a stochastic particle numerical method for 3d boltzmann equations without cutoff using the main ideas of tanaka the measure solution pt t of a 3 dimensional spatially homogeneous boltzmann equation of maxwellian molecules without cutoff is related to a poisson driven stochastic differential equation using this tool the convergence to pt t of solutions ptl t of approximating boltzmann equations with cutoff is proved then a result of graham meleard is used and allows us to approximate ptl t with the empirical measure tl n t of an easily simulable interacting particle system precise rates of convergence are given a numerical study lies at the end of the paper

divergence of eigenfunction expansions we shall use a result of rellich on the convergence of spectral projections to analyze mean and almost everywhere convergence of eigenfunction expansions the basic result is a transplantation theorem which reduces these questions on manifolds to questions on the boundedness of fourier multiplier operators on the cotangent space results such as these were long known in the context of fourier series where they were introduced by stein and weiss 121 various authors gave generalizations but the most far reaching was obtained by mitjagin 8 who applied his results to elliptic problems in this paper we shall give several extensions of mitjagin s results in the main part of the paper we shall employ these extensions and the results of kenig and tomas s 61 to analyze certain nonelliptic problems this paper is divided into four sections in the first we shall give mitjagin s original proof that one may transplant problems on the convergence of eigenfunction expansions in lp mean to fourier multiplier problems on i we shall also extend this to convergence of other summability methods and also to joint eigenexpansions of commuting selfadjoint operators

the cauchy problem for the radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand shilov smoothing effect abstract in this paper we study the cauchy problem for the radially symmetric homogeneous non cutoff boltzmann equation with maxwellian molecules the initial datum belongs to shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator and it is a small perturbation of maxwellian distribution the shubin space of the negative index contains the probability measures based on this spectral decomposition we construct the weak solution with shubin class initial datum we also prove that the cauchy problem enjoys gelfand shilov smoothing effect meaning that the smoothing properties are the same as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

on the rate of convergence to equilibrium in the becker doring equations abstract we provide an explicit rate of convergence to equilibrium for solutions of the becker doring equations using the energy energy dissipation relation the main difficulty is the structure of equilibria of the becker doring equations which do not correspond to a gaussian measure such that a logarithmic sobolev inequality is not available we prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e ct1 3

quantitative stability of the entropy power inequality we establish quantitative stability results for the entropy power inequality epi specifically we show that if uniformly log concave densities nearly saturate the epi then they must be close to gaussian densities in the quadratic kantorovich wasserstein distance furthermore if one of the densities is gaussian and the other is log concave or more generally has positive spectral gap then the deficit in the epi can be controlled in terms of the l 1 kantorovich wasserstein distance or relative entropy respectively as a counterpoint an example shows that the epi can be unstable with respect to the quadratic kantorovich wasserstein distance when densities are uniformly log concave on sets of measure arbitrarily close to one our stability results can be extended to non log concave densities provided certain regularity conditions are met the proofs are based on mass transportation

asymptotic of grazing collisions and particle approximation for the kac equation without cutoff the subject of this article is the kac equation without cutoff we first show that in the asymptotic of grazing collisions the kac equation can be approximated by a fokker planck equation the convergence is uniform in time and we give an explicit rate of convergence next we replace the small collisions by a small diffusion term in order to approximate the solution of the kac equation and study the resulting error we finally build a system of stochastic particles undergoing collisions and diffusion that we can easily simulate which approximates the solution of the kac equation without cutoff we give some estimates on the rate of convergence

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

entropy dissipation estimates for the linear boltzmann operator abstract we prove a linear inequality between the entropy and entropy dissipation functionals for the linear boltzmann operator with a maxwellian equilibrium background this provides a positive answer to the analogue of cercignani s conjecture for this linear collision operator our result covers the physically relevant case of hard spheres interactions as well as maxwellian kernels both with and without a cut off assumption for maxwellian kernels the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to 27 32 for more general kernels the proof relies on a comparison principle finally we also show that in the grazing collision limit our results allow to recover known logarithmic sobolev inequalities

