global solutions and time decay of the non cutoff vlasov maxwell boltzmann system in the whole space the global in time classical solutions to the cauchy problem of the vlasov maxwell boltzmann system near maxwellians are obtained under the lower regularity index assumption and the weaker smallness condition on the initial perturbation in comparison with the work duan et al in kinet relat models 6 1 159 204 2013 in particular we show the relation between time decay rates and spatial derivatives of solutions our analysis relies on a refined energy estimates and the interpolation techniques between negative sobolev norms and positive sobolev norms without linear decay analysis

large deviations hypotheses testing and source coding for finite markov chains let x n n geq 1 be a finite markov chain with transition probability matrix of strictly positive entries a large deviation theorem is proved for the empirical transition count matrix and is used to get asymptotically optimal critical regions for testing simple hypotheses about the transition matrix as a corollary the error exponent in the source coding theorem for x n is obtained these results generalize the corresponding results for the independent and identically distributed case

boltzmann diffusive limit beyond the navier stokes approximation given a normalized maxwellian and n 1 we establish the global in time validity of a diffusive expansion 0 1 f e t x v ef 1 t x v e 2 f 2 t x v e n f e n t x v for a solution f e to the rescaled boltzmann equation diffusive scaling 0 2 e 1 f e v x fe 1 eq f e f e inside a periodic box t 3 we assume that in the initial expansion 0 1 at t 0 the fluid parts of these f m 0 x v have arbitrary divergence free velocity fields as well as temperature fields for all 1 m n while f 1 0 x v has small amplitude in h 2 for m 2 these f m t x v are determined by a sequence of linear navier stokes fourier systems iteratively more importantly the remainder f e n t x v is proven to decay in time uniformly in e via a unified nonlinear energy method in particular our results lead to an error estimate for f 1 t x v the well known navier stokes fourier approximation and beyond the collision kernel q includes hard sphere the cutoff inverse power as well as the coulomb interactions

decay of the two species vlasov poisson boltzmann system specialized research fund for the doctoral program of higher education 20120121120023 natural science foundation of fujian province of china 2012j05011 national natural science foundation of china 11201389

on a micro macro model for polymeric fluids near equilibrium in this paper we study a micro macro model for polymeric fluid the system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation we employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure for the initial data not far from the equilibrium we prove the global existence and uniqueness of classical solutions to the system 2006 wiley periodicals inc

a new class of supermemory gradient methods in this paper we propose a new class of supermemory gradient methods for unconstrained optimization problems trust region approach is used in the new algorithms to guarantee the global convergence in each iteration the new algorithms generate a suitable trust region radius automatically and obtain the next iterate by solving a simple subproblem these algorithms converge stably and averagely due to using more iterative information at each iteration and can be reduced to quasi newton methods when the iterate is close to the optimal solution numerical results show that this new class of supermemory gradient methods is effective in practical computation

the same class of stationary solutions to some multidimensional kinetic systems with extensive background density abstract in this study we investigate the existence and uniqueness of the same class of stationary solutions for some kinetic systems comprising the vlasov poisson fokker planck system vlasov poisson boltzmann system and vlasov maxwell boltzmann system in arbitrary space dimensions n n 3 essentially the problem can be reduced to solving the problem of a second order elliptic equation with exponential nonlinearity this result had been proved in three spatial dimensions but the extension to a higher dimensional setting makes the existence proof nontrivial in particular it necessary to highlight that the requirement regarding the background density function is relaxed compared with previous studies in the case of three spatial dimensions

l1 and bv type stability of the boltzmann equation with external forces abstract based on the existence theory on the boltzmann equation with external forces in infinite vacuum in this paper we will study the l 1 and bv type stability of the classical solutions for small initial data the stability results generalize those for the boltzmann equation without force to the case with external force in particular we show that the stability can be established for the soft potentials directly while the stability for the hard potentials and hard sphere model is obtained through the construction of some nonlinear functionals the functionals thus constructed generalize those constructed in s y ha nonlinear functionals of the boltzmann equation and uniform stability estimates j differential equations 215 2005 178 205 for the case without force to capture the effect of the force term on the time evolution of the solutions

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

maximum entropy principles for disordered spins we transform nonstationary independent random fields with exponential radon nikodym factors and study the asymptotics of the transformed processes as applications we deduce conditional limit theorems for such random fields and we study a curie weiss type mean field model of a quenched mixed magnetic crystal this model has quenched site disorder and frustration but non random coupling constants we find a continuous phase transition with critical exponents equal to those of the classical curie weiss theory

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

asymptotics of likelihood ratio derivative estimators in simulations of highly reliable markovian systems we discuss the estimation of derivatives of a performance measure using the likelihood ratio method in simulations of highly reliable markovian systems we compare the difficulties of estimating the performance measure and of estimating its partial derivatives with respect to component failure rates as the component failure rates tend to 0 and the component repair rates remain fixed we first consider the case when the quantities are estimated using naive simulation i e when no variance reduction technique is used in particular we prove that in the limit some of the partial derivatives can be estimated as accurately as the performance measure itself this result is of particular interest in light of the somewhat pessimistic empirical results others have obtained when applying the likelihood ratio method to other types of systems however the result only holds for certain partial derivatives of the performance measure when using naive simulation more specifically we can estimate a certain partial

line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions many successful quasi newton methods for optimization are based on positive definite local quadratic approximations to the objective function that interpolate the values of the gradient at the current and new iterates line search termination criteria used in such quasi newton methods usually possess two important properties first they guarantee the existence of such a local quadratic approximation second under suitable conditions they allow one to prove that the limit of the component of the gradient in the normalized search direction is zero this is usually an intermediate result in proving convergence collinear scaling algorithms proposed initially by davidon in 1980 are natural extensions of quasi newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations and that they interpolate values of both the gradient and the function at the current and new iterates line search termination criteria that guarantee the existence of such a normal conic local approximation which also allow one to prove that the component of the gradient in the normalized search direction tends to zero are not known in this paper we propose such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions

the cauchy problem for the bgk equation with an external force abstract in this paper we study the cauchy problem for the bgk equation with an external force firstly we establish an l existence result for this equation and obtain some weighted l estimates then by means of the regularizing effects to the initial datum we construct the approximate solutions and obtain some uniform estimates of the approximate solutions finally by using compactness method and passing to the limit we prove the existence theorems of the l 1 and l p solutions and establish the propagation properties of the l p moments

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

acoustic limit for the boltzmann equation in the whole space abstract this paper is devoted to the following rescaled boltzmann equation in the acoustic time scaling in the whole space 0 1 t f v x f 1 q f f x r 3 t 0 with prescribed initial data f t 0 f 0 x v x r 3 for a solution f t x v f t x v to the rescaled boltzmann equation 0 1 in the whole space r 3 for all t 0 with initial data f 0 x v f 0 x v f 0 x v x v r 3 our main purpose is to justify the global in time uniform energy estimates of f t x v in and prove that f t x v converges strongly to f t x v whose dynamic is governed by the acoustic system

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

the boltzmann equation besov spaces and optimal time decay rates in rxn abstract we prove that k th order derivatives of perturbative classical solutions to the hard and soft potential boltzmann equation without the angular cut off assumption in the whole space r x n with n 3 converge in large time to the global maxwellian with the optimal decay rate of o t 1 2 k n 2 n r in the l x r l v 2 norm for any 2 r these results hold for any 0 n 2 as long as initially f 0 b 2 l v 2 in the hard potential case we prove faster decay results in the sense that if p f 0 b 2 l v 2 and i p f 0 b 2 1 l v 2 for n 2 n 2 2 then the solution decays the global maxwellian in l v 2 l x 2 with the optimal large time decay rate of o t 1 2

complete equivalence of the gibbs ensembles for one dimensional markov systems we show the equivalence of the gibbs ensembles at the level of measures for one dimensional markov systems with arbitrary boundary conditions that is the limit of the microcanonical gibbs ensemble is a gibbs measure with an interaction depending on the microcanonical constraint in fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins including the boundary spins are fixed when conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble we show that the limit is a gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures for this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions we establish analogous results for finite range potentials

new bounds and constructions for granular media coding improved lower and upper bounds on the size and the rate of grain correcting codes are presented the lower bound is gilbert varshamov like combined with a construction by gabrys et al and it improves on the previously best known lower bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval 0 0 0668 one of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval tau in 0 1 8 and meets the lower bound of 1 2 for tau geq 1 8 moreover in a nonasymptotic regime both upper bounds improve on the previously best known results on the largest size of t grain correcting codes of length n for certain values of n and t constructions of 1 grain correcting codes based on a partitioning technique are presented for lengths up to 18 finally a lower bound of 1 2 log 2 n on the minimum redundancy of infty grain detecting codes of length n is presented

