asymptotics of complete kahler metrics of finite volume on quasiprojective manifolds let x be a quasiprojective manifold given by the complement of a divisor d with normal crossings in a smooth projective manifold x using a natural compactification of x by a manifold with corners x we describe the full asymptotic behavior at infinity of certain complete kahler metrics of finite volume on x when these metrics evolve according to the ricci flow we prove that such asymptotic behaviors persist at later times by showing that the associated potential function is smooth up to the boundary on the compactification x however when the divisor d is smooth with kx d 0 so that the ricci flow converges to a kahler einstein metric we show that this kahler einstein metric has a rather different asymptotic behavior at infinity since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary

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

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

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

the boltzmann grad limit of a hard sphere system analysis of the correlation error we present a quantitative analysis of the boltzmann grad low density limit of a hard sphere system we introduce and study a set of functions the correlation errors measuring the deviations in time from the statistical independence of particles propagation of chaos in the context of the bbgky hierarchy a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization we show that provided k varepsilon alpha such an error flows to zero with the average density varepsilon for short times as varepsilon gamma k for some positive alpha gamma in 0 1 this provides an information on the size of chaos namely j different particles behave as dictated by the boltzmann equation even when j diverges as a negative power of varepsilon the result requires a rearrangement of lanford perturbative series into a cumulant type expansion and an analysis of many recollision events

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

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

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

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

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

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

critical behavior of random fibers with mixed weibull distribution a random fiber bundle model with a mixed weibull distribution is studied under the global load sharing scheme the mixed model consists of two sets of fibers the threshold strength of one set of fibers is randomly chosen from a weibull distribution with a particular weibull index and another set of fibers with a different index the mixing tunes the critical stress of the bundle and the variation of critical stress with the amount of mixing is determined using a probabilistic method where the external load is increased quasistatically in a special case which we illustrate the critical stress is found to vary linearly with the mixing parameter the critical exponents and power law behavior of burst avalanche size distribution is found to remain unaltered due to mixing

uniqueness of the self similar profile for a kinetic annihilation model abstract we prove the uniqueness of the self similar profile solution for a modified boltzmann equation describing probabilistic ballistic annihilation such a model describes a system of hard spheres such that whenever two particles meet they either annihilate with probability 0 1 or they undergo an elastic collision with probability 1 the existence of a self similar profile for smaller than an explicit threshold value 1 has been obtained in our previous contribution 6 we complement here our analysis of such a model by showing that for some explicit the self similar profile is unique for 0

dynamic critical behavior of failure and plastic deformation in the random fiber bundle model the random fiber bundle rfb model with the strength of the fibers distributed uniformly within a finite interval is studied under the assumption of global load sharing among all unbroken fibers of the bundle at any fixed value of the applied stress sigma load per fiber initially present in the bundle the fraction u t sigma of fibers that remain unbroken at successive time steps t is shown to follow simple recurrence relations the model is found to have stable fixed point u filled sigma for applied stress in the range 0 infinity the avalanche size distribution for this mean field dynamics of failure at sigma sigma c has been studied the elastic response of the rfb model has also been studied analytically for a specific probability distribution of fiber strengths where the bundle shows plastic behavior before complete failure following an initial linear response

fast numerical method for the boltzmann equation on non uniform grids we introduce a new fast numerical method for computing discontinuous solutions to the boltzmann equation and illustrate it by numerical examples a combination of adaptive grids for approximation of the distribution function and an approximate fast fourier transform on non uniform grids for computing smooth terms in the boltzmann collision integral is used

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

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

noise induced rupture process phase boundary and scaling of waiting time distribution a bundle of fibers has been considered here as a model for composite materials where breaking of the fibers occur due to a combined influence of applied load stress and external noise through numerical simulation and a mean field calculation we show that there exists a robust phase boundary between continuous no waiting time and intermittent fracturing regimes in the intermittent regime throughout the entire rupture process avalanches of different sizes are produced and there is a waiting time between two consecutive avalanches the statistics of waiting times follows a distribution and the avalanche distribution shows power law scaling similar to what has been observed in the case of earthquake events and bursts in fracture experiments we propose a prediction scheme that can tell when the system is expected to reach the continuous fracturing point from the intermittent phase