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation we first consider the boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates this is the simplest boltzmann equation having the compressible euler equations as a scaling limit for it we prove a stability result for theh theorem which says that when the entropy production is small the solution of the spatially homogeneous boltzmann equation is necessarily close to equilibrium in the entropie sense and therefore strongl 1 sense we use this to prove that solutions to the spatially homogeneous boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes every initial condition with finite entropy andp th velocity moment for some p 2 belongs to such a class we then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below which covers a wide class of slightly modified physical collision kernels these results are the basis of a study of the relation between scaling limits of solutions of the boltzmann equation and hydrodynamics which will be developed in subsequent papers the program is described here

exponential decay toward equilibrium via entropy methods for reaction diffusion equations abstract in this work we show how the entropy method enables to get in an elementary way and without linearization estimates of exponential decay towards equilibrium for solutions of reaction diffusion equations corresponding to a reversible reaction explicit rates of convergence combining the dissipative effects of diffusion and reaction are given

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

smoothing effect and cauchy problem for radially symmetric homogeneous boltzmann equation with debye yukawa potential of shubin class initial datum abstract in this paper we study the cauchy problem for the radially symmetric homogeneous non cutoff boltzmann equation with debye yukawa potential the initial datum belongs to shubin type space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator and it is a small perturbation of maxwellian distribution the shubin type space of negative index contains the probability measures based on the spectral decomposition we construct the weak solution with shubin type class initial datum and prove the smoothing effect for the solution to this cauchy problem

chaos ergodicity and equilibria in a quantum kac model abstract we introduce quantum versions of the kac master equation and the kac boltzmann equation we study the steady states of each of these equations and prove a propagation of chaos theorem that relates them the quantum kac master equation qkme describes a quantum markov semigroup p n t while the kac boltzmann equation describes a non linear evolution of density matrices on the single particle state space all of the steady states of the n particle quantum system described by the qkme are separable and thus the evolution described by the qkme is entanglement breaking the results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory and a quantitative study of the rate of destruction of entanglement in a class of quantum markov semigroups describing binary interactions

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

a maxwellian lower bound for solutions to the boltzmann equation we prove that the solution of the spatially homogeneous boltzmann equation is bounded pointwise from below by a maxwellian i e a function of the formc1 exp c2v2 this holds for any initial data with bounded mass energy and entropy and for any positive timet t0 the constantsc1 andc2 depend on the mass energy and entropy of the initial data and ont0 0 only

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

analysis of spectral methods for the homogeneous boltzmann equation the development of accurate and fast algorithms for the boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis recently several works were devoted to the derivation of spectrally accurate schemes for the boltzmann equation but very few of them were concerned with the stability analysis of the method in particular there was no result of stability except when the method is modified in order to enforce the posivity preservation which destroys the spectral accuracy in this paper we propose a new method to study the stability of homogeneous boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution this method takes advantage of the spreading property of the collision together with estimates on regularity and entropy production as an application we prove stability and convergence of spectral methods for the boltzmann equation when the discretization parameter is large enough with explicit bound

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

a counter example to cercignani s conjecture for the d dimensional kac model kac s d dimensional model gives a linear many particle binary collision model from which under suitable conditions the celebrated boltzmann equation in its spatially homogeneous form arise as a mean field limit the ergodicity of the evolution equation leads to questions about the relaxation rate in hope that such a rate would pass on the boltzmann equation as the number of particles goes to infinity this program starting with kac and his one dimensional spectral gap conjecture at 1956 finally reached its conclusion in the maxwellian case in a series of papers by authors such as janvresse maslen carlen carvalho loss and geronimo but the hope to get a limiting relaxation rate for the boltzmann equation with this linear method was already shown to be unrealistic although the problem is still important and interesting due to its connection with the linearized boltzmann operator a less linear approach via a many particle version of cercignani s conjecture is the grounds for this paper in our paper we extend recent results by the author from the one dimensional kac model to the d dimensional one showing that the entropy entropy production ratio n still yields a very strong dependency in the number of particles of the problem when we consider the general case

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

rate of convergence of the nanbu particle system for hard potentials and maxwell molecules we consider the numerically motivated nanbu stochastic particle system associated to the spatially homogeneous boltzmann equation for true hard potentials we establish a rate of propagation of chaos of the particle system to the unique solution of the boltzmann equation more precisely we estimate the expectation of the squared wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution the rate we obtain is almost optimal as a function of the number of particles but is not uniform in time

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

on the rate of entropy production for the boltzmann equation we show that there exists a wide class of distribution functions with moments of any order as close to their equilibrium values as we like which can provide an abnormally low rate of entropy production the result is valid for the boltzmann equation with any cross section v satisfying a mild restriction the functions are constructed in an explicit form and we discuss some applications of our results

boltzmann model for viscoelastic particles asymptotic behavior pointwise lower bounds and regularity we investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous boltzmann equation we prove the existence of a universal maxwellian intermediate asymptotic state with explicit rate of convergence towards it exponential lower pointwise bounds and propagation of regularity are also studied these results can be seen as a generalization of several classical results holding for the pseudo maxwellian and constant normal restitution models

quadratic pseudopotentials for gl n c principal sigma models abstract matrix pseudopotentials for the gl n c principal sigma model in two dimensional minkowski space are derived through the method of closed differential ideals the structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix riccati type a set of further constraints is deduced which leads to inner backlund transformations depending on two complex parameters through a geometrical interpretation of such matrix riccati equations and the associated constraints a linearization procedure is derived and the equations are reduced to those of zakharov mikhailov and shabat 7 10 an inductive procedure is applied to explicitly solve an iterated sequence of such backlund transformations through a purely algebraic transformation of solutions to the first step the resulting nonlinear superposition formula is used to demonstrate a permutability theorem and to analyse the appearance of singularities fr nous deduisons les pseudo potentiels matriciels pour le modele sigma principal a valeurs dans gl n c defini sur l espace de minkowski bi dimensionnel en utilisant la methode des ideaux differentiels fermes on restreint la structure des equations des pseudopotentiels de facon a ce que n apparaissent que des non linearites quadratiques du type riccati matriciel nous deduisons un ensemble de contraintes supplementaires menant a des transformations de backlund internes dependantes de deux parametres complexes gr ce a l interpretation geometrique de ces equations de riccati matricielles et des contraintes associees nous deduisons un procede de linearisation et les equations se reduisent a celles de zakharov mikhailov et shabat 7 10 un procede inductif est utilise pour resoudre explicitement une succession de transformations de backlund par une transformation purement algebrique des solutions de la premiere etape la formule de superposition non lineaire qui en resulte permet de demontrer un theoreme de permutabilite et d analyser l apparition de singularites

higher local conservation laws for nonlinear sigma models on symmetric spaces we show that for the classical two dimensional nonlinear model on a riemannian symmetric space of dimensionm and rankp there existp independent series of higher local conservation laws and we reduce the field equations of the model to a system of nonlinear partial differential equations possessing an associated lax pair and involvingm p independent variables

on the generalization of the boltzmann h theorem for a spatially homogeneous maxwell gas sufficient conditions are given for a convex and isotropic functional that are monotonically decreasing in time along the solution of the spatially homogeneous boltzmann equation for maxwellian molecules the conditions are valid for plane velocities and for axially symmetric solutions in any dimension explicit examples are given

uniqueness for a class of spatially homogeneous boltzmann equations without angular cutoff we consider the 3 dimensional spatially homogeneous boltzmann equation which describes the evolution in time of the velocity distribution in a gas where particles are assumed to undergo binary elastic collisions we consider a cross section bounded in the relative velocity variable without angular cutoff but with a moderate angular singularity we show that there exists at most one weak solution with finite mass and momentum we use a wasserstein distance although our result is far from applying to physical cross sections it seems to be the first one which deals with cross sections without cutoff for non maxwellian molecules

