about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

parameter uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems a second order initial value problem with a small parameter multiplying the first and second order derivatives is considered the precise knowledge about the behavior of the exact solution is analyzed based on the behavior of the exact solution a hybrid finite difference scheme on a shishkin mesh is proposed which is a combination of the second order difference scheme on the fine mesh and the modified midpoint upwind scheme on the coarse mesh by applying the truncation error estimate techniques and a difference analogue of gronwalls inequality we prove that the scheme is almost second order convergent for numerical solutions and scaled numerical derivatives numerical experiments support these theoretical results and indicate that the estimates are sharp

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

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

on the summation of fourier and bessel series abstract using fourier and hankel transform techniques and a complex contour integration method that arises in interpolating electrical signals h d helms and j b thomas proc ire 50 1962 179 184 m d rawn siam j appl math april 1989 we are able to derive many classical fourier summation formulae and many interesting new bessel summations that involve the zeros of the bessel functions of the first kind our method is similar in form to the one employed by p j forrester rocky mountain j math 13 1983 557 572 and m l glasser math comp 37 1981 499 501 finally we extend several of the classical fourier summations to character series in the spirit of b c berndt publ elektrotehnickog fak univ beograder no 381 409 1972 25 29

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

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

on some definite integrals and infinite sums containing products of special functions abstract a method is proposed for evaluation of some definite integrals and infinite sums containing products of bessel struve and other special functions

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

lattice sums of generalized hypergeometric functions representations and convergence criteria for one dimensional 1d and two dimensional 2d lattice sums of generalized hypergeometric functions pfp 1 are derived by using apparently new integral representations for the latter functions and 1d and 2d forms of the poisson summation formula these lattice sum representations concisely unify and generalize various specializations given previously by several authors and should prove useful in analysing finite size effects in systems subjected to non periodic boundary conditions

analytical evaluation of a class of phase modulated lattice sums making use of the poisson summation formula in m dimensions a class of phase modulated lattice sums is evaluated analytically the resulting formulas provide a broad generalization of the results reported previously and are of considerable use in the analysis of finite size effects in systems undergoing phase transitions especially when the boundary conditions imposed are other than periodic

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

schlomilch series that arise in diffraction theory and their efficient computation we are concerned with a certain class of schlomilch series that arise naturally in the study of diffraction problems when the scatterer is a semi infinite periodic structure by combining new results derived from integral representations and the poisson summation formula with known identities we obtain expressions which enable the series to be computed accurately and efficiently many of the technical details of the derivations are omitted they can however be obtained from linton 2005 schlomilch series that arise in diffraction theory and their efficient computation technical report loughborough university available online at http www staff lboro ac uk macml1 schlomilch techreport pdf

some solutions of the boltzmann equation without angular cutoff we show the existence of local or global in time solutions for the non homogeneous boltzmann equation this is done under the assumptions that initial data are smaller than a suitable maxwellian and that collisional cross sections do not satisfy grad s angular cutoff partial regularity in space velocity of the solutions constructed herein is also proved

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

a uniformly convergent scheme for a system of reaction diffusion equations in this work a system of two parabolic singularly perturbed equations of reaction diffusion type is considered the asymptotic behaviour of the solution and its partial derivatives is given a decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives to approximate the solution we consider the implicit euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform shishkin mesh we prove that this scheme is uniformly convergent with respect to the diffusion parameters having first order convergence in time and almost second order convergence in space in the discrete maximum norm numerical experiments illustrate the order of convergence proved theoretically

on the conjectures of henkel and weston recent conjectures of henkel and weston 1992 on certain two dimensional sums are examined and some significant generalizations of these sums are developed the results so obtained should be useful in analysing finite size effects in systems subjected to non periodic boundary conditions

regularization estimates and cauchy theory for inhomogeneous boltzmann equation for hard potentials without cut off in this paper we investigate the problems of cauchy theory and exponential stability for the inhomogeneous boltzmann equation without angular cut off we only deal with the physical case of hard potentials type interactions with a moderate angular singularity we prove a result of existence and uniqueness of solutions in a close to equilibrium regime for this equation in weighted sobolev spaces with a polynomial weight contrary to previous works on the subject all developed with a weight prescribed by the equilibrium it is the first result in this more physically relevant framework for this equation moreover we prove an exponential stability for such a solution with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation let us highlight the fact that a key point of the development of our cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo differential tools

asymptotic analysis of the spatially homogeneous boltzmann equation grazing collisions limit in the present work we consider the asymptotic problem of the spatially homogeneous boltzmann equation when almost all collisions are grazing that is the deviation angle theta of the collision is limited near zero i e theta le epsilon we show that by taking the proper scaling to the cross section which was used in 37 that is assuming begin aligned b epsilon v v sigma 2 1 s v v gamma epsilon 3 sin 1 theta left frac theta epsilon right 1 2s mathrm 1 theta le epsilon end aligned where theta langle theta frac upsilon upsilon upsilon upsilon sigma rangle the solution f epsilon of the boltzmann equation with initial data f 0 can be globally or locally expanded in some weighted sobolev space as begin aligned f epsilon f o epsilon end aligned where the function f is the solution of landau equation which is associated with the grazing collisions limit of boltzmann equation with the same initial data f 0 this gives the rigorous justification of the landau approximation in the spatially homogeneous case in particular if taking gamma 3 and s 1 epsilon in the cross section b epsilon we show that the above asymptotic formula still holds and in this case f is the solution of landau equation with the coulomb potential going further we revisit the well posedness problem of the boltzmann equation in the limiting process we show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution f epsilon thanks to the weak convergence results on the grazing collisions limit in 37 in other words we establish a unified framework to establish the well posedness results for both boltzmann and landau equations

on certain schlomilch type series abstract by using a form of the poisson summation formula together with a generalization due to srivastava and exton of the discontinuous integral of weber and schafheitlin and the heretofore not readily available cosine transform of the hypergeometric function 1 f 2 b 2 t 2 b 0 several new schlomilch and fourier type series are evaluated by specialization of the latter series numerous results appearing in the literature are obtained in a unified way

a finite element analysis of a coupled system of singularly perturbed reaction diffusion equations we consider a system of two coupled reaction diffusion equations when the parameters multiplying the second order derivatives in the equations are small their solutions exhibit boundary layers moreover when the parameters are of different magnitudes two distinct but overlapping boundary layers are present we study a finite element discretization on general layer adapted meshes including the frequently studied shishkin mesh and the bakhvalov mesh supporting numerical results are presented

second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems abstract we consider a coupled system of singularly perturbed first order ordinary differential equations the equations have small positive parameters of different magnitudes associated with them a decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method we apply the hodie high order difference approximation with identity expansions technique to construct a second order finite difference scheme and combine this with standard backward euler difference scheme in a special way on a piecewise uniform shishkin mesh to solve the system numerically it is proved that the method is second order up to a logarithmic factor convergent in the maximum norm uniformly in both perturbation parameters numerical experiments are conducted to support the theoretical results

a numerical study on the finite element solution of singularly perturbed systems of reaction diffusion problems we consider the approximation of singularly perturbed systems of reaction diffusion problems with the finite element method the solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters in this article we conduct a numerical study of several finite element methods applied to a model problem having as our goal their assessment and the identification of a high order scheme which approximates the solution at an exponential rate of convergence independently of the singular perturbation parameters

a parameter uniform numerical method for a system of singularly perturbed ordinary differential equations in this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered a parameter robust computational method is constructed and it is proved that it gives essentially first order parameter uniform convergence in the maximum norm numerical results are presented in support of the theory

global solutions in the critical besov space for the non cutoff boltzmann equation abstract the boltzmann equation is studied without the cutoff assumption under a perturbative setting a unique global solution of the cauchy problem of the equation is established in a critical chemin lerner space in order to analyze the collisional term of the equation a chemin lerner norm is combined with a non isotropic norm with respect to a velocity variable which yields an a priori estimate for an energy estimate together with local existence following from commutator estimates and the hahn banach extension theorem the desired solution is obtained

spectrum analysis of some kinetic equations we analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory the models include the classical boltzmann equation for hard potentials with or without angular cutoff and the landau equation with gamma geqq 2 as an application we show that the solutions to these two fundamental equations are asymptotically equivalent mod time decay rate t 5 4 as t to infty to that of the compressible navier stokes equations for initial data around an equilibrium state

