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

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

half space problem for the discrete boltzmann equation condensing vapor flow in the presence of a non condensable gas we consider a non linear half space problem related to the condensation problem for the discrete boltzmann equation and extend some known results for a single component gas to the case when a non condensable gas is present the vapor is assumed to tend to an assigned maxwellian at infinity as the non condensable gas tends to zero at infinity we assume that the vapor is completely absorbed and that the non condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution the conditions on the given distribution needed for the existence of a unique solution of the problem are investigated we also find exact solvability conditions and solutions for a simplified six four velocity model as the given distribution is a maxwellian at rest and study a simplified twelve six velocity model

nonlinear stability of rarefaction waves for the boltzmann equation it is well known that the boltzmann equation is related to the euler and navier stokes equations in the field of gas dynamics the relation is either for small knudsen number or for dissipative waves in the time asymptotic sense in this paper we show that rarefaction waves for the boltzmann equation are time asymptotic stable and tend to the rarefaction waves for the euler and navier stokes equations our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro macro decomposition of the boltzmann equation the expansion nature of the rarefaction waves and the suitable microscopic version of the h theorem are essential elements of our analysis

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

some remarks on a conjecture of de oliveira abstract let u n be the group of the unitary n n matrices let a diag 1 n b diag 1 n where 1 n 1 n are complex numbers a class of normal matrices satisfying the inclusion det a ubu u n co n i 1 i i s n where co is the convex hull of and s n is the symmetric group of degree n is obtained

symmetry of the linearized boltzmann equation and its application a symmetric relation of macroscopic quantities between two different steady problems of the linearized boltzmann equation is derived a few applications to half space problems are presented first then for the gas in bounded or unbounded domains such that solid bodies or condensed phases are confined in a finite region general representations of the mass momentum and heat fluxes through the boundary possibly at a point on or on a part of it are derived from the symmetric relation linked to the separability of boundary data this result implies a reduction of the original problem to a single elemental problem in the same domain as far as the fluxes are concerned many applications are also presented

a relative higher index theorem diffeomorphisms and positive scalar curvature abstract we prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity we apply this theorem to study riemannian metrics of positive scalar curvature on manifolds for every two metrics of positive scalar curvature on a closed manifold and a galois cover of the manifold we define a secondary higher index class non vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature in the special case where one metric is induced from the other by a diffeomorphism of the manifold we obtain a formula for computing this higher index class in particular it follows that the higher index class lies in the image of the baum connes assembly map

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

the higson roe exact sequence and l2 eta invariants abstract the goal of this paper is to solve the problem of existence of an l 2 relative eta morphism on the higson roe structure group using the cheeger gromov l 2 eta invariant we construct a group morphism from the higson roe maximal structure group constructed in 35 to the reals when we apply this morphism to the structure class associated with the spin dirac operator for a metric of positive scalar curvature we get the spin l 2 rho invariant when we apply this morphism to the structure class associated with an oriented homotopy equivalence we get the difference of the l 2 rho invariants of the corresponding signature operators we thus get new proofs for the classical l 2 rigidity theorems of keswani obtained in 41

variance reduced particle simulation of the boltzmann transport equation in the relaxation time approximation we present an efficient variance reduced particle simulation technique for solving the linearized boltzmann transport equation in the relaxation time approximation used for phonon electron and radiative transport as well as for kinetic gas flows the variance reduction is achieved by simulating only the deviation from equilibrium we show that in the limit of small deviation from equilibrium of interest here the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium in stark contrast to standard particle simulation methods our results demonstrate that a space dependent equilibrium distribution improves the variance reduction achieved especially in the collision dominated regime where local equilibrium conditions prevail we also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation time approximation a very simple collision algorithm with a clear physical interpretation can be formulated

matrices satisfying a conjecture of g n de oliveira on determinants abstract it is shown that the determinant of the sum of a positive definite hermitian matrix and a skew hermitian matrix with eigenvalues j and i j j 1 n respectively is contained in the convex hull of the points j 1 n j i j s n as usual s n denotes the symmetric group of degree n this result proves in a particular case a conjecture of g n de oliveira on determinants

the coarse geometric novikov conjecture and uniform convexity abstract the coarse geometric novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero the purpose of this paper is to prove the coarse geometric novikov conjecture for spaces which admit a coarse uniform embedding into a uniformly convex banach space

pointwise convergence to knudsen layers of the boltzmann equation in this paper green s functions for the boltzmann equation around a global maxwellian are used to construct the non characteristic nonlinear knudsen layers as well as their time asymptotic stability furthermore the detailed pointwise structures nonlinear wave couplings and wave interactions with boundary are studied

omc for scalar multiples of unitaries abstract we establish the following case of the determinantal conjecture of marcus m marcus derivations plucker relations and the numerical range indiana univ math j 22 1973 1137 1149 and de oliveira g n de oliveira research problem normal matrices linear and multilinear algebra 12 1982 153 154 let a and b be unitary n n matrices with prescribed eigenvalues a1 an and b1 bn respectively then for any scalars t and s det ta sb co j 1 n ta j sb j s n where s n denotes the group of all permutations of 1 n and co the convex hull taken in the complex plane

higher rho invariants and the moduli space of positive scalar curvature metrics abstract given a closed smooth manifold m which carries a positive scalar curvature metric one can associate an abelian group p m to the space of positive scalar curvature metrics on this manifold the group of all diffeomorphisms of the manifold naturally acts on p m the moduli group p m of positive scalar curvature metrics is defined to be the quotient abelian group of this action i e the coinvariant of the action p m measures the size of the moduli space of positive scalar curvature metrics on m in this paper we use the higher rho invariant and the finite part of the k theory of the group c algebra of 1 m to give a lower bound of the rank of the moduli group p m the key idea of our proof is the use of higher rho invariant this is a secondary invariant associated to dirac operators and in particular depends on the choice of riemannian metric we show that the higher rho invariant remains unchanged in a certain k theory group under the action of the diffeomorphism group allowing us to distinguish elements in p m

nonlinear stability of boundary layers of the boltzmann equation i the case m 1 this is a continuation of the paper 15 on nonlinear boundary layers of the boltzmann equation where the existence is established and shown to be strongly dependent on the mach number m of the maxwellian state at far field in this paper when m 1 we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers

second order knudsen layer analysis for the generalized slip flow theory ii curvature effects numerical analyses of the second order knudsen layer are carried out on the basis of the linearized boltzmann equation for hard sphere molecules under the diffuse reflection boundary condition the effects of the boundary curvature have been clarified in details thereby completing the numerical data required up to the second order of the knudsen number for the asymptotic theory of the boltzmann equation the generalized slip flow theory a local singularity appears as a result of the expansion at the level of the velocity distribution function when the curvature exists

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

the validity of the marcus de oliveira conjecture for essentially hermitian matrices abstract let x y be matrices of spectra x 1 x n and y 1 y n respectively for a wide class of pairs of essentially hermitian matrices x y we prove that the determinant of x y belongs to the convex hull of the n points n i 1 x i y i s n

a classification of well posed kinetic layer problems in the first part of this paper we study the half space boundary value problem for the boltzmann equation with an incoming distribution obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero we linearize it about a drifting maxwellian and prove that as conjectured by cercignani 4 the problem is well posed when the drift velocity u exceeds the sound speed c but that one respectively four five additional conditions must be imposed when 0 u c respectively c u 0 and u c in the second part we show that the well posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions

asymptotic theory for the time dependent behavior of a slightly rarefied gas over a smooth solid boundary a time evolution of a slightly rarefied monoatomic gas namely a gas for small knudsen numbers which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized boltzmann equation by a systematic asymptotic analysis a set of fluid dynamic type equations and its boundary conditions that describe the gas behavior up to the second order of the knudsen number are derived the developed theory covers a general intermolecular potential and a gas surface interaction it is shown that i the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation ii although the momentum equation is the stokes equation it contains a double laplacian of the leading order flow velocity as a source term at the second order iii a double laplacian source term also appears in the energy equation at the second order iv the slip and jump conditions are the same as those in the time independent case up to the first order and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the laplacian of temperature numerical values of all the slip and jump coefficients are obtained for a hard sphere gas by the use of a symmetric relation developed recently

invariant manifolds for steady boltzmann flows and applications we develop a theory of invariant manifolds for the steady boltzmann equation and apply it to the study of boundary layers and nonlinear waves the steady boltzmann equation is an infinite dimensional differential equation so the standard center manifold theory for differential equations based on spectral information does not apply here instead we employ a time asymptotic approach using the pointwise information of green s function for the construction of the linear invariant manifolds at the resonance cases when the mach number at the far field is around one of the critical values of 1 0 or 1 the truly nonlinear theory arises in such a case there are wave patterns combining the fast decaying knudsen type and slow varying fluid like waves the key knudsen manifolds consisting of only knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical mach numbers around mach number 1 the fluidlike waves are compressive and expansive waves and around the mach number 0 they are linear thermal layers the quantitative analysis of the fluid like waves is done using the reduction of dimensions to the center manifolds two scale nonlinear dynamics based on those on the knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns there are striking bifurcations in the transition of evaporation to condensation and in the transition of the milne s problem with a subsonic far field to one with a supersonic far field the analysis of these wave patterns allows us to understand the sone diagram for the study of the complete condensation boundary value problem the monotonicity of the boltzmann shock profiles a problem that initially motivated the present study is shown as a consequence of the quantitative analysis of the nonlinear fluid like waves

positive scalar curvature higher rho invariants and localization algebras abstract in this paper we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics the higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems the main result of the paper connects the higher index of the dirac operator on a spin manifold with boundary to the higher rho invariant of the dirac operator on the boundary where the boundary is endowed with a positive scalar curvature metric our result extends a theorem of piazza and schick 27 theorem 1 17

the novikov conjecture for groups with finite asymptotic dimension recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology page 28 14 more precisely the asymptotic dimension for a metric space is the smallest integer n such that for any r 0 there exists a uniformly bounded cover c ui iei of the metric space for which the r multiplicity of c is at most n 1 i e no ball of radius r in the metric space intersects more than n 1 members of c 14 the class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word length metric then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word length metrics cf section 6 this together with a result of gromov in 14 implies that finitely generated subgroups of gromov s hyperbolic groups have finite asymptotic dimension currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known it should also be noted that two different definitions of asymptotic dimension