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

typical bad differentiable extensions abstract by a 1923 result of v jarnik every differentiable map from a closed subset p of r into r has a differentiable extension f r r it has been recently proved by the authors that among such differentiable extensions of there is always one that is nowhere monotone on r p in particular the family e 1 r of bad differentiable extensions f r r of for which the set f 0 x r f x 0 is dense in r p is nonempty we notice here that e 1 r with a natural distance is a complete metric space and prove that actually a typical function in e 1 r is nowhere monotone on r p at the same time the set m r of functions f e 1 r which are monotone on some nonempty subinterval of every nonempty open u r p is dense in e 1 r this last statement remains true when the term monotone is replaced with either of the following three terms strictly increasing strictly decreasing or constant

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

local properties of self similar solutions to smoluchowski s coagulation equation with sum kernels the regularity of the scaling profiles to smoluchowski s coagulation equation is studied when the coagulation kernel k is given by k x y x y with 0 1 more precisely is c smooth on 0 and decays exponentially fast for large x furthermore the singular behaviour of x as x 0 is identified thus giving a rigorous proof of physical conjectures

fast conservative and entropic numerical methods for the boson boltzmann equation in this paper we derive accurate numerical methods for the quantum boltzmann equation for a gas of interacting bosons the schemes preserve the main physical features of the continuous problem namely conservation of mass and energy the entropy inequality and generalized bose einstein distributions as steady states these properties are essential in order to develop schemes that are able to capture the energy concentration behavior of bosons in addition we develop fast algorithms for the numerical evaluation of the resulting quadrature formulas which allow the final schemes to be computed only in o n2 log 2n operations instead of o n3

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

on the singularity formation and relaxation to equilibrium in 1d fokker planck model with superlinear drift abstract we consider a class of fokker planck equations with linear diffusion and superlinear drift enjoying a formal wasserstein like gradient flow structure with convex mobility function in the drift dominant regime the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component our approach which addresses the one dimensional case is based on a reformulation of the problem in terms of the pseudo inverse distribution function motivated by the structure of the equation in the new variables we establish a general framework for global in time existence uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate resp singular parabolic equations using as a key tool comparison principles and maximum arguments we then focus on a specific equation and study in more detail the regularity and dynamics of solutions in particular blow up behaviour formation of condensates i e dirac measures at zero and long time asymptotics are investigated as a consequence in the mass supercritical case solutions will blow up in l in finite time and understood in a generalised measure sense they will eventually have condensate we further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient the equations considered are motivated by a model for bosons introduced by kaniadakis and quarati 1994 which has a similar entropy structure and a critical mass if d 3

on the fundamental solution of a linearized uehling uhlenbeck equation in this paper we describe the fundamental solution of the equation that is obtained linearizing the uehling uhlenbeck equation around the steady state of kolmogorov type f k k 7 6 detailed estimates on its asymptotics are obtained

on a system of equations for the normal fluid condensate interaction in a bose gas abstract the existence of global solutions for a system of differential equations is proved and some of their properties are described the system involves a kinetic equation for quantum particles it is a simplified version of a mathematical description of a weakly interacting dilute gas of bosons in the presence of a condensate near the critical temperature

a joint generalization of whitneyʼs c1 extension theorem and aversa laczkovich preissʼ extension theorem abstract we prove a joint generalization of whitney s c 1 extension theorem and aversa laczkovich preiss extension theorem it can be roughly described as a theorem on extendibility of a differentiable function defined on a closed subset of r n to a differentiable function on r n with preserving the continuity of the derivative other results on c 1 and differentiable extensions from thick subsets of r n are also proved

from the von neumann equation to the quantum boltzmann equation ii identifying the born series in a previous paper ca1 the author studied a low density limit in the periodic von neumann equation with potential modified by a damping term the model studied in ca1 considered in dimensions d 3 is deterministic it describes the quantum dynamics of an electron in a periodic box actually on a torus containing one obstacle when the electron additionally interacts with say an external bath of photons the periodicity condition may be replaced by a dirichlet boundary condition as well in the appropriate low density asymptotics followed by the limit where the damping vanishes the author proved in ca1 that the above system is described in the limit by a linear space homogeneous boltzmann equation with a cross section given as an explicit power series expansion in the potential the present paper continues the above study in that it identifies the cross section previously obtained in ca1 as the usual born series of quantum scattering theory which is the physically expected result hence we establish that a von neumann equation converges in the appropriate low density scaling towards a linear boltzmann equation with cross section given by the full born series expansion we do not restrict ourselves to a weak coupling limit where only the first term of the born series would be obtained fermi s golden rule

scaling solutions of smoluchowski s coagulation equation we investigate the structure of scaling solutions of smoluchowski s coagulation equation of the formc k t s t k s t wherec k t is the concentration of clusters of sizek at timet s t is the average cluster size and x is a scaling function for the rate constantk i j in smoluchowski s equation we make the very general assumption thatk i j is a homogeneous function of the cluster sizesi andj k i j a k ai aj for alla 0 but we restrict ourselves to kernels satisfyingk i j j 0 asj we show that gelation occurs if 1 and does not occur if 1 for all gelling and nongelling models we calculate the time dependence ofs t and we derive an equation for x we present a detailed analysis of the behavior of x at large and small values ofx for all models we find exponential large x behavior x ax e x asx and for different kernelsk i j algebraic or exponential small x behavior x bx or x exp cx asx 0

finite time blow up and condensation for the bosonic nordheim equation the homogeneous bosonic nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous weakly interacting quantum gas of bosons we show the existence of classical solutions of the homogeneous bosonic nordheim equation that blow up in finite time we also prove finite time condensation for a class of weak solutions of the kinetic equation

on application of the lanczos method to solution of some partial differential equations abstract let a be a square symmetric n n matrix be a vector from r n and f be a function defined on the spectral interval of a the problem of computation of the vector u f a arises very often in mathematical physics we propose the following method to compute u first perform m steps of the lanczos method with a and define the spectral lanczos decomposition method sldm solution as um qf h e1 where q is the n m matrix of the m lanczos vectors and h is the m m tridiagonal symmetric matrix of the lanczos method we obtain estimates for u um that are stable in the presence of computer round off errors when using the simple lanczos method we concentrate on computation of exp ta when a is nonnegative definite error estimates for this special case show superconvergence of the sldm solution sample computational results are given for the two dimensional equation of heat conduction these results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time stepping schemes finally we consider application of sldm to hyperbolic and elliptic equations

uniqueness of self similar solutions to smoluchowski s coagulation equations for kernels that are close to constant we consider self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 1 we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small which is the first such uniqueness result for kernels that are not solvable our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures

on the boltzmann grad limit for the broadwell model of the boltzmann equation an example is given of a model dynamics for which the broadwell model of the boltzmann equation seems to appear in the formal stage of the boltzmann grad limit but actually does not

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

from the n body schrodinger equation to the quantum boltzmann equation a term by term convergence result in the weak coupling regime in this paper we analyze the asymptotic dynamics of a system of n quantum particles in a weak coupling regime particles are assumed statistically independent at the initial time

numerical solution of the smoluchowski kinetic equation and asymptotics of the distribution function we obtain the numerical solution of the smoluchowski kinetic equation for model kernels u m1 m2 varies as m1 m2 lambda and u m1 m2 varies as m1m2 lambda 2 0 lambda or 2 we show that the behaviour of the solution for the kernel u varies as m1 m2 lambda at 0 lambda or 1 and u varies as m1m2 lambda 2 at 0 lambda or 2 becomes self similar after some time the shape of the scaling function is analysed in particular a simple approximate expression for it at u varies as m1 m2 lambda is found an interesting result is obtained for u varies as m1m2 lambda 2 0 lambda 1 the asymptotic behaviour of the scaling function proved to be non power we develop the procedure for determining tcr the critical indices and the exponent of the power law asymptotics of the smoluchowski equation solution the concrete values of these quantities for the model kernels are obtained the stability conditions of the algorithm for numerical solving are analysed the possibilities afforded by numerical solution for investigation of the smoluchowski equation are discussed

