about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

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

some ideas about quantitative convergence of collision models to their mean field limit we consider a stochastic n particle model for the spatially homogeneous boltzmann evolution and prove its convergence to the associated boltzmann equation when n with non asymptotic estimates for any time t 0 we bound the distance between the empirical measure of the particle system and the measure given by the boltzmann evolution in a relevant hilbert space the control got is gaussian i e we prove that the distance is bigger than xn 1 2 with a probability of type o e x 2 the two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of lipschitz continuity for the boltzmann collision kernel we study more extensively the case where our hilbert space is the homogeneous negative sobolev space smash dot h s then we are only able to give bounds for maxwellian models however numerical computations tend to show that our results are useful in practice

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

regularity of renormalized solutions in the boltzmann equation with long range interactions it is well established that renormalized solutions of the boltzmann equation enjoy some kind of regularity or at least compactness in the velocity variable when the angular collision kernel is nonintegrable however obtaining explicit estimates in convenient and natural functional settings proves rather difficult in this work we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation as a direct consequence of our result we show that in the presence of long range interactions any renormalized solution f t x v to the boltzmann equation satisfies locally for every and for some s 0 depending on p we also provide an application of this new estimate to the hydrodynamic limit of the boltzmann equation without cutoff 2012 wiley periodicals inc

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

gelfand shilov smoothing properties of the radially symmetric spatially homogeneous boltzmann equation without angular cutoff we prove that the cauchy problem associated to the radially symmetric spatially homogeneous non cutoff boltzmann equation with maxwellian molecules enjoys the same gelfand shilov regularizing effect as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

moderate deviation principles for weakly interacting particle systems moderate deviation principles for empirical measure processes associated with weakly interacting markov processes are established two families of models are considered the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump markov processes with a countable state space for both cases the moderate deviation principle is formulated in terms of a large deviation principle ldp with an appropriate speed function for suitably centered and normalized empirical measure processes for the first family of models the ldp is established in the path space of an appropriate schwartz distribution space whereas for the second family the ldp is proved in the space of l 2 the hilbert space of square summable sequences valued paths proofs rely on certain variational representations for exponential functionals of brownian motions and poisson random measures

a convergence proof for bird s direct simulation monte carlo method for the boltzmann equation bird s direct simulation monte carlo method for the boltzmann equation is considered the limit as the number of particles tends to infinity of the random empirical measures associated with the bird algorithm is shown to be a deterministic measure valued function satisfying an equation close in a certain sense to the boltzmann equation a markov jump process is introduced which is related to bird s collision simulation procedure via a random time transformation convergence is established for the markov process and the random time transformation these results together with some general properties concerning the convergence of random measures make it possible to characterize the limiting behavior of the bird algorithm

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

the explicit construction of all harmonic two spheres in quaternionic projective spaces we show how to construct explicitly all harmonic maps or equivalently branched minimal immersions from the two sphere to a quaternionic projective space this forms part of a programme to determine explicitly all harmonic two spheres in various symmetric spaces

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

nonlinear markov semigroups and interacting levy type processes semigroups of positivity preserving linear operators on measures of a measurable space x describe the evolutions of probability distributions of markov processes on x their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions b x on x describe the evolutions of averages over the trajectories of these markov processes in this paper we introduce and study the general class of semigroups of non linear positivity preserving transformations on measures that is non linear markov or feller semigroups an explicit structure of generators of such groups is given in case when x is the euclidean space r d or more generally a manifold showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for markov models of interacting particles well posedness results for these equations are given together with applications to interacting particles dynamic law of large numbers and central limit theorem the latter being new already for the standard coagulation fragmentation models

a weak formulation of the boltzmann equation based on the fourier transform in this article we present an alternative formulation of the spatially homogeneous boltzmann equation rewriting the weak form of the equation with shifted test functions and using fourier techniques it turns out that the transformed problem contains only a three fold integral

convergence from boltzmann to landau processes with soft potential and particle approximations our aim in this paper is to show how a probabilistic interpretation of the boltzmann and landau equations gives a microscopic understanding of these equations we firstly associate stochastic jump processes with the boltzmann equations we consider then we renormalize these equations following asymptotics which make prevail the grazing collisions and prove the convergence of the associated boltzmann jump processes to a diffusion process related to the landau equation the convergence is pathwise and also implies a convergence at the level of the partial differential equations the best feature of this approach is the microscopic understanding of the transition between the boltzmann and the landau equations by an accumulation of very small jumps we deduce from this interpretation an approximation result for a solution of the landau equation via colliding stochastic particle systems this result leads to a monte carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from boltzmann to landau equations numerical results are given

on the size of chaos in the mean field dynamics we consider the error arising from the approximation of an n particle dynamics with its description in terms of a one particle kinetic equation we estimate the distance between the j marginal of the system and the factorized state obtained in a mean field limit as n our analysis relies on the evolution equation for the correlation error rather than on the usual bbgky hierarchy the rate of convergence is shown to be o j 2 n in any bounded interval of time size of chaos as expected from heuristic arguments our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data covering a stochastic jump processes converging to the homogeneous boltzmann and the povzner equation and b quantum systems giving rise to the hartree equation

chaos hypothesis for a system interacting through shared resources we study the chaos hypothesis for a wide class of pure jump multitype interacting systems the interaction may be strong there is no symmetry assumption and the system is not necessarily markovian we use interaction graphs and coupling and study in a precise way how a chain reaction is constituted by a series of direct interactions we obtain the chaos hypothesis in variation norm with speed of convergence and deduce from it convergence of general empirical measures we couple the interaction graph to a boltzmann tree and show that the variation norm between the processes constructed on each goes to zero this proves propagation of chaos in total variation with speed of convergence when the boltzmann trees converge under light symmetry assumptions we characterize the limit law by a nonlinear martingale problem

some solutions of the boltzmann equation without angular cutoff we show the existence of local or global in time solutions for the non homogeneous boltzmann equation this is done under the assumptions that initial data are smaller than a suitable maxwellian and that collisional cross sections do not satisfy grad s angular cutoff partial regularity in space velocity of the solutions constructed herein is also proved

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

a convergence proof for nanbu s simulation method for the full boltzmann equation it is shown that the discrete measures given by the nanbu simulation method converge with respect to the weak topology of measures to solutions of the boltzmann equation the main conditions for this result are that the cauchy problem for the boltzmann equation has a sufficiently smooth solution and that the discretization parameters cell size timestep and test particle number satisfy suitable constraints

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

internal categories and groupoids in congruence modular varieties abstract it is well known that there exist several simple descriptions of the internal categories and groupoids in various classical algebraic categories such as groups rings lie algebras etc one of those descriptions is extended to arbitrary congruence modular varieties we describe all internal categories satisfying a certain commutator condition which always holds for internal groupoids and for all internal categories in mal tsev congruence permutable varieties the results of this paper show a deep connection between internal category theory and commutator theory in universal algebra the so called kiss difference operation also plays a central role

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

classification of conformal minimal immersions of constant curvature from s 2 to hp 2 in this paper we study geometry of conformal minimal two spheres immersed in quaternionic projective spaces we firstly use bahy el dien and wood s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from s 2 to the quaternionic projective space hp 2 then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from s 2 to the quaternionic projective space hp 2

convergence of the stochastic weighted particle method for the boltzmann equation this paper studies convergence of the stochastic weighted particle method for the boltzmann equation first the method is extended by introducing new stochastic reduction procedures in order to control the number of simulation particles then under rather general conditions convergence to the solution of the boltzmann equation is proved finally numerical experiments are performed illustrating both convergence and considerable variance reduction for the specific problem of calculating tails of the velocity distribution

brownian approximation and monte carlo simulation of the non cutoff kac equation the non cutoff boltzmann equation can be simulated using the dsmc method by a truncation of the collision term however even for computing stationary solutions this may be very time consuming in particular in situations far from equilibrium by adding an appropriate diffusion to the dsmc method the rate of convergence when the truncation is removed may be greatly improved we illustrate the technique on a toy model the kac equation as well as on the full boltzmann equation in a special case

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

a new approach to quantitative propagation of chaos for drift diffusion and jump processes this paper is devoted the study of the mean field limit for many particle systems undergoing jump drift or diffusion processes as well as combinations of them the main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles as the number of particles goes to infinity to this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators consistency estimate together with fine stability estimates on the flow of the limiting nonlinear equation stability estimates then we apply this method to a boltzmann collision jump process for maxwell molecules to a mckean vlasov drift diffusion process and to an inelastic boltzmann collision jump process with stochastic thermal bath to our knowledge our approach yields the first such quantitative results for a combination of jump and diffusion processes

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

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

the limiting kinetic equation of the consistent boltzmann algorithm for dense gases this paper establishes a theoretical foundation for the consistent boltzmann algorithm cba by deriving the limiting kinetic equation the formulation is similar to the proof by one of the authors that the boltzmann equation is the limiting kinetic equation for direct simulation monte carlo w wagner j statist phys 66 1011 1992 for a simplified model distilled from cba the limiting equation is solved numerically and very good agreement with the predictions of the theory is found

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 fokker planck boltzmann equation we consider the boltzmann equation perturbed by fokker planck type operator to overcome the lack of strong a priori estimates and to define a meaningful collision operator we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the cauchy problem with large data the proof of stability and existence combines renormalization with an analysis of a defect measure

explicit equilibria in a kinetic model of gambling we introduce and discuss a nonlinear kinetic equation of boltzmann type which describes the evolution of wealth in a pure gambling process where the entire sum of wealths of two agents is up for gambling and randomly shared between the agents for this equation the analytical form of the steady states is found for various realizations of the random fraction of the sum which is shared to the agents among others the exponential distribution appears as steady state in case of a uniformly distributed random fraction while gamma distribution appears for a random fraction which is beta distributed the case in which the gambling game is only conservative in the mean is shown to lead to an explicit heavy tailed distribution

precrossed modules and galois theory the adjunction between crossed modules and precrossed modules over a fixed group can be seen as a special case of a more general adjunction between internal groupoids and internal reflexive graphs in a mal tsev variety by using the categorical galois theory we characterize the central extensions with respect to this latter adjunction in terms of the universal algebraic commutator in particular we get a description of the central extensions of precrossed modules and of precrossed rings this characterization provides a natural way to define a categorical notion of peiffer commutator c 2005 elsevier inc all rights reserved

time splitting error in dsmc schemes for the spatially homogeneous inelastic boltzmann equation the paper is concerned with the numerical treatment of the uniformly heated inelastic boltzmann equation by the direct simulation monte carlo dsmc method this technique is presently the most widely used numerical method in kinetic theory we consider three modifications of the dsmc method and study them with respect to their efficiency and convergence properties convergence is investigated with respect to both the number of particles and the time step the main issue of interest is the time step discretization error due to various splitting strategies a scheme based on the strang splitting strategy is shown to be of second order with respect to time step while there is only first order for the commonly used euler splitting scheme on the other hand a no splitting scheme based on appropriate markov jump processes does not produce any time step error it is established in numerical examples that the no splitting scheme is about two orders of magnitude more efficient than the euler splitting scheme the strang splitting scheme reaches almost the same level of efficiency as that of the no splitting scheme since the deterministic time step error vanishes sufficiently fast

the geometry of binary collisions and generalized radon transforms elastic collisions are characterized by the conservation of momentum and energy we consider some geometrical aspects of such collisions when the energy of one particle can be expressed in terms of the moments as the geometry of elastic collisions is essential for the regularizing property of the gain term in the boltzmann equation which was proved by p l lions we show how such results can be deduced from a regularity theorem for generalized radon transforms by sogge stein this is possible for and for we also show that the same technique cannot be used with other choices of

the boltzmann equation without angular cutoff in the whole space qualitative properties of solutions this is a continuation of our series of works for the inhomogeneous boltzmann equation we study qualitative properties of classical solutions the full regularization in all variables uniqueness non negativity and convergence rate to the equilibrium to be precise together with the results of parts i and ii about the well posedness of the cauchy problem around the maxwellian we conclude this series with a satisfactory mathematical theory for the boltzmann equation without angular cutoff

on conformal minimal immersions of constant curvature from s 2 to hp n we generalize here our general characterization of the harmonic sequence generated by conformal minimal immersions from s 2 to the quaternionic projective space hp n then we give a classification theorem of linearly full unramified conformal minimal immersions of constant curvature from s 2 to hp n under some conditions

stochastic particle approximations for generalized boltzmann models and convergence estimates fied boltzmann equation with general motion between collisions and nonlinear bounded jump collision operator and give the nonlinear martingale problem it solves we consider various linear interacting particle systems in order to approximate this nonlinear process we prove propagation of chaos in variation norm on path space with a precise rate of convergence using coupling and interaction graph techniques and a representation of the nonlinear process on a boltzmann tree no regularity nor uniqueness assumption is needed we then consider a nonlinear equation with both vlasov and boltzmann terms and give a weak pathwise propagation of chaos result using a compactness uniqueness method which necessitates some regularity these results imply functional laws of large numbers and extend to multitype models we give algorithms simulating or approximating the particle systems 1 framework and main results the boltzmann equation describes the evolution of the limit density f t x v of molecules of a rarefied gas at time t position x and speed v f is positive and f f t x v dx dv 1 it is given by

a categorical approach to commutator theory abstract we define a notion of commutator for equivalence relations in a mal cev category and show that it verifies a universal property

a pure jump markov process associated with smoluchowski s coagulation equation the aim of the present paper is to construct a stochastic process whose law is the solution of the smoluchowski s coagulation equation we introduce first a modified equation dealing with the evolution of the distribu tion q t dx of the mass in the system the advantage we take on this is that we can perform an unified study for both continuous and discrete models the integro partial differential equation satisfied by q t t 0 can be interpreted as the evolution equation of the time marginals of a markov pure jump process at this end we introduce a nonlinear poisson driven stochastic differential equation related to the smoluchowski equation in the following way if x t satisfies this stochastic equation then the law of x t satisfies the modified smoluchowski equation the nonlinear process is richer than the smoluchowski equation since it provides historical information on the particles existence uniqueness and pathwise behavior for the solution of this sde are studied finally we prove that the nonlinear process x can be obtained as the limit of a marcus lushnikov procedure 1 introduction the coagulation model governs various phenomena as for example polymerization aggregation of colloidal particles formation of stars and planets behavior of fuel mixtures in engines etc smoluchowski s coagulation equation models the dynamic of such phenomena and describes the evolution of a system of clusters which coalesce in order to form bigger clusters each cluster is identified by its size the only mechanism taken into account is the coalescence of two clusters other effects as multiple coagulation are neglected we assume also that the rate of these reactions depends on the sizes of clusters involved in the coagulation denoting by n k t the nonnegative concentration of clusters of size k at time t the discrete

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

entropy dissipation estimates for the landau equation in the coulomb case and applications abstract we present in this paper an estimate which bounds from below the entropy dissipation d f of the landau operator with coulomb interaction by a weighted h 1 norm of the square root of f as a consequence we get a weighted l t 1 l v 3 estimate for the solutions of the spatially homogeneous landau equation with coulomb interaction and the propagation of l 1 moments of any order for this equation we also present an application of our estimate to the landau equation with moderately soft potentials providing thus a new proof of some recent results of 30

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

a direct method for the regularity of the gain term in the boltzmann equation this paper presents an essentially simple and direct method to prove the regularizing properties of the collision gain term in the first equation of the boltzmann hierarchy some corollaries of the main results of this paper are given to improve the previous results on the regularity of the gain term in the boltzmann equation an extension of a new fundamental result concerning separated variable kernels is also given to some general kernels

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

a markov process associated with a boltzmann equation without cutoff and for non maxwell molecules tanaka 18 showed a way to relate the measure solution pt t of a spatially homogeneous boltzmann equation of maxwellian molecules without angular cutoff to a poisson driven stochastic differential equation pt is the flow of time marginals of the solution of this stochastic equation in the present paper we extend this probabilistic interpretation to much more general spatially homogeneous boltzmann equations then we derive from this interpretation a numerical method for the concerned boltzmann equations by using easily simulable interacting particle systems

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

equations de type de boltzmann spatialement homogenes we study the construction of a nonhomogeneous n valued markov process whose laws at fixed times evolve according to a boltzmann type equation spatially homogeneous we then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles

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

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

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

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

harmonic maps of the two sphere into a complex grassmann manifold ii where qt is a complex valued one form defined up to a factor of absolute value 1 this form qp defines a complex structure on m for x e m the space f x has an orthogonal space f x of dimension n k we denote by f x and f x their corresponding projective spaces of dimensions k 1 and n k 1

a feynman kac type formula for the deterministic and stochastic wave equations and other p d e s we establish a probabilistic representation for a wide class of linear deterministic p d e s with potential term including the wave equation in spatial dimensions 1 to 3 our representation applies to the heat equation where it is related to the classical feynman kac formula as well as to the telegraph and beam equations if the potential is a random spatially homogeneous gaussian noise then this formula leads to an expression for the moments of the solution