stability of the front under a vlasov fokker planck dynamics we consider a kinetic model for a system of two species of particles interacting through a long range repulsive potential and a reservoir at given temperature the model is described by a set of two coupled vlasov fokker plank equations the important front solution which represents the phase boundary is a stationary solution on the real line with given asymptotic values at infinity we prove the asymptotic stability of the front for small symmetric perturbations

exponential decay of the landau equation with potential forces abstract the cauchy problem of the landau equation with potential forces on torus is investigated the global existence of solutions with the symmetry of origin and the exponential convergence rate in time to the steady state are obtained for any initially smooth periodic origin symmetric small perturbation

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

global well posedness for the fokker planck boltzmann equation in besov chemin lerner type spaces abstract in this paper motivated by 16 we use the littlewood paley theory to establish some estimates on the nonlinear collision term which enable us to investigate the cauchy problem of the fokker planck boltzmann equation when the initial data is a small perturbation of the maxwellian equilibrium state under the grad s angular cutoff assumption the unique global solution for the hard potential case is obtained in the besov chemin lerner type spaces c 0 l 2 b 2 r s with 1 r 2 and s 3 2 or s 3 2 and r 1 besides we also obtain the uniform stability of the dependence on the initial data

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

variable length state splitting with applications to average runlength constrained arc codes a new class of constrained systems average runlength constraints arcs is defined by requiring that the sum of n consecutive run lengths be bounded above by a linear function of n in particular the running average runlength of every sequence in the system is bounded above by a constant a general result is given on the capacity of arc systems the state splitting algorithm is then improved for variable length graphs this is then applied to obtain high fixed rate codes from the free binary source to arc systems as an example a rate 1 2 d k

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

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

deriving collinear scaling algorithms as extension of quasi newton methods and the local convergence of dfp and bfgs related collinear scaling algorithms this paper is concerned with collinear scaling algorithms for unconstrained minimization where the underlying local approximants are forced to interpolate the objective function value and gradient at only the two most recent iterates by suitably modifying the procedure of sorensen 1980 for deriving such algorithms we show that two members of the algorithm class derived related to the dfp and bfgs methods respectively are locally and q superlinearly convergent this local analysis as well as the results they yield exhibit the same sort of duality exhibited by those of broyden dennis and more 1973 and dennis and more 1974 for the dfp and bfgs methods the results in this paper also imply the local and q superlinear convergence of collinear scaling algorithms of sorensen 1982 pp 154 156 related to the dfp and bfgs methods

the vlasov poisson landau system in mathbb r 3 _ x for the landau poisson system with coulomb interaction in mathbb r 3 x we prove the global existence uniqueness and large time convergence rates to the maxwellian equilibrium for solutions which start out sufficiently close

on the cauchy problem for the boltzmann equation in the whole space global existence and uniform stability in lξ2 hxn abstract based on a refined energy method in this paper we prove the global existence and uniform in time stability of solutions in the space l 2 h x n to the cauchy problem for the boltzmann equation around a global maxwellian in the whole space r 3 compared with the solution space used by the spectral analysis and the classical energy method the velocity weight functions or time derivatives need not be included in the norms of l 2 h x n which is realized by introducing some temporal interactive energy functionals to estimate the macroscopic dissipation rate the key proof is carried out in terms of the macroscopic equations together with the local conservation laws it is also found that the perturbed macroscopic variables actually satisfy the linearized compressible navier stokes equations with remaining terms only related to the microscopic part

deinterleaving finite memory processes via penalized maximum likelihood we study the problem of deinterleaving a set of finite memory markov processes over disjoint finite alphabets which have been randomly interleaved by a finite memory switch the deinterleaver has access to a sample of the resulting interleaved process but no knowledge of the number or structure of the component markov processes or of the switch we study conditions for uniqueness of the interleaved representation of a process showing that certain switch configurations as well as memoryless component processes can cause ambiguities in the representation we show that a deinterleaving scheme based on minimizing a penalized maximum likelihood cost function is strongly consistent in the sense of reconstructing almost surely as the observed sequence length tends to infinity a set of component and switch markov processes compatible with the original interleaved process furthermore under certain conditions on the structure of the switch including the special case of a memoryless switch we show that the scheme recovers all possible interleaved representations of the original process experimental results are presented demonstrating that the proposed scheme performs well in practice even for relatively short input samples

global weak solutions to the cometary flow equation with a self generated electric field abstract we study the cauchy problem of a cometary flow equation with a self generated electric field this kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation by a wave particle collision operator of the classical vlasov poisson system by asymptotic methods in kinetic theory global existence of nonnegative weak solutions to the cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

on the trend to equilibrium for the vlasov poisson boltzmann equation abstract the dynamics of dilute electrons and plasma can be modeled by vlasov poisson boltzmann equation for which the equilibrium state can be a global maxwellian in this paper we show that the rate of convergence to equilibrium is o t by using a method developed for the boltzmann equation without external force in l desvillettes c villani on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation invent math 159 2005 245 316 in particular the idea of this method is to show that the solution f cannot stay near any local maxwellians for long the improvement in this paper is to handle the effect from the external force governed by the poisson equation moreover by using the macro micro decomposition we simplify the estimation on the time derivatives of the deviation of the solution from the local maxwellian with same macroscopic components

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

negative sobolev spaces and the two species vlasov maxwell landau system in the whole space abstract a global solvability result of the cauchy problem of the two species vlasov maxwell landau system near a given global maxwellian is established by employing an approach different than that of 2 compared with that of 2 the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker our analysis does not rely on the decay of the corresponding linearized system and the duhamel principle and thus it can be used to treat the one species vlasov maxwell landau system for the case of 3 and the one species vlasov maxwell boltzmann system for the case of 1 1 to deduce the global existence results together with the corresponding temporal decay estimates

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

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

stability of the boltzmann equation with potential forces on torus abstract in this paper we are concerned with the stability of solutions to the cauchy problem of the boltzmann equation with potential forces on torus it is shown that the natural steady state with the symmetry of origin is asymptotically stable in the sobolev space with exponential rate in time for any initially smooth periodic origin symmetric small perturbation which preserves the same total mass momentum and mechanical energy for the non symmetric steady state it is also shown that it is stable in l 1 norm for any initial data with the finite total mass mechanical energy and entropy

the vlasov poisson boltzmann system without angular cutoff this paper is concerned with the vlasov poisson boltzmann system for plasma particles of two species in three space dimensions the boltzmann collision kernel is assumed to be angular non cutoff with 3 2s and 1 2 s 1 where s are two parameters describing the kinetic and angular singularities respectively we establish the global existence and convergence rates of classical solutions to the cauchy problem when initial data is near maxwellians this extends the results in duan et al j diff eqs 252 12 6356 6386 2012 math models methods appl sci 23 6 927 2013 for the cutoff kernel with 2 1 to the case 3 2 as long as the angular singularity exists instead and is strong enough i e s is close to 1 the proof is based on the time weighted energy method building also upon the recent studies of the non cutoff boltzmann equation in gressman and strain j amer math soc 24 3 771 847 2011 and the vlasov poisson landau system in guo j amer math soc 25 759 812 2012

boundary conditions for the microscopic fene models we consider the microscopic equation of finite extensible nonlinear elasticity fene models for polymeric fluids under a steady flow field it is shown that for the underlying fokker planck type of equations any preassigned distribution on the boundary will become redundant once the nondimensional number text it li frac hb k bt geq 2 where h is the elasticity constant sqrt b is the maximum dumbbell extension t is the temperature and k b is the usual boltzmann constant moreover if the probability density function is regular enough for its trace to be defined on the sphere m sqrt b then the trace is necessarily zero when text it li 2 these results are consistent with our numerical simulations as well as some recent well posedness results by preassuming a zero boundary distribution

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

entropy at a weight per symbol and embeddings of markov chains in 11 krieger gave a remarkable embedding result for shifts of finite type he proved that two simple necessary conditions strict inequality of the topological entropies of the shifts and the existence of an embedding for their periodic points are in fact sufficient for markov chains when the appropriate definitions of embedding and subsystem were supplied in 10 analogous necessary conditions strict inequality of functions and the existence of a weight preserving embedding for periodic points became apparent however an example in 10 also revealed these conditions to not be sufficient in this paper we bring to bear on embeddings of markov chains the ideas we introduced in 16 in fact we work in the closely related setting of triples z a a a f where 2 a aa is a shift of finite type and f is a function assigning weights to the edges of the graph associated with sa aa the func t ionf takes values in ir or more generally in n q for some qen the weight of a periodic orbit which corresponds to a cycle in the graph is then the sum of the weights of the edges lying on the cycle division by the length of the cycle gives the weight per symbol in 16 we continued the rational linear span of the set wps i of weights per symbol is a finite dimensional vector space vs the convex hull of wps i in v i is a polytope wpsi called the weight per symbol polytope consideration of the faces of wps i gives subsystems of r a aa f which are ordered according to the ordering of the faces the resulting nesting of subsystems is the scaffold of a o a f an embedding of a triple i a a a f in another 2 b ab g is a continuous shift commuting injection q 2 a 12 which preserves weights of periodic orbits the two necessary conditions mentioned above imply w p s i c w when the boundary c wpsi meets wpsg that is when a face f of wps i meets a face g of wpsg any embedding must carry the subsystem corresponding to f into that corresponding to g and this gives rise to further necessary conditons a discussion of these complications and a description of the example of 10 in these terms will be given in sect 7 before that the main embedding

a large step analytic center method for a class of smooth convex programming problems in this paper a large step analytic center method for smooth convex programming is proposed the method is a natural implementation of the classical method of centers it is assumed that the objective and constraint functions fulfil the so called relative lipschitz condition with lipschitz constant m 0 a great advantage of the method over the existing path following methods is that the steps can be made long by performing linesearches in this method linesearches are performed along the newton direction with respect to a strictly convex potential function when located far away from the central path when sufficiently close to this path a lower bound for the optimal value is updated it is proven that the number of iterations required by the algorithm to converge to an epsilon optimal solution is o 1 m 2 sqrt n ln epsilon or o 1 m 2 n ln epsilon depending on the updating scheme for the lower bound

optimal large time behavior of the vlasov maxwell boltzmann system in the whole space in this paper we study the large time behavior of classical solutions to the two species vlasov maxwell boltzmann system in the whole space input amssym bbb r 3 the existence of global in time nearby maxwellian solutions is known from strain in 2006 however the asymptotic behavior of these solutions has been a challenging open problem building on our previous work on time decay for the simpler vlasov poisson boltzmann system we prove that these solutions converge to the global maxwellian with the optimal decay rate of o t 3 2 3 2r in the l l norm for any 2 r if initial perturbation is smooth enough and decays in space velocity fast enough at infinity moreover some explicit rates for the electromagnetic field tending to 0 are also provided 2011 wiley periodicals inc

global classical solution of the vlasov maxwell landau system near maxwellians we consider a classical model in the kinetic theory of plasma the vlasov maxwell landau system global in time classical solutions near maxwellians are constructed for hard potential and soft potential i e 1 the construction of global solution is based on an energy method meanwhile global classical solutions of the vlasov poisson landau system near maxwellians for 1 and the exponential decay of such solutions are obtained

the vlasov poisson boltzmann system for the whole range of cutoff soft potentials abstract the dynamics of dilute electrons can be modeled by the fundamental one species vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electrostatic field for cutoff intermolecular interactions although there is some progress on the construction of global smooth solutions to its cauchy problem near maxwellians recently the problem for the case of very soft potentials remains unsolved by introducing a new time velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables it is shown in this manuscript that the cauchy problem of the one species vlasov poisson boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in 13 for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted sobolev spaces our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well posedness theory to the one species vlasov poisson boltzmann system near maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework

optimal time decay of the boltzmann system for gas mixtures abstract in this paper we are concerned with the boltzmann equation for the mixture of vapors of two gases in the whole space given the initial data of one gas near vacuum and the other near a global maxwellian state we obtain the global existence and optimal large time behaviors of the boltzmann system for such binary mixture the proof is based on kawashima s compensating function technique and a refined energy method

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

global existence and decay of solutions for soft potentials to the fokker planck boltzmann equation without cut off abstract this paper is concerned with the large time behavior of classical solutions near a given global maxwellian to the cauchy problem of the fokker planck boltzmann equation for non cutoff soft potentials our analysis is based on the coercivity of the fokker planck operator and elementary weighted energy method

the vlasov poisson landau system in a periodic box the classical vlasov poisson landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions such grazing collisions are modeled by the famous landau fokker planck collision kernel proposed by landau in 1936 we construct global unique solutions to such a system for initial data which have small weighted h norms but can have large h k 3 norms with high velocity moments our construction is based on accumulative study on the landau kernel in the past decade g1 sg1 3 with four extra ingredients to overcome the speci c mathematical di culties present in the vlasov poisson landau system a new exponential weight of electric potential to cancel the growth of the velocity a new velocity weight to capture the weak velocity di usion in the landau kernel a decay of the electric eld to close the energy estimate and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude

stability of rarefaction waves for the bipolar vlasov poisson boltzmann system abstract in this paper we construct the global solutions near a local maxwellian for the one dimensional bipolar vlasov poisson boltzmann system the macroscopic components of this local maxwellian are the approximate rarefaction wave solutions to the associated one dimensional compressible euler equations thus we prove the stability of the rarefaction waves for the bipolar vlasov poisson boltzmann system in the weighted function space moreover we obtain some time decay rates of the disparity between two species and the electric field which decay at an exponential rate

maxentropic markov chains corresp the markov chain that has maximum entropy for given first and second moments is determined the solution provides a discrete analog to the continuous gauss markov process

the global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball iii the 3 d boltzmann equation abstract this paper is a continuation of the works in 35 and 37 where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases respectively in this paper we are concerned with the global existence and large time behavior of compressible boltzmann gases in an infinitely expanding ball such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time due to the conservation of mass the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball which is easily observed in finite time in the present paper we will confirm this physical phenomenon for the boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function

type classes of context trees it is well known that a tree model does not always admit a finite state machine fsm representation with the same minimal number of parameters therefore known characterizations of type classes for fsms do not apply in general to tree models in this paper the type class of a sequence with respect to a given context tree is studied an exact formula is derived for the size of the class extending whittle s formula for type classes with respect to fsms the derivation is more intricate than in the fsm case since some basic properties of fsm types do not hold in general for tree types the derivation also yields an efficient enumeration of the tree type class a formula for the number of type classes with respect to is also derived the formula is asymptotically tight up to a multiplicative constant and also extends the corresponding result for fsms the asymptotic behavior of the number of type classes and of the size of a class is expressed in terms of the so called minimal canonical extension of t a tree that is generally larger than but smaller than its fsm closure

the vlasov poisson boltzmann system in the whole space the hard potential case abstract this paper is concerned with the cauchy problem on the vlasov poisson boltzmann system for hard potentials in the whole space when the initial data is a small perturbation of a global maxwellian a satisfactory global existence theory of classical solutions to this problem together with the corresponding temporal decay estimates on the global solutions is established our analysis is based on time decay properties of solutions and a new time velocity weight function which is designed to control the large velocity growth in the nonlinear term for the case of non hard sphere interactions

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

on the cauchy problem for the relativistic vlasov poisson fokker planck system abstract we consider the cauchy problem for the relativistic vlasov poisson fokker planck system in the whole space for perturbative initial data with suitable regularity we obtain the global classical solutions and prove the exponential time decay rate to the equilibrium around a global relativistic maxwellian

the error exponent for the noiseless encoding of finite ergodic markov sources a new approach to the classical fixed length noiseless source coding problem is proposed for the case of finite ergodic markov sources this approach is based on simple counting arguments the central notion of markov type a set containing all the source sequences having the same transition counts from letter to letter is introduced and the cardinality of such a set is evaluated via graph theoretical tools the error exponent is shown to be a weighted average of informational divergences and the universal character of the result is stressed as a corollary the classical source coding theorem determining the achievable rates is rederived

on the bgk equation with some force fields in lp space abstract the cauchy problem for the bgk equation with two kinds of force fields is studied in this paper the first one considers a given force field f t x v which is assumed to be divergence free with respect to the velocity variable and has an l p integrability it is shown that there exists a nonnegative distributional solution which extends a previous result obtained by the authors the second one deals with a lorentz field e t x v b t x with e t x and b t x belonging to l p space which gives a similar existence result to the cauchy problem

stability of the vlasov poisson boltzmann system in r3 abstract we construct global in time classical solutions to the cauchy problem for the 2 species vlasov poisson boltzmann system near maxwellian in the whole space r 3 moreover the convergence rate in time to the global maxwellian and the uniform in time stability of solutions are also obtained through energy method

a new energy method for the boltzmann equation an energy method for the boltzmann equation was proposed by liu yang and yu physica d 188 178 192 2004 based on the decomposition of the boltzmann equation and its solution around the local maxwellian the main idea is to rewrite the boltzmann equation as a fluid type dynamics system with the nonfluid component appearing in the source terms coupled with an equation for the time evolution of the nonfluid component in this paper we will elaborate this method and our main observation is that the microscopic projection of the local maxwellian with respect to a given global maxwellian is not linear but quadratic based on this and by analyzing the fluid type system using the analytic techniques for the system of conservation laws we can indeed control the conserved quantities u and 12u2 e of the boltzmann equation by the microscopic projection of the solution of the boltzmann equation with respect to the global maxwellian which is sufficient to deduce the energy estimates for the solution of

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

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

a universal finite memory source an irreducible parameterization for a finite memory source is constructed in the form of a tree machine a universal information source for the set of finite memory sources is constructed by a predictive modification of an earlier studied algorithm context it is shown that this universal source incorporates any minimal data generating tree machine in an asymptotically optimal manner in the following sense the negative logarithm of the probability it assigns to any long typical sequence generated by any tree machine approaches that assigned by the tree machine at the best possible rate

non isothermal boundary in the boltzmann theory and fourier law in the study of the heat transfer in the boltzmann theory the basic problem is to construct solutions to the following steady problem v cdot nabla x f frac 1 rm k rm n q f f qquad x v in omega times mathbf r 3 quad quad 0 1 f x v n x cdot v 0 f x v prime n x cdot v prime dv prime quad x in partial omega quad quad 0 2 where is a bounded domain in mathbf r d 1 leq d leq 3 kn is the knudsen number and mu theta frac 1 2 pi theta 2 x rm exp frac v 2 2 theta x is a maxwellian with non constant non isothermal wall temperature x based on new constructive coercivity estimates for both steady and dynamic cases for theta theta 0 leq delta ll 1 and any fixed value of kn we construct a unique non negative solution fs to 0 1 and 0 2 continuous away from the grazing set and exponentially asymptotically stable this solution is a genuine non equilibrium stationary solution differing from a local equilibrium maxwellian as an application of our results we establish the expansion f s mu theta 0 delta f 1 o delta 2 and we prove that if the fourier law holds the temperature contribution associated to f1 must be linear in the slab geometry

on global existence and trend to the equilibrium for the vlasov poisson fokker planck system with exterior confining potential we prove a global existence result with initial data of low regularity and prove the trend to the equilibrium for the vlasov poisson fokker planck system with small non linear term but with a possibly large exterior confining potential in dimension d 2 and d 3 the proof relies on a fixed point argument using sharp estimates at short and long time scales of the semi group associated to the fokker planck operator which were obtained by the first author

global well posedness for the microscopic fene model with a sharp boundary condition abstract we prove global well posedness for the microscopic fene model under a sharp boundary requirement the well posedness of the fene model that consists of the incompressible navier stokes equation and the fokker planck equation has been studied intensively mostly with the zero flux boundary condition in this article we show that for the well posedness of the microscopic fene model b 2 the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function under this condition it is shown that there exists a unique weak solution in a weighted sobolev space moreover such a condition still ensures that the distribution is a probability density the sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero no faster than the distance function

hypocoercivity for linear kinetic equations conserving mass we develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law local mass conservation is assumed at the level of the collision kernel while transport involves a confining potential so that the solution relaxes towards a unique equilibrium state our goal is to evaluate in an appropriately weighted l 2 norm the exponential rate of convergence to the equilibrium the method covers various models ranging from diffusive kinetic equations like vlasov fokker planck equations to scattering models like the linear boltzmann equation or models with time relaxation collision kernels corresponding to polytropic gibbs equilibria including the case of the linear boltzmann model in this last case and in the case of vlasov fokker planck equations any linear or superlinear growth of the potential is allowed

distributional and classical solutions to the cauchy boltzmann problem for soft potentials with integrable angular cross section this paper focuses on the study of existence and uniqueness of distributional and classical solutions to the cauchy boltzmann problem for the soft potential case assuming s n 1 integrability of the angular part of the collision kernel grad cut off assumption for this purpose we revisit the kaniel shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local maxwellian regime with possibly infinite initial mass we study the propagation of regularity using a recent estimate for the positive collision operator given in alonso et al in convolution inequalities for the boltzmann collision operator arxiv 0902 0507 math ap by e carneiro and the authors that allows us to show such propagation without additional conditions on the collision kernel finally an l p stability result with 1 p is presented assuming the aforementioned condition