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

from the boltzmann equation to the incompressible navier stokes equations on the torus a quantitative error estimate we investigate the boltzmann equation depending on the knudsen number in the navier stokes perturbative setting on the torus using hypocoer civity we derive a new proof of existence and exponential decay for solutions close to a global equilibrium with explicit regularity bounds and rates of convergence these results are uniform in the knudsen number and thus allow us to obtain a strong derivation of the incompressible navier stokes equations as the knudsen number tends to 0 moreover our method is also used to deal with other kinetic models finally we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time unless the initial layer conditions are satisfied

a second order hybrid finite difference scheme for a system of singularly perturbed initial value problems a system of coupled singularly perturbed initial value problems with two small parameters is considered the leading term of each equation is multiplied by a small positive parameter but these parameters may have different magnitudes the solution of the system has boundary layers that overlap and interact the structure of these layers is analyzed and this leads to the construction of a piecewise uniform mesh that is a variant of the usual shishkin mesh on this mesh a hybrid finite difference scheme is proved to be almost second order accurate uniformly in both small parameters numerical results supporting the theory are presented

convergence analysis of a finite difference scheme for a riemann liouville fractional derivative two point boundary value problem on an adaptive grid abstract a two point boundary value problem whose highest order term is a riemann liouville fractional derivative of order 2 with 0 1 is considered such problem is reformulated as a volterra integral equation of the second kind an integral discrete scheme is developed for this volterra integral equation on an adaptive grid that is constructed adaptively from a knowledge of the exact solution it is shown from a rigorous priori error analysis that the discrete solutions are uniformly convergent with respect to the parameter besides in order to establish the parameter of the volterra integral equation we construct a nonlinear optimization problem which is solved by the nelder mead simplex method numerical results are given to demonstrate the performance of presented method

parameter uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution in this paper we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection diffusion two point boundary value problem the solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc length monitor function it is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter in addition we prove the uniform first order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain numerical results are presented that demonstrate the sharpness of our results

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

from boltzmann s equation to the incompressible navier stokes fourier system with long range interactions we establish a rigorous demonstration of the hydrodynamic convergence of the boltzmann equation towards a navier stokes fourier system under the presence of long range interactions this convergence is obtained by letting the knudsen number tend to zero and has been known to hold at least formally for decades it is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered however these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption which requires that the cross sections be integrable indeed as soon as long range intermolecular forces are present non integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas in this long range setting the boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case in fact the diperna lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the boltzmann equation are the so called renormalized solutions with a defect measure developed by alexandre and villani our work overcomes the new mathematical difficulties coming from the consideration of long range interactions by proving the hydrodynamic convergence of the alexandre villani solutions towards the leray solutions

from newton s law to the linear boltzmann equation without cut off we provide a rigorous derivation of the linear boltzmann equation without cut off starting from a system of particles interacting via a potential with infinite range as the number of particles n goes to infinity under the boltzmann grad scaling more particularly we will describe the motion of a tagged particle in a gas close to global equilibrium the main difficulty in our context is that due to the infinite range of the potential a non integrable singularity appears in the angular collision kernel making no longer valid the single use of lanford s strategy our proof relies then on a combination of lanford s strategy of tools developed recently by bodineau gallagher and saint raymond to study the collision process and of new duality arguments to study the additional terms associated with the long range interaction leading to some explicit weak estimates

on certain two dimensional schlomilch series by using representations for one dimensional 1d schlomilch series lommel s expansion for bessel functions of the first kind and elementary series manipulations several algebraic formulae for two dimensional 2d schlomilch series are derived one of these formulae contains as a special case an identity involving a certain trigonometric lattice sum conjectured by henkel and weston 1993 which has recently been derived by boersma and de doelder 1993 using the 2d poisson summation formula the latter method is used to give an alternative derivation for 2d schlomilch series further it is shown that representation by 2d schlomilch series of a null function is not unique in addition representations for several other 1d type series are given

a class of bessel summations these quantities have a host of useful properties which are concisely summarized in 3 in particular l s x is an entire function except for the trivial character x n 1 for which l s 1 c s has a simple pole at s 1 we shal express s x as a power series indicate various sets of parameters for which it can be summed in closed form and give some examples the mellin transform of s x is 2

a moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction diffusion problems with arbitrary small diffusion terms abstract in this paper a system of time dependent boundary layer originated reaction dominated problems with diffusion parameters of different magnitudes is considered for numerical analysis the presence of these parameters lead to the boundary layer phenomena here an optimal order uniformly accurate boundary layer adaptive method moving mesh method is proposed this method is able to capture the layer phenomena without using a priori information of the solution the problem is discretized by a modified implicit euler scheme in time direction for the present system adaptive mesh generation is required in space due to the singularly perturbed nature of the problem for this purpose a positive error monitor function is used whose equidistribution will move the mesh points towards the boundary layers parameter uniform error estimates are derived to show that the convergence rate is optimal with respect to the problem discretization numerical experiments strongly verify the theoretical findings and confirm the efficiency and accuracy of the proposed method

higher order global solution and normalized flux for singularly perturbed reaction diffusion problems in this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction diffusion two point boundary value problems the hoc higher order compact finite difference scheme developed in gracia et al 2001 4 and which is constructed on an appropriate piecewise uniform shishkin mesh has been considered to find an almost fourth order convergent solution at mesh points using these values piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined it has been shown that the global solution and the global normalized flux are also uniformly convergent moreover for the global solution the order of uniform convergence in the whole domain is optimal i e it is the same as this one obtained at mesh points whereas for the global normalized flux the uniform convergence is almost third order except at midpoints of the mesh where it is also almost fourth order theoretical error bounds have been provided along with some numerical examples which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux

uncertainty principle and kinetic equations abstract a large number of mathematical studies on the boltzmann equation are based on the grad s angular cutoff assumption however for particle interaction with inverse power law potentials the associated cross sections have a non integrable singularity corresponding to the grazing collisions smoothing properties of solutions are then expected on the other hand the uncertainty principle established by heisenberg in 1927 has been developed so far in various situations and it has been applied to the study of the existence and smoothness of solutions to partial differential equations this paper is the first one to apply this celebrated principle to the study of the singularity in the cross sections for kinetic equations precisely we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations in particular we give some estimates on the regularity of solutions in sobolev spaces w r t all variables for both linearized and nonlinear space inhomogeneous boltzmann equations without angular cutoff as well as the linearized space inhomogeneous landau equation

a revisiting of the l 2 stability theory of the boltzmann equation near global maxwellians we study the l 2 stability theory of the boltzmann equation near a global maxwellian when an initial datum is a perturbation of a global maxwellian we show that the l 2 distance between two classical solutions can be controlled by the initial data in a lipschitz manner which illustrates the lipschitz continuity of the solution operator for the boltzmann equation in l 2 topology our local in time l 2 stability results cover cutoff very soft potentials as well as non cutoff hard and soft potentials these cases were not treated in the previous work ha et al in arch ration mech anal 197 657 688 2010 thus our results together with the results in ha et al 2010 complete the l 2 stability theory for the boltzmann equation near a global maxwellian for this l 2 stability estimate we use the coercivity estimate of the linearized collision operator the smallness of perturbation in a mixed lebesgue norm and strichartz type estimates of perturbation we also show that for all classical solutions available in the literature the lipschitz constant can be chosen as independent of time to obtain the uniform l 2 stability of the boltzmann equation