crossover behavior in a mixed mode fiber bundle model we introduce a mixed mode load sharing scheme in a fiber bundle model this model reduces exactly to equal load sharing els and local load sharing lls models at the two extreme limits of a single load sharing parameter we identify two distinct regimes a the mean field regime where the els mode dominates and b the short range regime dominated by the lls mode the crossover behavior is explored through a numerical study of the strength variation the avalanche statistics susceptibility and relaxation time variations the correlations among the broken fibers and their cluster analysis analyzing the moments of the cluster size distributions we locate the crossover point of these regimes we thus conclude that even in one dimension the fiber bundle model shows crossover behavior from mean field to short range interactions

stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres we consider the spatially homogeneous boltzmann equation for inelastic hard spheres in the framework of so called constant normal restitution coefficients alpha in 0 1 in the physical regime of a small inelasticity that is alpha in alpha 1 for some constructive alpha in 0 1 we prove uniqueness of the self similar profile for given values of the restitution coefficient alpha in alpha 1 the mass and the momentum therefore we deduce the uniqueness of the self similar solution up to a time translation

asymptotics of kahler einstein metrics on quasi projective manifolds and an extension theorem on holomorphic maps such a framed manifold is also called canonically polarized let x x c be the complement of c in x according to results of r kobayashi h tsuji g tian and s t yau x possesses a uniquely determined complete kahlereinstein metric x of constant negative ricci curvature 1 in this paper the asymptotic behavior of this metric will be studied since kx c c kc 0 there exists also a kahler einstein metric c on c with ricci curvature 1 using a canonical section of c as a local coordinate function on x we restrict the complete kahler einstein metric x to the locally defined sets c 0 0 the notion of locally uniform convergence of x c 0 where 0 0 makes sense and we can state

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

optimal local well posedness theory for the kinetic wave equation abstract we prove local existence and uniqueness results for the space homogeneous 4 wave kinetic equation in wave turbulence theory we consider collision operators defined by radial but general dispersion relations satisfying suitable bounds and we prove two local well posedness theorems in nearly critical weighted spaces

on the global existence of mild solutions to the boltzmann equation for small data in ld we develop a new theory of existence of global solutions to the boltzmann equation for small initial data these new mild solutions are analogous to the mild solutions for the navier stokes equations the existence comes as a result of the study of the competing phenomena of dispersion due to the transport operator and of singularity formation due to the nonlinear boltzmann collision operator it is the joint use of the so called dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions

relaxation dynamics in strained fiber bundles under an applied external load the global load sharing fiber bundle model with individual fiber strength thresholds sampled randomly from a probability distribution will relax to an equilibrium state or to complete bundle breakdown the relaxation can be viewed as taking place in a sequence of steps in the first step all fibers weaker than the applied stress fail as the total load is redistributed on the surviving fibers a group of secondary fiber failures occur etc for a bundle with a finite number of fibers the process stops after a finite number of steps tf by simulation and theoretical estimates it is determined how tf depends upon the stress the initial load per fiber both for subcritical and supercritical stress the two sided critical divergence is characterized by an exponent 1 2 independent of the probability distribution of the fiber thresholds

burst avalanches in solvable models of fibrous materials we review limiting models for fracture in bundles of fibers with statistically distributed thresholds for breakdown of individual fibers during the breakdown process avalanches consisting of simultaneous rupture of several fibers occur and the distribution d delta of the magnitude delta of such avalanches is the central characteristics in our analysis for a bundle of parallel fibers two limiting models of load sharing are studied and contrasted the global model in which the load carried by a bursting fiber is equally distributed among the surviving members and the local model in which the nearest surviving neighbors take up the load for the global model we investigate in particular the conditions on the threshold distribution which would lead to anomalous behavior i e deviations from the asymptotics d delta sim delta 5 2 known to be the generic behavior for the local model no universal power law asymptotics exists but we show for a particular threshold distribution how the avalanche distribution can nevertheless be explicitly calculated in the large bundle limit

regularization properties of the 2d homogeneous boltzmann equation without cutoff we consider the 2 dimensional spatially homogeneous boltzmann equation for hard potentials we assume that the initial condition is a probability measure that has some exponential moments and is not a dirac mass we prove some regularization properties for a class of very hard potentials the solution instantaneously belongs to hr for some r in 1 2 depending on the parameters of the equation our proof relies on the use of a well suited malliavin calculus for jump processes

