance of the limiting gaussian distribution is determined by a time dependent bogoliubov transformation describing the dynamics of initial coherent states in a fock space representation of the system

the derivation of the mathbb t 3 t 3 energy critical nls from quantum many body dynamics we derive the 3d energy critical quintic nls from quantum many body dynamics with 3 body interaction in the mathbb t 3 periodic setting due to the known complexity of the energy critical setting previous progress was limited in comparison to the 2 body interaction case yielding energy subcritical cubic nls previously the only result for the 3d energy critical case was hong et al j funct anal 270 1 34 67 2016 which proved the uniqueness part of the argument in the case of small solutions in the main part of this paper we develop methods to prove the convergence of the bbgky hierarchy to the infinite gross pitaevskii gp hierarchy and separately the uniqueness of large gp solutions since the trace estimate used in the previous proofs of convergence is the false endpoint trace estimate in our setting we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de finetti theorem for the large solution uniqueness argument we discover the new hierarchical uniform frequency localization hufl property for the gp hierarchy and use it to prove a new type of uniqueness theorem the hufl property reduces to a new statement even for nls with the help of colliander et al ann math 2 167 3 767 865 2008 and ionescu and pausader duke math j 161 1581 1612 2012 which proved the global well posedness for the quintic nls this new uniqueness theorem establishes global uniqueness

rigorous derivation of the gross pitaevskii equation with a large interaction potential consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 hn n 0i cn and that the one particle density matrix converges to a projection as n then we prove that the k particle density matrices of n t factorize in the limit n moreover the one particle orbital wave function solves the time dependent gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant proportional to the scattering length of the potential v in 12 we proved the same statement under the condition that the interaction potential v is sufficiently small in the present work we develop a new approach that requires no restriction on the size of the potential

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

l p properties for gaussian random series let c c n n n be an arbitrary sequence of l 2 n and let f c be a random series of the type f c n n g n c n e n where g n n n is a sequence of independent n c 0 1 gaussian random variables and e n n n an orthonormal basis of l 2 y the finite measure space y m being arbitrary by using the equivalence of gaussian moments and an integrability theorem due to fernique we show that a necessary and sufficient condition for f c to belong to l p y m p 2 oo for any c l 2 n almost surely is that sup n n e n l p y m oo one of the main motivations behind this result is the construction of a nontrivial gibbs measure invariant under the flow of the cubic defocusing nonlinear schrodinger equation posed on the open unit disc of r 2

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

on a tensor cross product based formulation of large strain solid mechanics abstract this paper describes in detail the formulation of large strain solid mechanics based on the tensor cross product originally presented by r de boer 1982 and recently re introduced by bonet et al 2015a and bonet et al 2015b the paper shows how the tensor cross product facilitates the algebra associated with the area and volume maps between reference and final configurations these maps together with the fibre map make up the fundamental kinematic variables in polyconvex elasticity the algebra proposed leads to novel expressions for the tangent elastic operator which neatly separates material from geometrical dependencies the paper derives new formulas for the spatial and material stress and their corresponding elasticity tensors these are applied to the simple case of a mooney rivlin material model the extension to transversely isotropic material models is also considered

mean field dynamics singular potentials and rate of convergence we consider the time evolution of a system of n identical bosons whose interaction potential is rescaled by n 1 we choose the initial wave function to describe a condensate in which all particles are in the same one particle state it is well known that in the mean field limit n the quantum n body dynamics is governed by the nonlinear hartree equation using a nonperturbative method we extend previous results on the mean field limit in two directions first we allow a large class of singular interaction potentials as well as strong possibly time dependent external potentials second we derive bounds on the rate of convergence of the quantum n body dynamics to the hartree dynamics

potts model for exaggeration of a simple rumor transmitted by recreant rumormongers a simple rumor transmitted by recreant rumormongers is considered quantitatively the simple message contained in the rumor is represented by a simple proposition that has been universally quantified the operations to change the proposition by rumormongers are established to describe the rumor s transmission along different channels mathematically a spin chain is introduced in which spins represent the operations the addition of spins is established according to the laws of operations the result of a rumor s transmission is given by the chain s spin sum the model which is favorable for a social prognostication can determine quantitatively how the social guide and the competition among various opinions affect the exaggeration of the simple rumor transmitted by recreant rumormongers it proves that the rumor forms potts like spin chains in the case with maximum information entropy the approximate calculation shows that the rumor may be aggrandized little or aggrandized normally even sometimes catastrophically moreover the exaggeration is greater when the guide is larger and the competition is lower

a tracial quantum central limit theorem we prove a central limit theorem for non commutative random variables in a von neumann algebra with a tracial state any non commutative polynomial of averages of i i d samples converges to a classical limit the proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the baker campbell hausdorff expansion the result can be considered a generalization of johansson s theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity math co 9906120 math pr 9909104

quantum dynamics with mean field interactions a new approach we propose a new approach for the study of the time evolution of a factorized n particle bosonic wave function with respect to a mean field dynamics with a bounded interaction potential the new technique which is based on the control of the growth of the correlations among the particles leads to quantitative bounds on the difference between the many particle schrodinger dynamics and the one particle nonlinear hartree dynamics in particular the one particle density matrix associated with the solution to the n particle schrodinger equation is shown to converge to the projection onto the one dimensional subspace spanned by the solution to the hartree equation with a speed of convergence of order 1 n for all fixed times

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

the gross pitaevskii hierarchy on general rectangular tori in this work we study the gross pitaevskii hierarchy on general rational and irrational rectangular tori of dimensions two and three this is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear schrodinger equation we prove a conditional uniqueness result for the hierarchy in two dimensions this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear schrodinger equation from the dynamics of many body quantum systems on irrational tori this question was posed as an open problem in the previous work of kirkpatrick schlein and staffilani

invariant gibbs measure evolution for the radial nonlinear wave equation on the 3d ball abstract we establish new global well posedness results along gibbs measure evolution for the nonlinear wave equation posed on the unit ball in r 3 via two distinct approaches the first approach invokes the method established in the works bourgain 1994 1996 3 4 5 based on a contraction mapping principle and applies to a certain range of nonlinearities the second approach allows to cover the full range of nonlinearities admissible to treatment by gibbs measure working instead with a delicate analysis of convergence properties of solutions the method of the second approach is quite general and we shall give applications to the nonlinear schrodinger equation on the unit ball in subsequent works bourgain and bulut 2013 10 11

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

construction of a gibbs measure associated to the periodic benjamin ono equation we define a finite borel measure of gibbs type supported by the sobolev spaces of negative indexes on the circle the measure can be seen as a limit of finite dimensional measures these finite dimensional measures are invariant by the ode s which correspond to the projection of the benjamin ono equation posed on the circle on the first n n 1 modes in the trigonometric bases

gross pitaevskii equation as the mean field limit of weakly coupled bosons we consider the dynamics of n boson systems interacting through a pair potential n 1va xi xj where va x a 3v x a we denote the solution to the n particle schrodinger equation by n t recall that the gross pitaevskii gp equation is a nonlinear schrodinger equation and the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices open image in new window solves the gp hierarchy under the assumption that a n for 0 3 5 we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy with the coupling constant in the nonlinear term of the gp equation given by v x dx the uniqueness of the solutions of this hierarchy remains an open question

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2