layer adapted methods for quasilinear singularly perturbed delay differential problems abstract in this work we consider a class of initial value problems for quasilinear singularly perturbed first order delay differential equations to solve this class of problems numerically we consider two finite difference schemes the backward euler scheme and a high order hybrid scheme which is a blend of the trapezoidal scheme and the backward euler scheme we derive general convergence results for both the schemes based on which a number of layer adapted meshes can be constructed and analyzed as consequences of these results we establish uniform convergence of the schemes on certain layer adapted meshes numerical experiments confirm our theoretical findings

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

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

a numerical method for a system of singularly perturbed reaction diffusion equations a dirichlet problem for a system of two coupled singularly perturbed reaction diffusion ordinary differential equations is examined a numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed numerical results are presented which illustrate the theoretical results

the boltzmann equation without angular cutoff in the whole space qualitative properties of solutions this is a continuation of our series of works for the inhomogeneous boltzmann equation we study qualitative properties of classical solutions the full regularization in all variables uniqueness non negativity and convergence rate to the equilibrium to be precise together with the results of parts i and ii about the well posedness of the cauchy problem around the maxwellian we conclude this series with a satisfactory mathematical theory for the boltzmann equation without angular cutoff

the boltzmann equation with frictional force for soft potentials in the whole space abstract this paper is concerned with the cauchy problem of the boltzmann equation with frictional force for both cutoff and non cutoff soft potentials in the whole space and our main purpose is to establish its global solvability result near a given global maxwellian and to deduce the temporal convergence rates of such a global solution toward the global maxwellian when initial perturbation is sufficiently small the analysis is based on the time weighted energy method building also upon the recent studies of the cutoff vlasov poisson boltzmann system 14 15 the non cutoff vlasov poisson boltzmann system 8 and non cutoff vlasov maxwell boltzmann system 9

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

estimates for the large time behavior of the landau equation in the coulomb case this work deals with the large time behaviour of the spatially homogeneous landau equation with coulomb potential firstly we obtain a bound from below of the entropy dissipation d f by weighted relative fisher information of f with respect to the associated maxwellian distribution which leads to a variant of cercignani s conjecture thanks to a logarithmic sobolev inequality secondly we prove the propagation of polynomial and stretched exponential moments with an at most linearly growing in time rate as an application of these estimates we show the convergence of any h or weak solution to the landau equation with coulomb potential to the associated maxwellian equilibrium with an explicitly computable rate assuming initial data with finite mass energy entropy and some higher l1 moment more precisely if the initial data have some large enough polynomial l1 moment then we obtain an algebraic decay if the initial data have a stretched exponential l1 moment then we recover a stretched exponential decay

schlomilch series and grating sums we consider sums over the set of positive integers relevant to construction of periodic green s functions for diffraction gratings and similar problems and provide a general formula for a combination of bessel functions of complex order and complex powers of distance from the origin this general formula is investigated in a number of particular cases and in particular we provide expressions which enable sums of functions with neumann series to be re expressed as combinations of hypergeometric series we also investigate sums of neumann functions of integer order using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions we also exhibit sums which may be evaluated analytically and recurrence formulae linking sums

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

solving a partially singularly perturbed initial value problem on shishkin meshes a coupled first order system of one singularly perturbed and one non perturbed ordinary differential equation with prescribed initial conditions is considered a shishkin piecewise uniform mesh is constructed and used in conjunction with a classical finite difference operator to form a new numerical method for solving this problem it is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain uniformly with respect to the singular perturbation parameter numerical results are presented in support of the theory

an adaptive moving grid method for a system of singularly perturbed initial value problems a system of first order singularly perturbed initial value problems is considered the system is discretized by a backward euler difference scheme for which a priori error analysis in the maximum norm is constructed it is shown from the a priori error bound that there exists a mesh with n subintervals that gives optimal error bound of o n 1 which is robust with respect to the perturbation parameters a partly heuristic argument based on a priori error analysis leads to a suitable monitor function based on an a posteriori error bound a first order rate of convergence independent of all perturbation parameters is established a linear and a nonlinear examples are tested and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method

the boltzmann equation without angular cutoff in the whole space i global existence for soft potential it is known that the singularity in the non cutoff cross section of the boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable by defining and analyzing a non isotropy norm which precisely captures the dissipation in the linearized collision operator we first give a new and precise coercivity estimate for the non cutoff boltzmann equation for general physical cross sections then the cauchy problem for the boltzmann equation is considered in the framework of small perturbation of an equilibrium state in this part for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator we derive some new functional estimates on the nonlinear collision operator together with the coercivity estimates we prove the global existence of classical solutions for the boltzmann equation in weighted sobolev spaces

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

on the boltzmann equation for long range interactions we study the boltzmann equation without grad s angular cutoff assumption we introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables angular singularities occur as soon as one is interested in long range interactions while singularities in the relative velocity variable occur in the study of soft potentials in particular coulomb interaction together with several new estimates this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by lions appearance of strong compactness under general fully realistic assumptions 2001 john wiley sons inc

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

a new regularization mechanism for the boltzmann equation without cut off we apply recent results on regularity for general integro differential equations to derive a priori estimates in holder spaces for the space homogeneous boltzmann equation in the non cut off case we also show an a priori estimate in l infty which applies in the space inhomogeneous case as well provided that the macroscopic quantities remain bounded

efficient computation of schlomilch type series a class of schlomilch type series which are functions of two spatial parameters x and y are transformed into series which are more suitable for numerical computation when x is small the resulting formulae involve a free parameter a which does not affect the value of the original series but does affect the convergence characteristics of the new representation by choosing a appropriately we can improve the convergence of the series significantly and calculations are presented which demonstrate the efficiency of the new representations when evaluating the series numerically mellin transforms are used to derive the asymptotic form of the series in the limit as x 0 y constant and this also leads to efficient methods for computing certain image series

a class of generalized hypergeometric summations abstract summations over the positive integers n of the generalized hypergeometric expressions 1 n p f p 1 n 2 x 2 x 0 are derived in closed form the specialization p 0 for example reduces to known results for schlomilch series in addition we record the apparently not readily available sine and cosine transforms of p f p 1 b 2 x 2 b 0 the latter of which is used together with a form of the poisson summation formula to deduce the aforementioned results

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

uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction diffusion systems in this paper 1d parabolic systems of two singularly perturbed equations of reaction diffusion type are examined for the time discretization we consider two additive or splitting schemes defined on a uniform mesh and for the space discretization we use the classical central difference approximation defined on a shishkin mesh the uniform convergence of both the semidiscrete and the fully discrete problems is proved the additive schemes are used to solve a test problem and the results obtained with these schemes and the standard discretization using the backward euler method are compared also numerical results are presented in the case of systems of three equations all the numerical results show the advantage in computational cost of the additive schemes compared to the standard discretization

sonine gegenbauer type integrals abstract representations for complete and incomplete sonine gegenbauer type integrals are given in terms of generalized hypergeometric functions in two and three variables and in terms of neumann type series certain interesting reductions of these integrals are also given in addition an application to a problem in diffraction theory is provided

rapidly convergent representations for green functions for laplace equation representations for green s functions for laplace equation in domains with infinite boundaries are obtained by integrating solutions to appropriate heat conduction problems with respect to time by using different representations for these heat equation solutions for small and large times the changeover being determined by an arbitrary positive parameter a a one parameter family of formulae for the required green function is derived and by varying a the convergence characteristics of this new representation can be altered letting a zero results in known eigenfunction expansions and in those situations in which they exist letting a recovers known image series representations the method which is essentially equivalent to ewald summation is applied to two types of problem first it is applied to potential flow between parallel planes and in a rectangular channel and second to two and three dimensional water wave problems in which the depth is constant in all cases the results of computations