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

the vlasov poisson boltzmann system with angular cutoff for soft potentials abstract this paper is concerned with the cauchy problem of the vlasov poisson boltzmann system near a given global maxwellian with angular cutoff for a class of soft potentials in three space dimensions and the main purpose here is to derive the global solvability of such a problem without the neutral condition which was imposed in duan et al 2013 11 on the initial perturbation our analysis is based on a time weighted energy method and some delicate estimates on its solutions which show that the l 2 r x 3 r 3 norms of the solutions with higher order x derivatives enjoy sharper time decay rates

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

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

a branch and cut algorithm without binary variables for nonconvex piecewise linear optimization we give a branch and cut algorithm for solving linear programs lps with continuous separable piecewise linear cost functions plfs models for plfs use continuous variables in special ordered sets of type 2 sos2 traditionally sos2 constraints are enforced by introducing auxiliary binary variables and other linear constraints on them alternatively we can enforce sos2 constraints by branching on them thus dispensing with auxiliary binary variables we explore this approach further by studying the inequality description of the convex hull of the feasible set of lps with plfs in the space of the continuous variables and using the new cuts in a branch and cut scheme without auxiliary binary variables we give two families of valid inequalities the first family is obtained by lifting the convexity constraints the second family consists of lifted cover inequalities finally we report computational results that demonstrate the effectiveness of our cuts and that branch and cut without auxiliary binary variables is significantly more practical than the traditional mixed integer programming approach

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

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

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

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

facets of the complementarity knapsack polytope we present a polyhedral study of the complementarity knapsack problem traditionally complementarity constraints are modeled by introducing auxiliary binary variables and additional constraints and the model is tightened by introducing strong inequalities valid for the resulting mip we use an alternative approach in which we keep in the model only the continuous variables and we tighten the model by introducing inequalities that define facets of the convex hull of the set of feasible solutions in the space of the continuous variables to obtain the facet defining inequalities we extend the concepts of cover and cover inequality commonly used in 0 1 programming for this problem and we show how to sequentially lift cover inequalities we obtain tight bounds for the lifting coefficients and we present two families of facet de ning inequalities that can be derived by lifting cover inequalities we show that unlike 0 1 knapsack polytopes in which different facet defining inequalities can be derived by fixing variables at 0 or 1 and then sequentially lifting cover inequalities valid for the projected polytope any sequentially lifted cover inequality for the complementarity knapsack polytope can be obtained by fixing variables at 0

a polyhedral study of the semi continuous knapsack problem we study the convex hull of the feasible set of the semi continuous knapsack problem in which the variables belong to the union of two intervals such structure arises in a wide variety of important problems e g blending and portfolio selection we show how strong inequalities valid for the semi continuous knapsack polyhedron can be derived and used in a branch and cut scheme for problems with semi continuous variables to demonstrate the effectiveness of these inequalities which we call collectively semi continuous cuts we present computational results on real instances of the unit commitment problem as well as on a number of randomly generated instances of linear programming with semi continuous variables

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

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

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

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

lifted inequalities for 0 1 mixed integer programming basic theory and algorithms we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables we develop a lifting theory for the continuous variables in particular we present a pseudo polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use

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

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

asymptotic uncorrelation for mexican needlets abstract we recall mexican needlets from d geller a mayeli continuous wavelets on compact manifolds math z 262 4 2009 895 927 d geller a mayeli nearly tight frames and space frequency analysis on compact manifolds math z 263 2 2009 234 264 we derive an estimate for certain types of legendre series which we apply to the statistical properties of mexican needlets more precisely we shall show that under isotropy assumption the mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated as the frequency parameter goes to infinity this property is important in the analysis of spherical random fields in particular in connection to the analysis of cosmic microwave background cmb radiation data

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

a polyhedral study of the cardinality constrained knapsack problem a cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive this structure occurs for example in areas such as finance location and scheduling traditionally cardinality constraints are modeled by introducing auxiliary 0 1 variables and additional constraints that relate the continuous and the 0 1 variables we use an alternative approach in which we keep in the model only the continuous variables and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables to derive the valid inequalities we extend the concepts of cover and cover inequality commonly used in 0 1 programming to this class of problems and we show how cover inequalities can be lifted to derive facet defining inequalities we present three families of non trivial facet defining inequalities that are lifted cover inequalities finally we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0 1 variables over the traditional mip approach for this class of problems

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

the piecewise linear optimization polytope new inequalities and intersection with semi continuous constraints we give new facets and valid inequalities for the separable piecewise linear optimization splo knapsack polytope we also extend the inequalities to the case in which some of the variables are semi continuous finally we give computational results that demonstrate their efficiency in solving difficult instances of splo and splo with semi continuous constraints

optimal decay estimates on the linearized boltzmann equation with time dependent force and their applications although the decay in time estimates of the semi group generated by the linearized boltzmann operator without forcing have been well established there is no corresponding result for the case with general external force this paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted sobolev spaces for the linearized boltzmann equation with a time dependent external force no time decay assumption is made on the force the proof is based on both the energy method through the macro micro decomposition and the lp lq estimates from the spectral analysis the decay estimates thus obtained are applied to the study on the global existence of the cauchy problem to the nonlinear boltzmann equation with time dependent external force and source precisely for space dimension n 3 the global existence and decay rates of solutions to the cauchy problem are obtained under the condition that the force and source decay in time with some rates this time decay restriction can be removed for space dimension n 5 moreover the existence and asymptotic stability of the time periodic solution are given for space dimension n 5 when the force and source are time periodic with the same period

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

a generalized assignment problem with special ordered sets a polyhedral approach we study a generalized assignment problem that arises in production scheduling in which special ordered sets of type ii appear naturally in the formulation we derive three families of facet defining valid inequalities and we show that they cut off all infeasible vertices of the lp relaxation we also give the complete facetial description for a particular case we then use the inequalities as cuts in a branch and cut scheme and we report computational results that demonstrate the superiority of branch and cut over branch and bound on this class of problems

global hilbert expansion for the vlasov poisson boltzmann system the dynamics of an electron gas in a constant ion background can be decribed by the vlasov poisson boltzmann system at the kinetic level or by the compressible euler poisson system at the fluid level we prove that any solution of the vlasov poisson boltzmann system near a smooth local maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the euler poisson system as the mean free path e goes to zero we use a recent l2 l framework in the boltzmann theory to control the higher order remainder in the hilbert expansion uniformly in e and globally in time

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

mixed integer models for the stationary case of gas network optimization a gas network basically consists of a set of compressors and valves that are connected by pipes the problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost efficiently such that all demands of the gas network are satisfied this problem leads to a complex mixed integer nonlinear optimization problem we describe techniques for a piece wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program we study sub polyhedra linking these piece wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms suitable branching strategies complementing the separation algorithms are also presented our computational results demonstrate the success of this approach

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

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

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

semi continuous network flow problems we consider semi continuous network flow problems that is a class of network flow problems where some of the variables are restricted to be semi continuous we introduce the semi continuous inflow set with variable upper bounds as a relaxation of general semi continuous network flow problems two particular cases of this set are considered for which we present complete descriptions of the convex hull in terms of linear inequalities and extended formulations we consider a class of semi continuous transportation problems where inflow systems arise as substructures for which we investigate complexity questions finally we study the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi continuous transportation problems

large scale portfolio optimization this paper describes a practical algorithm for large scale mean variance portfolio optimization the emphasis is on developing an efficient computational approach applicable to the broad range of portfolio models employed by the investment community what distinguishes these from the usual quadratic program is i the form of the covariance matrix arising from the use of factor and scenario models of return and ii the inclusion of transactions limits and costs a third aspect is the question of whether the problem should be solved parametrically in the risk reward trade off parameter or separately for several discrete values of we show how the parametric algorithm can be made extremely efficient by sparsifying the covariance matrix with the introduction of a few additional variables and constraints and by treating the transaction cost schedule as an essentially nonlinear nondifferentiable function then we show how these two seemingly unrelated approaches can be combined to yield good approximate solutions when minimum trading size restrictions buy or sell at least a certain amount or not at all are added in combination these approaches make possible the parametric solution of problems on a scale not heretofore possible on computers where cpu time and storage are the constraining factors

cutting planes for the multistage stochastic unit commitment problem as renewable energy penetration rates continue to increase in power systems worldwide new challenges arise for system operators in both regulated and deregulated electricity markets to solve the security constrained coal fired unit commitment problem with intermittent generation due to renewables and uncertain load in order to ensure system reliability and maintain cost effectiveness in this paper we study a security constrained coal fired stochastic unit commitment model which we use to enhance the reliability unit commitment process for day ahead power system operations in our approach we first develop a deterministic equivalent formulation for the problem which leads to a large scale mixed integer linear program then we verify that the turn on off inequalities provide a convex hull representation of the minimum up down time polytope under the stochastic setting next we develop several families of strong valid inequalities mainly through lifting schemes in particular by exploring sequence independent lifting and subadditive approximation lifting properties for the lifting schemes we obtain strong valid inequalities for the ramping and general load balance polytopes finally branch and cut algorithms are developed to employ these valid inequalities as cutting planes to solve the problem our computational results verify the effectiveness of the proposed approach