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

phase transition in fiber bundle models with recursive dynamics we study the phase transition in a class of fiber bundle models in which the fiber strengths are distributed randomly within a finite interval and global load sharing is assumed the dynamics is expressed as recursion relations for the redistribution of the applied stress and the evolution of the surviving fraction of fibers we show that an irreversible phase transition of second order occurs from a phase of partial failure to a phase of total failure when the initial applied stress just exceeds a critical value the phase transition is characterized by static and dynamic critical properties we calculate exactly the critical value of the initial stress for three models of this kind each with a different distribution of fiber strengths we derive exact expressions for the order parameter the susceptibility to changes in the initial applied stress and the critical relaxation of the surviving fraction of fibers for all the three models the static and dynamic critical exponents obtained from these expressions are found to be universal

failure avalanches in fiber bundles for discrete load increase the statistics of burst avalanche sizes n during failure processes in a fiber bundle follows a power law d n approximately n xi for large avalanches the exponent xi depends upon how the avalanches are provoked while it is known that when the load on the bundle is increased in a continuous manner the exponent takes the value xi 52 we show that when the external load is increased in discrete and not too small steps the exponent value xi 3 is relevant our analytic treatment applies to bundles with a general probability distribution of the breakdown thresholds for the individual fibers the preasymptotic size distribution of avalanches is also considered

breaking rate minimum predicts the collapse point of overloaded materials as a model of composite materials we choose a bundle of fibers with stochastically distributed breaking thresholds for the individual fibers the fibers are assumed to share the load equally and to obey hookean elasticity right up to the breaking point we study the evolution of the fiber breaking rate at a constant load in excess of the critical load the analysis shows that the breaking rate reaches a minimum when the system is half way from its complete collapse

effect of discontinuity in the threshold distribution on the critical behavior of a random fiber bundle the critical behavior of a random fiber bundle model with mixed uniform distribution of threshold strengths and global load sharing rule is studied with a special emphasis on the nature of distribution of avalanches for different parameters of the distribution the discontinuity in the threshold strength distribution of fibers nontrivially modifies the critical stress as well as puts a restriction on the allowed values of parameters for which the recursive dynamics approach holds good the discontinuity leads to a nonuniversal behavior in the avalanche size distribution for smaller values of avalanche size we observe that apart from the mean field behavior for larger avalanches a new behavior for smaller avalanche size is observed as a critical threshold distribution is approached the phenomenological understanding of the above result is provided using the exact analytical result for the avalanche size distribution most interestingly the prominence of nonuniversal behavior in avalanche size distribution depends on the system parameters

flory like statistics of fracture in the fiber bundle model as obtained via kolmogorov dispersion for turbulence a conjecture it has long been conjectured that rapid fracture propagation dynamics in materials and turbulent motion of fluids are two manifestations of the same physical process the universality class of turbulence kolmogorov dispersion in particular is conjectured to be identifiable with the flory statistics for linear polymers self avoiding walks on lattices these help us to relate fracture statistics to those of linear polymers flory statistics the statistics of fracture in the fiber bundle model fbm are now well studied and many exact results are now available for the equal load sharing els scheme yet the correlation length exponent in this model was missing and we show here how the correspondence between fracture statistics and the flory mapping of kolmogorov statistics for turbulence helps us to make a conjecture about the value of the correlation length exponent for fracture in the els limit of fbm and also about the upper critical dimension in addition the fracture avalanche size exponent values at lower dimensions as estimated from such mapping to flory statistics also compare well with the observations

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

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

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

estimates for the boltzmann collision operator via radial symmetry and fourier transform abstract we extend the l p theory of the boltzmann collision operator by using classical techniques based in the carleman representation and fourier analysis allied to new ideas that exploit the radial symmetry of this operator we are then able to greatly simplify existent technical proofs in this theory extend the range and obtain explicit sharp constants in some convolution like inequalities for the gain part of the boltzmann collision operator

bounding codimension one subvarieties and a general inequality between chern numbers we extend the miyaoka yau inequality for a surface to an arbitrary nonuniruled normal complex projective variety eliminating the hypothesis that the variety must be minimal the in equality is sharp in dimension three and is also sharp among minimal varieties for nonminimal varieties in dimension four or higher an error term is picked up which can be controlled as a consequence we bound codimension one subvarieties in a variety of general type linearly in terms of their chern numbers in particular we show that there are only a finite number of smooth fano abelian and calabi yau subvarieties of codimension one in any variety of general type

quasi projectivity of moduli spaces of polarized varieties by means of analytic methods the quasi projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of nonuniruled polarized varieties

failure properties of loaded fiber bundles having a lower cutoff in fiber threshold distribution the presence of lower cutoff in fiber threshold distribution may affect the failure properties of a bundle of fibers subjected to external load we investigate this possibility both in an equal load sharing els model and in a local load sharing lls one using analytic as well as numerical methods in the els model the critical strength gets modified and beyond a certain lower cutoff level the whole bundle fails instantly brittle failure after the first fiber ruptures although the dynamic exponents for the order parameter susceptibility and relaxation time remain unchanged the avalanche size distribution shows a gradual deviation from the mean field power law a similar instant failure situation occurs in the lls model at a lower cutoff level which reduces to that of the equivalent els model at higher high enough dimensions also the system size variation of the bundle s strength and the avalanche statistics show strong dependence on the lower cutoff level

