a microscopic model for stefan s melting and freezing problem we study a class of one dimensional interacting particle systems with random boundaries as a microscopic model for stefan s melting and freezing problem we prove that under diffusive rescaling these particle systems exhibit a hydrodynamic behavior described by the solution of a cauchy stefan problem

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

derivation of the hydrodynamical equation for one dimensional ginzburg landau model the hydrodynamical behavior of one dimensional scalar ginzburg landau model with conservation law is investigated the dynamics of the system is given by solving a stochastic partial differential equation under appropriate space time scaling a deterministic limit is obtained and the limit is described by a certain nonlinear diffusion equation

equilibrium fluctuations for a system of harmonic oscillators with conservative noise we investigate the harmonic chain forced by a multiplicative noise the evolution is given by an infinite system of stochastic differential equations total energy and deformation are preserved the conservation of momentum is destroyed by the noise gaussian product measures are the extremal stationary states of this model equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized ornstein uhlenbeck processes

derivation of a hydrodynamic equation for ginzburg landau models in an external field the lattice approximation to a time dependent ginzburg landau equation is investigated in the presence of a small external field the evolution law conserves the spin but is not reversible a nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit the proof extends to certain stochastically perturbed hamiltonian systems

on the regularity of solutions to a class of nonlinear dispersive equations we shall study special regularity properties of solutions to some nonlinear dispersive models the goal is to show how regularity on the initial data is transferred to the solutions this will depend on the spaces where regularity is measured

fluctuations of one dimensional ginzburg landau models in nonequilibrium we study the fluctuation of one dimensional ginzburg landau models in nonequilibrium along the hydrodynamic diffusion limit the hydrodynamic limit has been proved to be a nonlinear diffusion equation by fritz guo papanicolaou varadhan etc we proved that if the potential is uniformly convex then the fluctuation process is governed by an ornstein uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

hydrodynamic scaling limits with deterministic initial configurations we discuss the hydrodynamic scaling limits starting with deterministic configurations for different models certain estimates on the entropy of the system are derived

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

an infinite dimensional stochastic differential equation with state spacec ℝ we consider a time evolution of unbounded continuous spins on the real line the evolution is described by an infinite dimensional stochastic differential equation with local interaction introducing a condition which controls the growth of paths at infinity we can construct a diffusion process taking values inc in view of quantum field theory this is a time dependent model ofp 1 field in parisi and wu s scheme

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

on the propagation of regularities in solutions of the fifth order kadomtsev petviashvili ii equation abstract in this paper we study special properties of solutions of the initial value problem ivp associated to the fifth order kadomtsev petviashvili ii equation mainly for initial data h 1 r 2 whose restriction belongs to h m x 0 r for some m z m 2 and x 0 r we prove that the restriction of the corresponding solution u t belongs to h m b r for any b r and any t 0 in other words the extra regularity on the data propagates in the solutions in the direction of the dispersion

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

the scaling limit for a stochastic pde and the separation of phases we investigate the problem of singular perturbation for a reaction diffusion equation with additive noise or a stochastic partial differential equation of ginzburg landau type under the situation that the reaction term is determined by a potential with double wells of equal depth as the parameter e the temperature of the system tends to 0 the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit we derive a stochastic differential equation which describes the random movement of the phase separation point the proof consists of two main steps we show that the solution stays near a manifoldm e of minimal energy configurations based on a lyapunov type argument then the limit equation is identified by introducing a nice coordinate system in a neighborhood ofm e

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

a novel nonlinear approach to suppress resonant vibrations a novel nonlinear approach to suppress resonant vibrations is presented by employing a single degree of freedom transmissibility system which utilises a nonlinear damping element the study shows that a nonlinear damping element can reduce the output energy at the driving frequency and at the same time spread the output signal energy over a wider range of harmonics it is also shown that the reduction becomes larger as the nonlinear damping characteristic gets stronger and in most cases the power at the harmonics in the output spectrum is much less when the nonlinear damping characteristic is an odd function hence an odd nonlinear damping element can be designed and introduced between the incoming signal and the system of interest to suppress resonant vibrations an expression is derived to express the transmitted force spectrum in terms of the nonlinear generalised frequency response functions to clearly reveal how the energy at the excitation frequency is modified by the nonlinearity

on the diffusive nature of entropy flow in infinite systems remarks to a paper by guo papanicolau varadhan the hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy free energy arguments the methods of 13 are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise following 14 15 and 3 5 we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended the flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems 4

hydrodynamic limit for a system with finite range interactions we study a system of interacting diffusions the variables present the amount of charge at various sites of a periodic multidimensional lattice the equilibrium states of the diffusion are canonical gibbs measures of a given finite range interaction under an appropriate scaling of lattice spacing and time we derive the hydrodynamic limit for the evolution of the macroscopic charge density