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

nontrivial polydispersity exponents in aggregation models we consider the scaling solutions of smoluchowski s equation of irreversible aggregation for a non gelling collision kernel the scaling mass distribution f s diverges as s tau when s 0 tau is non trivial and could until now only be computed by numerical simulations we develop here new general methods to obtain exact bounds and good approximations of tau for the specific kernel kdd x y x 1 d y 1 d d describing a mean field model of particles moving in d dimensions and aggregating with conservation of mass s r d r is the particle radius perturbative and nonperturbative expansions are derived for a general kernel we find exact inequalities for tau and develop a variational approximation which is used to carry out the first systematic study of tau d d for kdd the agreement is excellent both with the expansions we derived and with existing numerical values finally we discuss a possible application to 2d decaying turbulence

differentiable extension theorem a lost proof of v jarnik abstract we present a beautiful but relatively unknown theorem that every differentiable function f p r with p r being closed admits differentiable extension f r r we present an elementary proof of this result based on a construction sketched in a hard to access 1923 paper 4 of v jarnik using this construction we also obtain an elegant version of whitney extension theorem characterizing when such an f admits continuously differentiable extension

global validity of the boltzmann equation for a three dimensional rare gas in vacuum we consider a system ofn hard disks in 2 in the boltzmann grad limit i e n d 0 n d 1 0 whered is the diameter of the disks if is sufficiently small and if the joint distribution densities factorize at time zero we prove that the time evolved one particle distribution converges for all times to the solution of the boltzmann equation with the same initial datum

lanczos method for heat conduction analysis an analysis technique that uses the lanczos vectors to construct a reyleigh ritz solution for the transient heat conduction problem is presented the lanczos vectors are derived from the heat supply vector and incorporate all the information peculiar to a given problem they are orthonormal with respect to the heat capacitance matrix and are generated with a minimum of computational effort furthermore an error bound is derived that can be used to terminate the lanczos algorithm as soon as the number of vectors required to obtain a desired degree of accuracy has been generated a summary of the algorithm is presented and the accuracy and efficiency of the method is illustrated using a numerical example

singular solutions for the uehling uhlenbeck equation in this paper we prove the existence of solutions of the uehling uhlenbeck equation that behave like k 7 6 as k 0 from the physical point of view such solutions can be thought as particle distributions in the space of momentum having a sink or a source of particles with zero momentum our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function k 7 6 that we obtained in an earlier paper

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

similarity solution of coagulation equation with an inverse kernel abstract we analyse a model for the aggregation of polystyrene particles which arises in an electrorheology system in which linear clusters grow upon the application of an alternating electromagnetic field we consider a coagulation kernel involving negative powers of cluster sizes and investigate the reduction of the governing equations to a similarity solution in the large time limit comparison between the experimental results and the theory presented here shows a good collapse of the data onto a single curve which matches the theoretical results particularly well at the larger cluster sizes

on coupling kinetic and schrodinger equations abstract we consider in this paper a system coupling a linear quantum boltzmann equation and a defocussing cubic nonlinear schrodinger equation the schrodinger equation reflects the dynamics of the wave function of the bose einstein condensate and the kinetic part of the system describes the evolution of the density function of the thermal cloud an existence and uniqueness result for the system is supplied we also prove the convergence to equilibrium of the density function of the thermal cloud and a scattering theory for the wave function of the condensate

weakly nonlinear schrodinger equation with random initial data it is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data while for the general case a proof of the kinetic limit remains open we report on first progress as wave equation we consider the nonlinear schrodinger equation discretized on a hypercubic lattice since this is a hamiltonian system a natural choice of random initial data is distributing them according to the corresponding gibbs measure with a chemical potential chosen so that the gibbs field has exponential mixing the solution t x of the nonlinear schrodinger equation yields then a stochastic process stationary in x d and t if denotes the strength of the nonlinearity we prove that the space time covariance of t x has a limit as 0 for t 2 with fixed and sufficiently small the limit agrees with the prediction from kinetic theory

on the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear schrodinger equation we study the mathematical properties of a kinetic equation derived in escobedo and velazquez arxiv 1305 5746v1 math ph which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear schrodinger equation in particular we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass we also prove that any nontrivial initial datum yields the instantaneous onset of a condensate by which we mean that for any nontrivial solution the mass of the origin is strictly positive for any positive time furthermore we show that the only stationary solutions with finite total measure are dirac masses at the origin we finally construct solutions with finite energy where the energy is transferred to infinity in a self similar manner

approach to self similarity in smoluchowski s coagulation equations we consider the approach to self similarity or dynamical scaling in smoluchowski s equations of coagulation for the solvable kernels k x y 2 x y and xy in addition to the known self similar solutions with exponential tails there are one parameter families of solutions with algebraic decay whose form is related to heavy tailed distributions well known in probability theory for k 2 the size distribution is mittag leffler and for k x y and k xyit is a power law rescaling of a maximally skewed stable levy distribution we characterize completely the domains of attraction of all self similar solutions under weak convergence of measures our results are analogous to the classical characterization of stable distributions in probability theory the proofs are simple relying on the laplace transform and a fundamental rigidity lemma for scaling limits c 2004 wiley periodicals inc

the lanczos optimization of a splitting up method to solve homogeneous evolutionary equations abstract let the cauchy problem for a symmetrical homogeneous ode system be solved by a difference scheme and let s be the required number of matrix vector operations with the finite difference matrix in classical schemes s is proportional to the number of time steps the lanczos method is used to decrease s without essential increase of error a theoretical estimate is given which shows approximately the s advantage of such an approach its application to the 2d heat conduction equation is considered one and two cyclic alternating direction difference schemes are used some numerical experiments show that the arithmetical costs are reduced by a factor 3 up to 60 with respect to the classical approach combination of a splitting scheme and the lanczos method is also proposed for the computation of the lower part of the spectrum and for solving some other problems

an analytical solution of coagulation processes with collision induced fragmentation we propose a single species coagulation model with collision induced fragmentation in which a pair of clusters can coagulate into a larger one if their encounter is a complete inelastic collision otherwise one of them will fragment into two smaller clusters due to a destructive collision consider a simple system with the constant coagulation kernel i i j i and the size dependent fragmentation kernel j l i j 2jlu we then investigate analytically the kinetic behavior of such a system by means of the rate equation approach it is indicated that the results are crucially dependent on the value of index u in the case of u 1 and i 2j u here x is the riemann zeta function the cluster size distribution ak t approaches the conventional scaling form and the system will evolve permanently in time while in other cases ak t always takes the modified scaling form moreover a balance can be established between coagulation and collision induced breakage processes and thus the system can eventually evolve to a steady state

self similar spreading in a merging splitting model of animal group size in a recent study of certain merging splitting models of animal group size degond et al in j nonlinear sci 27 2 379 424 2017 it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense corresponding to unbounded growth of group size in the present paper we show that for any such initial distribution with a power law tail the solution approaches a self similar spreading form a one parameter family of such self similar solutions exists with densities that are completely monotone having power law behavior in both small and large size regimes with different exponents

a survey of numerical solutions to the coagulation equation we present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form aij i j i j and monodisperse initial conditions the results confirm that there are three classes of rate coefficients with qualitatively different solutions for 1 and 1 the numerical solution evolves in an orderly fashion and tends towards a self similar solution at large time t the properties of the numerical solution in the scaling limit agree with the analytic predictions of van dongen and ernst in particular for the subset with 0 and 1 the numerical solution tends towards a self similar solution as t approaches a finite time t0 the mass spectrum nk develops at t0 a power law tail nk k at large masses that violates mass conservation and runaway growth gelation is expected to start at tcrit t0 in the limit the initial number of particles n0 the exponent is in general less than the analytic prediction 3 2 and t0 k 1 n0a11 with k 1 2 if 1 1 for 1 the behaviours of the numerical solution are similar to those found in a previous paper by us they strongly suggest that there are no self consistent solutions at any time and that runaway growth is instantaneous in the limit n0 they also indicate that the time tcrit for the onset of runaway growth decreases slowly towards zero with increasing n0

global existence of solutions of the boltzmann equation for bose einstein particles with anisotropic initial data abstract in this paper we prove the global in time existence and uniqueness of solutions of the spatially homogeneous boltzmann equation for bose einstein particles for the hard sphere model for bounded anisotropic initial data the main idea of our proof is as follows we first establish an intermediate equation which is closely related to the original equation and is relatively easily proven to have global in time and unique solutions then we use the multi step iterations of the collision gain operator to obtain a desired uniform l bound for solutions of the intermediate equation so that if an initial datum is sufficiently small relative to the inverse of the planck constant which belongs to the case of very high temperature then the corresponding solution of the intermediate equation becomes the solution of the original equation

well posedness of smoluchowski s coagulation equation for a class of homogeneous kernels the uniqueness and existence of measure valued solutions to smoluchowski s coagulation equation are considered for a class of homogeneous kernels denoting by 2 0 the degree of homogeneity of the coagulation kernel a measure valued solutions are shown to be unique under the sole assumption that the moment of order of the initial datum is finite a similar result was already available for the kernels a x y 2 x y and xy and is extended here to a much wider class of kernels by a different approach the uniqueness result presented herein also seems to improve previous results for several explicit kernels furthermore a comparison principle and a contraction property are obtained for the constant kernel

self similar solutions with fat tails for smoluchowski s coagulation equation with locally bounded kernels the existence of self similar solutions with fat tails for smoluchowski s coagulation equation has so far only been established for the solvable kernels and the diagonal one in this paper we prove the existence of such self similar solutions for continuous kernels k that are homogeneous of degree gamma in infty 1 and satisfy k x y c x y more precisely for any rho in gamma 1 we establish the existence of a continuous weak nonnegative self similar profile with decay x 1 rho as x

a uniqueness result for self similar profiles to smoluchowski s coagulation equation revisited in this note we indicate how to correct the proof of a uniqueness result in 6 for self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 frac 1 2 under the additional assumption in comparison to 6 that k has an analytic extension to mathbb c setminus infty 0 and that the precise asymptotic behaviour of k at the origin is prescribed we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small the complete details of the proof are available in 4 in addition we give here the proof of a uniqueness result for a related but simpler problem that appears in the description of self similar solutions for x rightarrow infty

some considerations on the derivation of the nonlinear quantum boltzmann equation ii the low density regime in this paper we analyse the asymptotic dynamics of a system of n identical quantum particles in a low density regime our approach follows the strategy introduced by the authors in a previous work 2 to treat the simpler weak coupling regime the time evolution of the wigner transform of the one particle reduced density matrix is represented by means of a perturbative series the expansion is obtained upon iterating the duhamel formula in the spirit of the paper by lanford 32 for short times and small interaction potential we rigorously prove that a subseries of the complete perturbative series converges to the solution of the nonlinear boltzmann equation that is physically relevant in the context an important point is that we completely identify the cross section entering the limiting boltzmann equation as being the born series expansion of quantum scattering

existence of self similar solutions to smoluchowski s coagulation equation the existence of self similar solutions to smoluchowski s coagulation equation has been conjectured for several years by physicists and numerical simulations have confirmed the validity of this conjecture still there was no existence result up to now except for the constant and additive kernels for which explicit formulae are available in this paper the existence of self similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

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

derivation of the boltzmann equation for a fermi gas we consider the time evolution of a fermi gas with two body interaction for an initial state which is translation invariant and sufficiently clustering we puth h 0 v we take the limit 0 t such that 2t and show that a the limiting state does not depend on thep point correlations of forp 2 b has vanishingp point correlations forp 2 and c the two point function that determines satisfies the boltzmann equation to avoid nonessential technical difficulties we consider the case of a fermi gas on a lattice

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

on the dynamics of finite temperature trapped bose gases abstract the system that describes the dynamics of a bose einstein condensate bec and the thermal cloud at finite temperature consists of a nonlinear schrodinger nls and a quantum boltzmann qb equations in such a system of trapped bose gases at finite temperature the qb equation corresponds to the evolution of the density distribution function of the thermal cloud and the nls is the equation of the condensate the quantum boltzmann collision operator in this temperature regime is the sum of two operators c 12 and c 22 which describe collisions of the condensate and the non condensate atoms and collisions between non condensate atoms above the bec critical temperature the system is reduced to an equation containing only a collision operator similar to c 22 which possesses a blow up positive radial solution with respect to the l norm cf 29 on the other hand at the very low temperature regime only a portion of the transition temperature t b e c the system can be simplified into an equation of c 12 with a different much higher order transition probability which has a unique global classical positive radial solution with weighted l 1 norm cf 3 in our model we first decouple the qb which contains c 12 c 22 and the nls equations then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system different from the case considered in 29 due to the presence of the bec the collision integrals are associated to sophisticated energy manifolds rather than spheres since the particle energy is approximated by the bogoliubov dispersion law moreover the mass of the full system is not conserved while it is conserved for the case considered in 29 a new theory is then supplied

dynamical scaling in smoluchowski s coagulation equations uniform convergence smoluchowski s coagulation equation is a fundamental mean field model of clustering dynamics we consider the approach to self similarity or dynamical scaling of the cluster size distribution for the solvable rate kernels k x y 2 x y and xy in the case of continuous cluster size distributions we prove uniform convergence of densities to a self similar solution with exponential tail under the regularity hypothesis that a suitable moment have an integrable fourier transform for discrete size distributions we prove uniform convergence under optimal moment hypotheses our results are completely analogous to classical local convergence theorems for the normal law in probability theory the proofs rely on the fourier inversion formula and the solution for the laplace transform by the method of characteristics in the complex plane

doubly paradoxical functions of one variable abstract this paper concerns three kinds of seemingly paradoxical real valued functions of one variable the first two defined on r are the celebrated continuous nowhere differentiable functions known as weierstrass s monsters and everywhere differentiable nowhere monotone functions simultaneously smooth and very rugged to which we will refer as differentiable monsters the third kind was discovered only recently and consists of differentiable functions f defined on a compact perfect subset x of r which has derivative equal zero on its entire domain making it everywhere pointwise contractive while counterintuitively f maps x onto itself the goal of this note is to show that this pointwise shrinking globally stable map f can be extended to functions f g r r which are differentiable and weierstrass s monsters respectively thus we pack three paradoxical examples into two functions the construction of f is based on the following variant of jarnik s extension theorem for every differentiable function f from a closed p r into r there exists its differentiable extension f r r such that f is nowhere monotone on r p

long time convergence of the bose einstein condensation we study long time behavior of the bose einstein condensation of measure valued solutions f t of the space homogeneous and velocity isotropic boltzmann equation for bose einstein particles at low temperature we prove that if f 0 ge 0 is a non singular borel measure on mathbb r ge 0 satisfying a very low temperature condition and that the ratio f 0 0 varepsilon varepsilon alpha is sufficiently large for all varepsilon in 0 r for some constants 0 0 then there exists a solution f t of the equation on 0 infty with the initial datum f 0 such that f t 0 converges to the expected bose einstein condensation as t rightarrow infty we also show that such initial data f 0 exist extensively

blow up rate estimates for the solutions of the bosonic boltzmann nordheim equation in this paper we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic boltzmann nordheim equation near the blow up we obtain some estimates on the blow up rate of the solutions and prove that as long as a solution is bounded above by the critical singularity 1x the equilibrium solutions behave like this power law near the origin it remains bounded in the uniform norm in sec iii of the paper we prove a local existence result for a class of measure valued mild solutions which is of independent interest and which allows us to solve the boltzmann nordheim equation for some classes of unbounded densities