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

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

regularity of solutions to the spatially homogeneous boltzmann equation for non maxwellian molecules without angular cutoff abstract in this paper we use the pseudo differential calculus to analyze the smoothing property of weak solutions to the spatially homogeneous boltzmann equation precisely we show that for the non maxwellian molecules with debye yukawa potential if the positive weak solution is lipschitz continuous in the velocity variable then it lies in the sobolev space h loc r 3 and hence it is automatically smooth

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

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

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

criticality in the dynamic failure model to probe the interplay of different mechanisms for criticality we study the dynamic failure model each element of which is allowed to break or to be healed with given conditional probabilities under an external load the system evolves to the stationary state which may exhibit breakages of various sizes in addition as the healing parameter is varied the system undergoes a discontinuous transition between the functioning state and the failed one in the regimes with appropriate amounts of healing and fluctuations of the surviving fraction scale invariance in spatial and temporal correlations is observed manifested by the power spectrum of the breakage rate of elements and by the cluster size distribution of broken elements

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

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

prediction of the collapse point of overloaded materials by monitoring energy emissions a bundle of many fibers with stochastically distributed breaking thresholds is considered as a model of composite materials the fibers are assumed to share the load equally and to obey hookean elasticity up to the breaking point the bundle is slightly overloaded which leads to complete failure we study the properties of emission bursts in which an amount of energy e is released the analysis shows that the size of the energy bursts has a minimum when the system is halfway from the collapse point

well posedness of spatially homogeneous boltzmann equation with full range interaction in the present work we give the coercivity estimates for the boltzmann collision operator q without angular cut off to clarify in which case the functional langle q g f f rangle will become the truly sub elliptic based on this observation and commutator estimates in alexandre et al arch rat mech anal 198 39 123 2010 the upper bound estimates for the collision operator in chen and he arch rat mech anal 201 2 501 548 2011 and the stability results in desvillettes and mouhot arch rat mech anal 193 2 227 253 2009 in the function space l 1 q cap h n we establish global well posedness or local well posedness for the spatially homogeneous boltzmann equation with full range interaction covering most of physical collision kernels

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

simulation of finite size effects of the fiber bundle model abstract in theory the macroscopic fracture of materials should correspond with the thermodynamic limit of the fiber bundle model however the simulation of a fiber bundle model with an infinite size is unrealistic to study the finite size effects of the fiber bundle model fiber bundle models of various size are simulated in detail the effects of system size on the constitutive behavior critical stress maximum avalanche size avalanche size distribution and increased step number of external load are explored the simulation results imply that there is no feature size or cut size for macroscopic mechanical and statistical properties of the model the constitutive curves near the macroscopic failure for various system size can collapse well with a simple scaling relationship simultaneously the introduction of a simple extrapolation method facilitates the acquisition of more accurate simulation results in a large limit system which is better for comparison with theoretical results

random fiber bundle with many discontinuities in the threshold distribution we study the breakdown of a random fiber bundle model rfbm with n discontinuities in the threshold distribution using the global load sharing scheme in other words n 1 different classes of fibers identified on the basis of their threshold strengths are mixed such that the strengths of the fibers in the ith class are uniformly distributed between the values sigma2i 2 and sigma2i 1 where 1 or i or n 1 moreover there is a gap in the threshold distribution between ith and i 1 th class we show that although the critical stress depends on the parameter values of the system the critical exponents are identical to that obtained in the recursive dynamics of a rfbm with a uniform distribution and global load sharing the avalanche size distribution on the other hand shows a nonuniversal non power law behavior for smaller values of avalanche sizes which becomes prominent only when a critical distribution is approached we establish that the behavior of the avalanche size distribution for an arbitrary n is qualitatively similar to a rfbm with a single discontinuity in the threshold distribution n 1 especially when the density and the range of threshold values of fibers belonging to strongest n 1 th class is kept identical in all the cases