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

on a direct approach to quasideterminant solutions of a noncommutative kp equation a noncommutative version of the modified kp equation and a family of its solutions expressed as quasideterminants are discussed the origin of these solutions is explained by means of darboux transformations and the solutions are verified directly we also verify directly an explicit connection between quasideterminant solutions of the noncommutative mkp equation and the noncommutative kp equation arising from the miura transformation

kinetic approach to long time behavior of linearized fast diffusion equations we show that the rate of convergence towards the self similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self similar solution at a fixed time as a consequence we find an improved rate of convergence to self similarity in terms of a fourier based distance between two solutions the results are based on the asymptotic equivalence of a collisional kinetic model of boltzmann type with a linear fokker planck equation with nonconstant coefficients and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous boltzmann equation for maxwell molecules

quasideterminant solutions of the generalized heisenberg magnet model in this paper we present the darboux transformation for the generalized heisenberg magnet ghm model based on the general linear lie group gl n and construct multi soliton solutions in terms of quasideterminants further we relate the quasideterminant multi soliton solutions obtained by means of darboux transformation with those obtained by the dressing method we also discuss the model based on the lie group su n and obtain explicit soliton solutions of the model based on su 2

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

explicit exponential convergence to equilibrium for nonlinear reaction diffusion systems with detailed balance condition abstract the convergence to equilibrium of mass action reaction diffusion systems arising from networks of chemical reactions is studied the considered reaction networks are assumed to satisfy the detailed balance condition and have no boundary equilibria we propose a general approach based on the so called entropy method which is able to quantify with explicitly computable rates the decay of an entropy functional in terms of an entropy entropy dissipation inequality based on the totality of the conservation laws of the system as a consequence follows convergence to the unique detailed balance equilibrium with explicitly computable convergence rates the general approach is further detailed for two important example systems a single reversible reaction involving an arbitrary number of chemical substances and a chain of two reversible reactions arising from enzyme reactions

equisummability of eigenfunction expansions under analytic multipliers in this paper we present some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of w the criteria to be developed originate in some cases from general banach space arguments and in others from the more specific spatial nature of the differential operators involved the importance of our analysis of l equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded we apply these criteria to give a simple proof of an equisummability result of gurarie and kon 4 for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set the key technique is to analyze kernels of the resolvents and use radial bounds of the resolvents developed in 3 the prototypical case for equisummability is found in the study of equiconvergence for differential operators classically haar 51 walsh lo birkhoff 2 and tamarkin lo have shown that the difference between expansions with respect to eigenfunctions of a sturm liouville operator or birkhoff series and the ordinary fourier series tends to zero uniformly in every finite interval in this case the multiplier or summator function is the characteristic function in some one dimensional cases where such equiconvergence fails stone 9 levitan and sargsjan 6 and benzinger 1 have showed equisummability for riesz typical means of eigenfunction expansions of differential operators where the summator function is

self similar asymptotics for a modified maxwell boltzmann equation in systems subject to deformations in this paper we study a generalized class of maxwell boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix a this class of equations arises for instance from the analysis of homoenergetic solutions for the boltzmann equation considered by many authors since 1950s our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix a the main result of this paper is that for sufficiently small norm of a any non negative solution with finite second moment tends to a self similar solution of relatively simple form for large values of time we also prove that the higher order moments of the self similar profile are finite under further smallness condition on the matrix a

on the well posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity we prove an inequality on the kantorovich rubinstein distance which can be seen as a particular case of a wasserstein metric between two solutions of the spatially homogeneous boltzmann equation without angular cutoff but with a moderate angular singularity our method is in the spirit of 7 we deduce some well posedness and stability results in the physically relevant cases of hard and moderately soft potentials in the case of hard potentials we relax the regularity assumption of 6 but we need stronger assumptions on the tail of the distribution namely some exponential decay we thus obtain the first uniqueness result for measure initial data in the case of moderately soft potentials we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy for very moderately soft potentials plus sometimes an additionnal moment condition we thus improve significantly on all previous results where weighted sobolev spaces were involved

stability and exponential convergence for the boltzmann equation we prove existence uniqueness and stability for solutions of the nonlinear boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function the results are given for a range of spaces including l1 and extend previous results in l for the non homogeneous equation as well as the more developed lp theory for the spatially homogeneous boltzmann equation

existence and regularity of a solution of a kac equation without cutoff using the stochastic calculus of variations we consider a generalized kac equation without cutoff with which we associate a non standard nonlinear stochastic differential equation we adapt the techniques in bichteler and jacod 2 to prove that the law of a solution of the stochastic differential equation has a density which is a solution of the kac equation the initial law is very general we only assume it has second order moments and is not the dirac mass at 0 we thus generalize the analytical results of existence of a solution of this equation if we furthermore assume existence of all moments for the initial law we obtain as a corollary using the proof in desvillettes 6 that the density is smooth we prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations adapting the methods in 1

supports of measure solutions for spatially homogeneous boltzmann equations we prove that the support of the unique measure solution for the spatially homoge neous boltzmann equation in r3 is the whole space if the initial distribution is not a dirac measure and has 4 order moment more precisely we obtain the lower bound of exponential type for the probability of any small ball in 3 relative to the measure solution

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

cp n 1 harmonic maps and the weierstrass problem a weierstrass type system of equations corresponding to the cpn 1 harmonic maps is presented the system constitutes a further generalization of our previous construction j math phys 44 328 2003 it consists of four first order equations for three complex functions which are shown to be equivalent to the cpn 1 harmonic maps when the harmonic maps are holomorphic or antiholomorphic one of the functions vanishes and the system reduces to the previously given generalization of the weierstrass problem we also discuss a possible interpretation of our results and show that in our new case the induced metric is proportional to the total energy density of the map and not only to its holomorphic part as was the case in the previous generalizations

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

quasideterminant solutions of a non abelian toda lattice and kink solutions of a matrix sine gordon equation two families of solutions of a generalized non abelian toda lattice are considered these solutions are expressed in terms of quasideterminants constructed by means of darboux and binary darboux transformations as an example of the application of these solutions we consider the 2 periodic reduction to a matrix sine gordon equation in particular we investigate the interaction properties of polarized kink solutions

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

superposition of solutions to backlund transformations for the su n principal σ model we show that the backlund transformations for the su n principal model may be linearized using a geometrical interpretation of these equations involving the minimal orbit of su n n in the grassmann manifold gn c2n linearization puts the equations in zakharov mikhailov shabat zms form using this form of the equations we prove inductively a nonlinear superposition law and a permutability theorem for iterated backlund transformations analogous to known results in the theory of the sine gordon and kdv equations from the superposition law we get an explicit form for multisoliton solutions to the model

a strengthened central limit theorem for smooth densities abstract we consider the central limit theorem for the probability density function n x of the standardized sum of independent and identically distributed random variables with finite variance and regular probability density function by showing boundedness of different convex functionals along the sequence fn we prove convergence in various norms to the normal density

strong convergence towards self similarity for one dimensional dissipative maxwell models we prove the propagation of regularity uniformly in time for the scaled solutions of the one dimensional dissipative maxwell models introduced in d ben avraham e ben naim k lindenberg a rosas self similarity in random collision processes phys rev e 68 2003 r050103 this result together with the weak convergence towards the stationary state proven in l pareschi g toscani self similarity and power like tails in nonconservative kinetic models j stat phys 124 2 4 2006 747 779 implies the strong convergence in sobolev norms and in the l1 norm towards it depending on the regularity of the initial data as a consequence the original nonscaled solutions are also proved to be convergent in l1 towards the corresponding self similar homogeneous cooling state the proof is based on the uniform in time control of the tails of the fourier transform of the solution and it holds for a large range of values of the mixing parameters in particular in the case of the one dimensional inelastic boltzmann equation the result does not depend of the degree of inelasticity

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule

darboux transformation and exact multisolitons of cpn nonlinear sigma model abstract exact multisolitons of c p n sigma model are obtained by using darboux transformation in terms of quasideterminant darboux matrix the multisoliton solutions are shown to be expressed in terms of quasideterminants of gelfand and retakh 1991 10 the method of quasideterminant darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the c p n sigma model by means of nonlinear superposition principle of backlund transformations explicit expressions of one two and three soliton solutions of c p 1 model have been computed by using properties of quasideterminants

estimates for the large time behavior of the landau equation in the coulomb case this work deals with the large time behaviour of the spatially homogeneous landau equation with coulomb potential firstly we obtain a bound from below of the entropy dissipation d f by weighted relative fisher information of f with respect to the associated maxwellian distribution which leads to a variant of cercignani s conjecture thanks to a logarithmic sobolev inequality secondly we prove the propagation of polynomial and stretched exponential moments with an at most linearly growing in time rate as an application of these estimates we show the convergence of any h or weak solution to the landau equation with coulomb potential to the associated maxwellian equilibrium with an explicitly computable rate assuming initial data with finite mass energy entropy and some higher l1 moment more precisely if the initial data have some large enough polynomial l1 moment then we obtain an algebraic decay if the initial data have a stretched exponential l1 moment then we recover a stretched exponential decay

long time asymptotics of kinetic models of granular flows we analyze the long time asymptotics of certain one dimensional kinetic models of granular flows which have been recently introduced in 22 in connection with the quasi elastic limit of a model boltzmann equation with dissipative collisions and variable coefficient of restitution these nonlinear equations classified as nonlinear friction equations split naturally into two classes depending on whether or not the temperature of their similarity solutions homogeneous cooling states reduce to zero in finite time for both classes we show uniqueness of the solution by proving decay to zero in the wasserstein metric of any two solutions with the same mass and mean velocity furthermore if the temperature of the similarity solution decays to zero in finite time we prove by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support that the temperature of any other solution with initially bounded support must also decay to zero in finite time

regularity and long time behavior of the boltzmann equation for granular gases abstract this paper investigates regularity of solutions of the boltzmann equation with dissipative collisions in a thermal bath in the case of pseudo maxwellian approximation we prove that for any initial datum f 0 in the set of probability density with zero bulk velocity and finite temperature the unique solution of the equation satisfies f t h r 3 for all t 0 furthermore for any t 0 0 and s 0 the h s norm of f t is bounded for t t 0 as a consequence the exponential convergence to the unique steady state is also established under the same initial condition

sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation we derive a new lower bound for the entropy dissipation associated with the spatially homogeneous boltzmann equation this bound is expressed in terms of the relative entropy with respect to the equilibrium and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy with an explicit rate our result gives a considerable refinement of the analogous estimate by carlen and carvalho 9 10 under very little additional assumptions our proof takes advantage of the structure of boltzmann s collision operator with respect to the tensor product and its links with fokker planck and landau equations several variants are discussed

explicit forms of backlund transformations for the grassmannian and cp n 1 sigma models a fairly wide class of classical solutions of the euclidean two dimensional grassmannian and cpn 1 sigma models has been constructed explicitly and elementarily by the present author starting from these classical solutions we derive the explicit forms of the backlund transformations following harnad et al properties of the backlund transformations for the grassmannian models are discussed in particular a simple interpretation of the backlund transformations for the cpn 1 model is obtained generalization of these results to the noncompact grassmannian sigma models is straightforward

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

entropy dissipation and moment production for the boltzmann equation leth f m flog f m dv be the relative entropy off and the maxwellian with the same mass momentum and energy and denote the corresponding entropy dissipation term in the boltzmann equation byd f q f f logf dv an example is presented which shows that d f h f m can be arbitrarily small this example is a sequence of isotropic functions and the estimates are very explicitly given by a simple formula ford which holds for such functions the paper also gives a simplified proof of the so called povzner inequality which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision that inequality is then used to prove that f v v s dt c t wheref is the solution of the spatially homogeneous boltzmann equation herec t is an explicitly given function dependings and the mass energy and entropy of the initial data

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

radial bounds for perturbations of elliptic operators abstract elliptic operators a m b x d a multi index with leading term positive and constant coefficient and with lower order coefficients b x l r l with n r defined on r n or a quotient space r n r n u u r n are considered it is shown that the lp spectrum of a is contained in a parabolic region of the complex plane enclosing the positive real axis uniformly in p outside the kernel of the resolvent of a is shown to be uniformly bounded by an l1 radial convolution kernel some consequences are a can be closed in all lp 1 p and is essentially self adjoint in l2 if it is symmetric a generates an analytic semigroup e ta in the right half plane strongly lp and pointwise continuous at t 0 a priori estimates relating the leading term and remainder are obtained and summability ea e 0 0 with analytic is proved for l p with convergence in lp and on the lebesgue set of more comprehensive summability results are obtained when a has constant coefficients

tanaka theorem for inelastic maxwell models we show that the euclidean wasserstein distance is contractive for inelastic homogeneous boltzmann kinetic equations in the maxwellian approximation and its associated kac like caricature this property is as a generalization of the tanaka theorem to inelastic interactions even in the elastic classical boltzmann equation we give a simpler proof of the tanaka theorem than the ones in 29 31 consequences are drawn on the asymptotic behavior of solutions in terms only of the euclidean wasserstein distance

exponential convergence to equilibrium for the homogeneous boltzmann equation for hard potentials without cut off this paper deals with the long time behavior of solutions to the spatially homogeneous boltzmann equation the interactions considered are the so called non cut off with a moderate angular singularity and non mollified hard potentials we prove an exponential in time convergence towards the equilibrium improving results of villani commun math phys 234 3 455 490 2003 where a polynomial decay to equilibrium is proven the basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a l 1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by gualdani et al http hal archives ouvertes fr ccsd 00495786 2013 we then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish

equilibration in the kac model using the gtw metric d_2 we use the fourier based gabetta toscani wennberg metric d 2 to study the rate of convergence to equilibrium for the kac model in 1 dimension we take the initial velocity distribution of the particles to be a borel probability measure mu on mathbb r n that is symmetric in all its variables has mean vec 0 and finite second moment let mu t dv denote the kac evolved distribution at time t and let r mu be the angular average of mu we give an upper bound to d 2 mu t r mu of the form min left b e frac 4 lambda 1 n 3 t d 2 mu r mu right where lambda 1 frac n 2 2 n 1 is the gap of the kac model in l 2 and b depends only on the second moment of mu we also construct a family of schwartz probability densities f 0 n mathbb r n rightarrow mathbb r with finite second moments that shows practically no decrease in d 2 f 0 t r f 0 for time at least frac 1 2 lambda with lambda the rate of the kac operator we also present a propagation of chaos result for the partially thermostated kac model in tossounian and vaidyanathan j math phys 56 8 083301 2015

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

a class of asymptotic preserving schemes for the fokker planck landau equation we present a class of asymptotic preserving ap schemes for the nonhomogeneous fokker planck landau nfpl equation filbet and jin 16 designed a class of ap schemes for the classical boltzmann equation by penalization with the bgk operator so they become efficient in the fluid dynamic regime we generalize their idea to the nfpl equation with a different penalization operator the fokker planck operator that can be inverted by the conjugate gradient method we compare the effects of different penalization operators and conclude that the fokker planck fp operator is a good choice such schemes overcome the stiffness of the collision operator in the fluid regime and can capture the fluid dynamic limit without numerically resolving the small knudsen number numerical experiments demonstrate that the schemes possess the ap property for general initial data with numerical accuracy uniformly in the knudsen number

quasideterminant solutions of an integrable chiral model in two dimensions the darboux transformation is used to obtain multisoliton solutions of the chiral model in two dimensions the matrix solutions of the principal chiral model and its lax pair are expressed in terms of quasideterminants the iteration of the darboux transformation gives the quasideterminant multisoliton solutions of the model it has been shown that the quasideterminant multisoliton solution of the chiral model is the same as obtained by zakharov and mikhailov using the dressing method based on the matrix riemann hilbert problem

on spatially homogeneous solutions of a modified boltzmann equation for fermi dirac particles the paper considers a modified spatially homogeneous boltzmann equation for fermi dirac particles bfd we prove that for the bfd equation there are only two classes of equilibria the first ones are fermi dirac distributions the second ones are characteristic functions of the euclidean balls and they can be simply classified in terms of temperatures t 2 5tf and t 2 5tf where tf denotes the fermi temperature in general we show that the l bound 0 f 1 e derived from the equation for solutions implies the temperature inequality t 2 5tf and if t 2 5tf then f trend towards fermi dirac distributions if t 2 5tf then f are the second equilibria in order to study the long time behavior we also prove the conservation of energy and the entropy identity and establish the moment production estimates for hard potentials

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions

global l p properties for the spatially homogeneous boltzmann equation this paper studies the lp behavior for 1 p of solutions of the nonlinear spatially homogeneous boltzmann equation for a class of collision kernels including inverse kth power forces with k 5 and angular cut off the following topics are treated differentiability in lp together with global boundedness in time for lp moments that exist initially translation continuity in lp uniformly in time and strong convergence to equilibrium