local stability and hydrodynamical limit of spitzer s one dimensional lattice model an infinite system of ordinary differential equations is considered the right hand side is just the negative gradient of potential energy of a one dimensional system of unbounded spins interacting by a symmetric and convex pair potential constant configurations are stationary points and the mean spin is conserved it is shown that each of these stationary points has its own domain of attraction the initial distribution need not be translation invariant as a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

a harmonic probing algorithm for the multi input volterra series it is shown how the conventional harmonic probing algorithm of bedrosian and rice can be extended to deal with the multi input multi output form of the volterra functional series example calculations are presented for both continuous time and discrete time multi input non linear systems subject to multiple sinusoidal inputs

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

on the steady state response and stability of non linear oscillators using harmonic balance abstract the harmonic balance hb method is used to study the steady state periodic response as well as its stability of a softening duffing oscillator the objectives of the work are 1 to show how the qualitative nature of the hb solution changes if one uses two harmonics rather than one in the approximate solution and 2 to show that the analysis of the mathieu equation which governs local stability of the steady state response must be based on an approximate rather than an exact analysis of this mathieu equation thus floquet theory incorrectly describes the stability of the approximate solution

on the hydrodynamic limit of a ginzburg landau lattice model the law of large numbers in arbitrary dimensions a lattice system of interacting diffusion processes is investigated the evolution is attractive and time reversible the spin satisfies a conservation law it is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation

macroscopic energy diffusion for a chain of anharmonic oscillators we study the energy diffusion in a chain of anharmonic oscillators where the hamiltonian dynamics is perturbed by a local energy conserving noise we prove that under diffusive rescaling of space time energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation we also give variational expressions for the thermal diffusivity and some upper and lower bounds

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

a class of stochastic evolutions that scale to the porous medium equation a class of reversible markov jump processes on a periodic lattice is described and a result about their scaling behavior stated under diffusion scaling the empirical measure converges to a solution of the porous medium equation on thed dimensional torus the process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbors through a zero range pressure mechanism with conservation of total stick length

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

on the propagation of regularity for solutions of the fractional korteweg de vries equation abstract we consider the initial value problem ivp for the fractional korteweg de vries equation fkdv 0 1 t u d x x u u x u 0 x t r 0 1 u x 0 u 0 x it has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena more precisely it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution this property was found originally in solutions of korteweg de vries kdv equation and it has been also verified in other dispersive equations as the benjamin ono bo equation recently it has been shown that the solutions of the dispersive generalized benjamin ono dgbo equation this is 2 3 in 0 1 also satisfy the propagation of regularity phenomena this is achieved by introducing a commutator decomposition to handle the dispersive part in the equation following the approach used in the dgbo we prove that the solutions of the fkdv also satisfies the propagation of regularity phenomena consequently this type of regularity travels with infinite speed to its left as time evolves

a particle model for spinodal decomposition we study a one dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration we prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift there is a critical valuea c ofa fora a c the density values are unrestricted while fora a c they should all be to the right or to the left of a given interval a the diffusion coefficient of the limiting equation can be continued analytically to a and in the interior of a it has negative values which should correspond to particle aggregation phenomena we also show that the dynamics can be obtained as a limit of a kawasaki evolution associated to a kac potential the coefficienta plays the role of the inverse temperature the critical value ofa coincides with the critical inverse temperature in the van der waals limit and a with the spinodal region it is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic the system evolves according to an integrodifferential equation the instanton solutions of this equation as studied by dal passo and de mottoni are then related to the phase transition region in the thermodynamic phase diagram analogies with the cahn hilliard equations are also discussed

time dependent critical fluctuations of a one dimensional local mean field model one dimensional stochastic ising systems with a local mean field interaction kac potential are investigated it is shown that near the critical temperature of the equilibrium gibbs distribution the time dependent process admits a scaling limit given by a nonlinear stochastic pde the initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way earlier results of bertini presutti rudiger saada are improved the proof is based on an energy inequality obtained by coupling the glauber dynamics to its voter type linear approximation

generalized frequency response function matrix for mimo non linear systems recursive algorithms are derived to compute the generalized frequency response function matrix of multi input multioutput mimo non linear systems as an analytical map from both non linear differential equation models and narx non linear auto regressive models with exogenous inputs models of the system the algorithm is computationally compact and exposes the explicit relationship between the model parameters and the elements of the generalized frequency response function matrix and can thus provide important insights into the behaviour of non linear systems

on the propagation of regularities in solutions of the benjamin ono equation abstract we shall deduce some special regularity properties of solutions to the ivp associated to the benjamin ono equation mainly for datum u 0 h 3 2 r whose restriction belongs to h m b for some m z m 2 and b r we shall prove that the restriction of the corresponding solution u t belongs to h m for any r and any t 0 therefore this type of regularity of the datum travels with infinite speed to its left as time evolves

hydrodynamics of the voter model we study the voter model on mathbb z d d geqq 3 for a sequence mu varepsilon of initial states which have a gradient in the mean magnetization of the order varepsilon varepsilon rightarrow 0 we prove that the magnetization field m varepsilon f t varepsilon d sum f varepsilon x eta x varepsilon 2 t tends to a deterministic field m f t int dqf q m q t as varepsilon rightarrow 0 m q t is the solution of the diffusion equation the fluctuations of m varepsilon f t around its mean are given by an infinite dimensional non homogeneous ornstein uhlenbeck process in the limit as varepsilon rightarrow 0 locally i e around varepsilon 1 q varepsilon 2 t the voter model is in equilibrium with parameter m q t

microscopic theory of isothermal elastodynamics this paper examines random perturbations of the anharmonic chain of coupled oscillators the microscopic system has two conservation laws and its hyperbolic scaling limit results in the quasi linear wave equation p system of isothermal isentropic elasticity in the shock regime the compensated compactness method is used lastly results from j w shearer and d serre are applied

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

heat flow in an exactly solvable model a chain of one dimensional oscillators is considered they are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors the total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures the stationary measures are obtained when the chain is finite the thermodynamic limit is then considered approach to the gibbs distribution is proven and a linear temperature profile is obtained

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

stationary states of random hamiltonian systems we investigate the ergodic properties of hamiltonian systems subjected to local random energy conserving perturbations we prove for some cases e g anharmonic crystals with random nearest neighbor exchanges or independent random reflections of velocities that all translation invariant stationary states with finite entropy per unit volume are microcanonical gibbs states the results can be utilized in proving hydrodynamic behavior of such systems

vibration reduction of beams under successive traveling loads by means of linear and nonlinear dynamic absorbers abstract the goal of the present work is to assess the performances of dynamic vibration absorbers dva in suppressing the vibrations of a simply supported beam subjected to an infinite sequence of regularly spaced concentrated moving loads in particular several types of dva are considered linear cubic higher odd order monomials and piecewise linear stiffness linear cubic and linear quadratic viscous damping the purpose is to clarify if nonlinear dvas show improvements with respect to the classical linear devices the dynamic scenario is deeply investigated in a wide range of operating conditions spanning the parameter space of the dva damping stiffness nonlinear stiffness can lead to complex dynamics such as quasi periodic chaotic and sub harmonic responses moreover acting on the stiffness nonlinearity no improvement is found with respect to the linear dva a nonlinear non symmetric dissipation in the dva leads to a great reduction of the beam response the reduction is larger with respect to the linear dva

the hydrodynamic limit for a system with interactions prescribed by ginzburg landau type random hamiltonian as a microscopic model we consider a system of interacting continuum like spin field overr d its evolution law is determined by the ginzburg landau type random hamiltonian and the total spin of the system is preserved by this evolution we show that the spin field converges under the hydrodynamic space time scalling to a deterministic limit which is a solution of a certain nonlinear diffusion equation this equation describes the time evolution of the macroscopic field the hydrodynamic scaling has an effect of the homogenization on the system at the same time

on the propagation of regularity and decay of solutions to the benjamin equation in this paper we investigate some special regularities and decay properties of solutions to the initial value problem of the benjamin equation the main result shows that for initial datum u0 hs r with s 3 4 if the restriction of u0 belongs to hl x0 for some l z and x0 r then the restriction of the corresponding solution u t belongs to hl for any r and any t 0 t consequently this type of regularity travels with infinite speed to its left as time evolves

one dimensional hard rod caricature of hydrodynamics we give here a rigorous deduction of the hydrodynamic equation which holds in the hydrodynamic limit for a model system of one dimensional identical hard rods interacting through elastic collisions the equation should be considered as the analog of the euler equation of real hydrodynamics owing to the degeneracy of the model it is written in terms of a functiong q v t expressing the density of particles with velocityv at the pointq at timet for this equation we prove an existence and uniqueness theorem in some natural class of functions our main result is the proof that if 0 is a class of initial states which are homogeneous on a scale much less than e 1 and if the corresponding particle densities tend ase 0 in the proper scale to the initial hydrodynamic densitygo q v then under some general assumptions on the states and ong0 the particle densities of the evolved states at timee 1t tend ase 0 to the unique solution of the hydrodynamic equation with initial conditiong0 the proof is completed by exhibiting a large class of initial families 0 which possess the required properties

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies