the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

uniform homeomorphisms of banach spaces and asymptotic structure we give a general result on the behavior of spreading models in banach spaces which coarse lipschitz embed into asymptotically uniformly convex spaces we use this result to study the uniqueness of the uniform structure in p sums of finite dimensional spaces for 1 p in particular we give some new examples of spaces with unique uniform structure

constructing smoothing functions in smoothed particle hydrodynamics with applications this paper presents a general approach to construct analytical smoothing functions for the meshfree lagrangian and particle method of smoothed particle hydrodynamics the approach uses integral form of function representation and applies taylor series expansion to the sph function and derivative approximations the constructing conditions are derived systematically which not only interpret the consistency condition of the method but also describe the compact supportness requirement of the smoothing function examples of sph smoothing function are constructed including some existing ones with this approach a new quartic smoothing function with some advantages is constructed and is applied to the one dimensional shock problem and a one dimensional tnt detonation problem the good agreement between the sph results and those from other sources shows the effectiveness of the approach and the newly constructed smoothing function in numerical simulations

on the geometry of the countably branching diamond graphs abstract in this article the bi lipschitz embeddability of the sequence of countably branching diamond graphs d k k n is investigated in particular it is shown that for every e 0 and k n d k embeds bi lipschiztly with distortion at most 6 1 e into any reflexive banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm on the other hand it is shown that the sequence d k k n does not admit an equi bi lipschitz embedding into any banach space that has an equivalent asymptotically midpoint uniformly convex norm combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces within the class of reflexive banach spaces with an unconditional asymptotic structure applications to bi lipschitz embeddability into l p spaces and to some problems in renorming theory are also discussed

solving the inverse stokes problems by the modified collocation trefftz method and laplacian decomposition in this paper a boundary type meshfree algorithm is proposed to accurately and stably deal with the two dimensional inverse stokes problems which are highly ill conditioned based on the laplacian decomposition the stokes equations are recast as three laplace equations then the modified collocation trefftz method mctm one of the most promising boundary type meshless methods is adopted to solve these three laplace equations the mctm can stabilize the numerical scheme and obtain highly accurate results by utilizing the characteristic length accordingly the numerical solutions of these three laplace equations are expressed by linear combination of the modified t complete functions the unknown coefficients in the solution expressions are found by enforcing the satisfactions of the boundary conditions at the boundary collocation points three numerical examples are provided to show the efficacy and stability of the proposed meshless method besides noises are added into the boundary conditions to demonstrate the stability of the proposed scheme for dealing with the inverse stokes problems

velocity boundary condition at solid walls in rarefied gas calculations maxwell s famous slip boundary condition is often misapplied in current rarefied gas flow calculations e g in hypersonics microfluidics for simulations of gas flows over curved or moving surfaces this means crucial physics can be lost we give examples of such cases we also propose a higher order boundary condition based on maxwell s general equation and the constitutive relations derived by burnett unlike many other higher order slip conditions these are applicable to any form of surface geometry it is shown that these maxwell burnett boundary conditions are in reasonable agreement with the limited experimental data available for poiseuille flow and can also predict sone s thermal stress slip flow a phenomenon which cannot be captured by conventional slip boundary conditions

dynamic correction to moment approximations another question comes from the approximate character of the grad equations and is discussed in frames of the eit while the grad equations are strictly hyperbolic at any level n i e predicting a finite speed of propagation will this feature will be preserved in the further corrections these two questions are special cases of a more general one namely how does one derive a closed description with a given number of moments such a description is sometimes called mesoscopic 7 since it occupies an intermediate level between the hydrodynamic macroscopic and the kinetic microscopic levels of description in this paper we aim at deriving the mesoscopic dynamics of thirteen moments in the simplest case when the kinetic description satisfies the linearized boltzmann equation our approach will be based on the two assumptions i the mesoscopic dynamics of thirteen moments exists and is invariant with respect to the microscopic dynamics ii the 13m grad approximation is a suitable first approximation to this mesoscopic dynamics the assumption i is realized as the invariance equation for the unknown mesoscopic distribution function following the assumption ii we solve the invariance equation iteratively taking the 13m grad approximation for the input approximation and consider the first iteration further we refer to this as to the dynamic correction to distinguish from constructing another ansatz w e demonstrate that the correction results in the exact ce transport coefficients we also demonstrate how the dynamic correction modifies the hyperbolicity of the grad equations a similar viewpoint on derivation of hydrodynamics was earlier developed in 8 we will return to a comparison below first we review the boltzmann equation and the 13m grad approximation we denote as n0 u050 and p0 the equilibrium values of the hydrodynamic parameters n is the number density u is the average velocity and p5nkbt is the pressure the global maxwell distribution function f is

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

nonpositive curvature is not coarsely universal we prove that not every metric space embeds coarsely into an alexandrov space of nonpositive curvature this answers a question of gromov geometric group theory cambridge university press cambridge 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov space of nonnegative curvature as shown by andoni et al ann sci ec norm super 4 51 3 657 700 2018 we establish this statement by proving that a metric space which is q barycentric for some q in 1 infty has metric cotype q with sharp scaling parameter our proof utilizes nonlinear metric space valued martingale inequalities and yields sharp bounds even for some classical banach spaces this allows us to evaluate the bi lipschitz distortion of the ell infty grid m infty n 1 ldots m n vert cdot vert infty into ell q for all q in 2 infty from which we deduce the following discrete converse to the fact that ell infty n embeds with distortion o 1 into ell q for q o log n a rigidity theorem of ribe ark mat 14 2 237 244 1976 implies that for every n in mathbb n there exists m in mathbb n such that if m infty n embeds into ell q with distortion o 1 then q is necessarily at least a universal constant multiple of log n ribe s theorem does not give an explicit upper bound on this m but by the work of bourgain geometrical aspects of functional analysis 1985 86 springer berlin 1987 it suffices to take m n and this was the previously best known estimate for m we show that the above discretization statement actually holds when m is a universal constant

solving parabolic and hyperbolic equations by the generalized finite difference method classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics an evolution of the method of finite differences has been the development of generalized finite difference gfd method that can be applied to irregular grids of points in this paper the extension of the gfd to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one two or three space dimensions the convergence of the method has been studied and the truncation errors over irregular grids are given different examples have been solved using the explicit finite difference formulae and the criterion of stability this has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one two or three space dimensions the numerical results show the accuracy obtained over irregular grids this paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time dependent equations

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

instabilities in the chapman enskog expansion and hyperbolic burnett equations it is well known that the classical chapman enskog procedure does not work at the level of burnett equations the next step after the navier stokes equations roughly speaking the reason is that the solutions of higher equations of hydrodynamics burnett s etc become unstable with respect to short wave perturbations this problem was recently attacked by several authors who proposed different ways to deal with it we present in this paper one of possible alternatives first we deduce a criterion for hyperbolicity of burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases then we discuss in more detail the problem of truncation of the chapman enskog expansion and show that the way of truncation is not unique the general idea of changes of coordinates based on analogy with the theory of dynamical systems leads finally to nonlinear hyperbolic burnett equations hbes without using any information beyond the classical burnett equations it is proved that hbes satisfy the linearized h theorem the linear version of the problem is studied in more detail the complete chapman enskog expansion is given for the linear case a simplified proof of the slemrod identity for burnett coefficients is also given

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

a note on the application of the generalized finite difference method to seismic wave propagation in 2d this paper shows the application of generalized finite difference method gfdm to the problem of seismic wave propagation we investigated stability and star dispersion in 2d we obtained independent stability conditions and star dispersion of the phase velocity for the p and s waves also p and s waves group velocity dispersion have been obtained

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

generalized burnett hydrodynamics equations of hydrodynamics derived from the boltzmann equation beyond the navier stokes level are studied by a method proposed earlier by the author the main question we consider is the following what is the most natural replacement for classical ill posed burnett equations it is shown that in some sense it is a two parameter set of generalized burnett equations gbes derived in this paper some equations of this class are even simpler than original burnett equations the region of stability in the space of parameters and other properties of gbes are discussed

thermal transpiration for the linearized boltzmann equation the phenomena of thermal transpiration due to the boundary temperature gradient is studied on the level of the linearized boltzmann equation for the hard sphere model we construct such a flow for a highly rarefied gas between two plates and also in a circular pipe it is shown that the flow velocity parallel to the plates is proportional to the boundary temperature gradient for a highly rarefied gas that is for a sufficiently large knudsen number the flow velocity between two plates is of the order of log and the flow velocity in a pipe is of finite order our analysis is based on certain pointwise estimates of the solutions of the linearized boltzmann equation 2006 wiley periodicals inc

