on measure solutions of the boltzmann equation part i moment production and stability estimates the spatially homogeneous boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy we prove the existence of emph weak measure solutions with and without angular cutoff on the collision kernel the proof in particular makes use of an approximation argument based on the mehler transform moment production estimates in the usual form and in the exponential form are obtained for these solutions finally for the grad angular cutoff we also establish uniqueness and strong stability estimate on these solutions

global boundedness of moments of solutions of the boltzmann equation for forces of infinite range this paper proves the global boundedness in time of higher moments to a weak solution of the non linear space homogeneous boltzmann equation for inverse kth power forces with k 5 in the course of the proof a new collisional estimate of v s v s v s v s is obtained where v v are the velocities before and v v are the velocities after a binary collision

the boltzmann equation for bose einstein particles regularity and condensation we study regularity and finite time condensation of distributional solutions of the space homogeneous and velocity isotropic boltzmann equation for bose einstein particles for the hard sphere model global in time existence of distributional solutions had been proven before here we prove that the equation is locally and can be globally in time well posed for the class of distributional solutions having finite moment of the negative order 1 2 and solutions in this class with regular initial data are mild solutions in their regularity time intervals by observing a necessary condition on the initial data for the absence of condensation at some finite time we also propose a sufficient condition on the initial data for the occurrence of condensation at all large time and then using a positivity of a partial collision integral we prove further that the critical time of condensation can be strictly positive

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

moment inequalities for the boltzmann equation and applications to spatially homogeneous problems some inequalities for the boltzmann collision integral are proved these inequalities can be considered as a generalization of the well known povzner inequality the inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous boltzmann equation for a wide class of intermolecular forces we obtain simple necessary and sufficient conditions on the potential for the uniform boundedness of all moments for potentials with compact support the following statement is proved if all moments of the initial distribution function are bounded by the corresponding moments of the maxwelliana exp bv2 then all moments of the solution are bounded by the corresponding moments of the other maxwelliana1 exp b1 t v2 for anyt 0 moreoverb t const for hard spheres an estimate for a collision frequency is also obtained

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

asymptotic analysis of the spatially homogeneous boltzmann equation grazing collisions limit in the present work we consider the asymptotic problem of the spatially homogeneous boltzmann equation when almost all collisions are grazing that is the deviation angle theta of the collision is limited near zero i e theta le epsilon we show that by taking the proper scaling to the cross section which was used in 37 that is assuming begin aligned b epsilon v v sigma 2 1 s v v gamma epsilon 3 sin 1 theta left frac theta epsilon right 1 2s mathrm 1 theta le epsilon end aligned where theta langle theta frac upsilon upsilon upsilon upsilon sigma rangle the solution f epsilon of the boltzmann equation with initial data f 0 can be globally or locally expanded in some weighted sobolev space as begin aligned f epsilon f o epsilon end aligned where the function f is the solution of landau equation which is associated with the grazing collisions limit of boltzmann equation with the same initial data f 0 this gives the rigorous justification of the landau approximation in the spatially homogeneous case in particular if taking gamma 3 and s 1 epsilon in the cross section b epsilon we show that the above asymptotic formula still holds and in this case f is the solution of landau equation with the coulomb potential going further we revisit the well posedness problem of the boltzmann equation in the limiting process we show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution f epsilon thanks to the weak convergence results on the grazing collisions limit in 37 in other words we establish a unified framework to establish the well posedness results for both boltzmann and landau equations

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 measure solutions of the boltzmann equation part ii rate of convergence to equilibrium abstract the paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous boltzmann equation for hard potentials with angular cutoff we prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy the proof is based on the regularizing property of the iterated collision operators exponential moment production estimates and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data we also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the maxwellian equilibrium the constants in these convergence rates depend only on the collision kernel and conserved quantities mass momentum and energy we finally use these convergence rates in order to deduce global in time strong stability of measure solutions

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

conservation of energy entropy identity and local stability for the spatially homogeneous boltzmann equation for nonsoft potential collision kernels with angular cutoff we prove that under the initial condition f0 v 1 v 2 logf0 v l1 r3 the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous boltzmann equation in the class l 0 l12 r3 c1 0 l1 r3 where l1s r3 f f v 1 v 2 s 2 l1 r3 and in this class the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy moreover for hard potentials and the hard sphere model a local stability result for conservative solutions i e satisfying the conservation of mass momentum and energy is obtained as an application of the local stability a sufficient and necessary condition on the initial data f0 such that the conservative solutions f belong to l1loc 0 l12 r3 is also given

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

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

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

a generating integral for matrix elements of the coulomb green s function the generating integral il e d3r d3r exp r r rr l 1 yl m g r r e yl m is evaluated where the coulomb green s function g is the inverse e h 1 with h the hydrogenic hamiltonian 1 2 2 zr 1 and yl m is a spherical harmonic the result can be used for the evaluation of matrix elements of g with respect to wave functions of the form yl m f l p r where f l p r n l p exp ar 2 ar ll p ar with l p a laguerre polynomial and n l p a normalizing factor for general e the result is given in terms of a special case of the hypergeometric function which satisfies an inhomogeneous linear first order ordinary differential equation for e en where en z2 2n2 is a hydrogenic bound state energy g is replaced by the generalized green s function generalized inverse and the results are given in closed form in terms of elementary functions

on the boltzmann equation for long range interactions we study the boltzmann equation without grad s angular cutoff assumption we introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables angular singularities occur as soon as one is interested in long range interactions while singularities in the relative velocity variable occur in the study of soft potentials in particular coulomb interaction together with several new estimates this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by lions appearance of strong compactness under general fully realistic assumptions 2001 john wiley sons inc

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

study of interacting particle systems the transition to the oscillatory behavior of a prey predator model abstract a prey predator system with intelligent pursuit and evasion is studied by means of computer simulations it is found that the model exhibits a coexistence state between prey and predators in this state and depending on the parameters the system may evolve toward two different regimes characterized by the behavior of the prey predator population whitin the oscillatory fixed point regime the specie s density changes periodically remains constant respectively the transition between such regimes is rationalized in terms of a dynamic percolation process

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