the green s function and large time behavior of solutions for the one dimensional boltzmann equation we study the pointwise behavior of the green s function of the boltzmann equation our results reveal the particle and fluid aspects of the equation the particle aspect is represented by singular waves these waves are carried by transport equations and dominate the short time behavior of the solution we devise a picard type iteration for constructing the increasingly regular particlelike waves the fluidlike waves reveal the dissipative behavior of the type of navier stokes equations as usually seen by the chapman enskog expansion these waves are constructed as part of the long wave expansion in the spectrum of the fourier mode for the space variable the fluidlike waves represent the long time behavior of the solution as an application we obtain the pointwise description of the large time behavior of the convergence to the global maxwellian when the initial perturbation is not necessarily smooth in our analysis of the exchanges of the microscopic velocity decay and space decay we make essential use of the hard sphere models c 2004 wiley periodicals inc contents

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

theta lifting of nilpotent orbits for symmetric pairs we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study the theta lifting of nilpotent k c orbits where k is a maximal compact subgroup of g and we describe the precise k c module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair g k as an application we prove sphericality and normality of the closure of certain nilpotent k c orbits obtained in this way we also give integral formulas for their degrees

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 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

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

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

special functions of matrix argument i algebraic induction zonal polynomials and hypergeometric functions hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis multivariate statistics quantum physics molecular chemistry and number theory this paper presents a general theory of such functions for real division algebras these functions which generalize the classical hypergeometric functions are defined by infinite series on the space s s n f of all n x n hermitian matrices over the division algebra f the theory depends intrinsically upon the representation theory of the general linear group g gl n f of invertible n x n matrices over f and the theme of this work is the full exploitation of the inherent group theory the main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of g to decompose the space of polynomial functions on s and to describe the zonal polynomials from which the hypergeometric functions are constructed detailed descriptions of the convergence properties of the series expansions are given and integral representations are provided future papers in this series will develop the fine structure of these functions 0 introduction we begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument by fine structure we allude to the analogues of such classical results as series expansions integral formulas asymptotics differential equations summation formulas addition theorems composition formulas and recurrence relations this first paper in which we simultaneously treat real complex and quaternionic analysis is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras not only as a framework for the body of detailed results to follow in later papers but also to clarify the representationtheoretic foundation for such a theory although these hypergeometric functions are of interest on purely analytic grounds they arise in a wide range of applications indeed various classes of received by the editors may 19 1986 1980 malhema1zics s ject claszficain 1985 ren primary 22e30 22e45 33a75 43a85 43a90 62h10 secondary 20g20 32a07 32m15 44a10 62e15 key w and phrases generalized hypergeometric functions zonal polynomials representation theory algebraic induction multivariate statistics general linear group generalized gamma functions pochhammer symbols laplace transforms maximal compact subgroup invariant polynomials positive cones symmetric spaces schur functions special functions of matrix argument the first author is on leave from the university of wyoming the second author was partially supported by the national science foundation under grant mcs 8403381 and by the research council of the university of north carolina b1987 american mathematical society 0002 9947 87 1 00 25 per page

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

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

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

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

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

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

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

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

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

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

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

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

on the evaluation of some selberg like integrals abstract several methods of evaluation are presented for a family i n d p of selberg like integrals that arise in the computation of the algebraic geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real lie group adapting the technique of nishiyama ochiai and zhu we present an explicit evaluation in terms of certain iterated sums over permutation groups the resulting formula however is only valid when the integrand involves an even power of the vandermonde determinant we then apply to the general case the theory of symmetric functions and obtain an evaluation of the integral i n d p as a product of polynomial of fixed degree times a particular product of gamma factors thereby identifying the asymptotics of the integrals with respect to their parameters lastly we derive a recursive formula for evaluation of another general class of selberg like integrals by applying some of the technology of generalized hypergeometric functions

formality of dg algebras after kaledin abstract we provide proper foundations and proofs for the main results of kaledin 2007 ka the results include a flat base change for formality and behavior of formality in flat families of a and dg algebras