the trefftz method using fundamental solutions for biharmonic equations in this paper the trefftz method of fundamental solution fs called the method of fundamental solution mfs is used for biharmonic equations the bounds of errors are derived for the mfs with almansi s fundamental solutions denoted as the mafs in bounded simply connected domains the exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions respectively the stability analysis of the mafs is also made for circular domains numerical experiments are carried out for both smooth and singularity problems the numerical results coincide with the theoretical analysis made when the particular solutions satisfying the biharmonic equation can be found the method of particular solutions mps is always superior to the mfs and the mafs based on numerical examples however if such singular particular solutions near the singular points do not exist the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points based on the computed results the mfs using the greedy adaptive techniques may provide more accurate solutions for singularity problems moreover the numerical solutions by the mafs with almansi s fs are slightly better in accuracy and stability than those by the traditional mfs hence the mafs with the afs is recommended for biharmonic equations due to its simplicity

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

short note the method of fundamental solutions for 2d and 3d stokes problems a numerical scheme based on the method of fundamental solutions mfs is proposed for the solution of 2d and 3d stokes equations the fundamental solutions of the stokes equations stokeslets are adopted as the sources to obtain flow field solutions the present method is validated through other numerical schemes for lid driven flows in a square cavity and a cubic cavity test results obtained for a rectangular cavity with wave shaped bottom indicate that the mfs is computationally efficient than the finite element method fem in dealing with irregular shaped domain the paper also discusses the effects of number of source points and their locations on the numerical accuracy

boundary singularity for thermal transpiration problem of the linearized boltzmann equation we study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory logarithmic singularity has been demonstrated through the asymptotic and computational analysis the goal of this paper is to confirm this logarithmic singularity through exact analysis we use an iterated scheme with the gain part of the collision operator as a source the iterated scheme is appropriate for large knudsen numbers considered here and yields an explicit leading term

the method of fundamental solutions for annular shaped domains this paper consists of two parts in the first part we combine the analysis of bogomolny a bogomolny fundamental solutions method for elliptic boundary value problems siam journal on numerical analysis 22 1985 644 669 comodi and mathon m i comodi r mathon a boundary approximation method for fourth order problems mathematical models and methods in applied sciences 1 1991 437 445 and li et al z c li r mathon p sermer boundary methods for solving elliptic equations with singularities siam journal on numerical analysis 24 1987 487 498 for the method of fundamental solutions mfs and new error bounds are derived for the bounded simply connected in a readable approach a factor of o n is removed in the error bounds of bogomolny 1985 in the second part we extend the analysis for the annular domain s a by the mfs the other fundamental solutions are needed whose source nodes may also be located uniformly on a circle inside the domain s a as in the above reference error bounds are derived in detail to display the polynomial convergence rates

from boltzmann s kinetic theory to euler s equations abstract the incompressible euler equations are obtained as a weak asymptotics of the boltzmann equation in the fast relaxation limit the knudsen number kn goes to zero when both the mach number ma defined as the ratio between the bulk velocity and the speed of sound and the inverse reynolds number kn ma which measures the viscosity of the fluid go to zero the entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled boltzmann equation to its expected limit provided that it is sufficiently smooth it thus leads to some strong convergence result one of the main points to be understood is how to deal with the corrections to the weak limit i e the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves

from boltzmann to euler hilbert s 6th problem revisited abstract this article addresses the hydrodynamic limit of the boltzmann equation namely the compressible euler equations of gas dynamics an exact summation of the chapman enskog expansion originally given by gorban and karlin is the key to the analysis an appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the korteweg de vries burgers equations as a small parameter tends to zero

slip boundary conditions for the compressible navier stokes equations the slip boundary conditions for the compressible navier stokes equations are derived systematically from the boltzmann equation on the basis of the chapman enskog solution of the boltzmann equation and the analysis of the knudsen layer adjacent to the boundary the resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard sphere molecules as well as the bhatnagar gross krook model these formulas which can be applied to specific problems immediately help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis

singularity of the velocity distribution function in molecular velocity space we study the boundary singularity of the solutions to the boltzmann equation in the kinetic theory the solution has a jump discontinuity in the microscopic velocity zeta on the boundary and a secondary singularity of logarithmic type around the velocity tangential to the boundary zeta n sim 0 where zeta n is the component of molecular velocity normal to the boundary pointing to the gas we demonstrate this secondary singularity by obtaining an asymptotic formula for the derivative of the solution on the boundary with respect to zeta n that diverges logarithmically when zeta n sim 0 our study is for the thermal transpiration problem between two plates for the hard sphere gases with sufficiently large knudsen number and with the diffuse reflection boundary condition the solution is constructed and its singularity is studied by an iteration procedure

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

the coarse lipschitz geometry of ell_p oplus ell_q we show that for 1 p with p 2 the space l p 0 1 is not uniformly homeomorphic to ell p oplus ell 2 we also show that if 1 p 2 q the space ell p oplus ell q has unique uniform structure answering a question of johnson lindenstrauss and schechtman geom funct anal 6 430 470 1996

solving inverse stokes problems by modified collocation trefftz method in this paper the two dimensional inverse stokes problems governed by bi harmonic equations are stably solved by the modified collocation trefftz method mctm in some practical applications of the stokes problems part of the boundary conditions cannot be measured in advance so the mathematical descriptions of such problems are known as the inverse stokes problems when numerical simulation is adopted for solutions of the inverse stokes problems the solutions will become extremely unstable which means that small perturbations in the boundary conditions will result in large errors of the final results hence we adopted the mctm for stably and efficiently analyzing the inverse stokes problems the mctm is one kind of boundary type meshless methods so the mesh generation and the numerical quadrature can be avoided besides the numerical solution is expressed as a linear combination of t complete functions modified by a characteristic length by enforcing the satisfactions of the boundary conditions at every boundary node a system of linear algebraic equations will be yielded the unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix the numerical solutions and their derivatives can be easily obtained by linear summation three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two dimensional inverse stokes problems

boundary singularity of moments for the linearized boltzmann equation we study the boundary singularity for stationary solutions of the linearized boltzmann equation with hard sphere potential an asymptotic estimate for the gradient of the moments is established which shows the logarithmic singularity near the boundary our formula holds for the solutions of the milne and kramers problems obtained by bardos caflish nicolaenko commun pure appl math 49 323 452 1986 our theorem requires the holder continuity of the boundary data in particular it applies to the complete condensation problem for half space

thermal creep of a rarefied gas on the basis of non linear korteweg theory the study of thermal transpiration more commonly called thermal creep is accomplished by use of korteweg s theory of capillarity incorporation of this theory into the balance laws of continuum mechanics allows resolution of boundary value problems via solutions to systems of ordinary differential equations the problem was originally considered by maxwell in his classic 1879 paper maxwell phil trans roy soc london 170 231 256 1879 in that paper maxwell derived what is now called the burnett higher order contribution to the cauchy stress but was not able to solve his newly derived system of partial differential equations in this paper the authors note that a more appropriate higher order contribution to the cauchy stress follows from korteweg s 1901 theory korteweg arch neerl sci exactes nat ser ii 6 1 24 1901 the appropriateness of korteweg s theory is based on the exact summation of the chapman enskog expansion given by gorban and karlin the resulting balance laws are solved exactly qualitatively and numerically and the results are qualitatively similar to the numerical and exact results given by aoki et al loyalka et al and struchtrup et al

meshless local petrov galerkin mlpg approximation to the two dimensional sine gordon equation during the past few years the idea of using meshless methods for numerical solution of partial differential equations pdes has received much attention throughout the scientific community and remarkable progress has been achieved on meshless methods the meshless local petrov galerkin mlpg method is one of the truly meshless methods since it does not require any background integration cells the integrations are carried out locally over small sub domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions in this paper the mlpg method for numerically solving the non linear two dimensional sine gordon sg equation is developed a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of proposed method to deal with the unsteady non linear problems in large domains

a multiple scale pascal polynomial for 2d stokes and inverse cauchy stokes problems abstract the polynomial expansion method is a useful tool for solving both the direct and inverse stokes problems which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the stokes differential equations and the specified boundary conditions in this paper we propose two novel numerical algorithms based on a third first order system and a third third order system to solve the direct and the inverse cauchy problems in stokes flows by developing a multiple scale pascal polynomial method of which the scales are determined a priori by the collocation points to assess the performance through numerical experiments we find that the multiple scale pascal polynomial expansion method mspem is accurate and stable against large noise

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme