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

nonlinear stability of boundary layer solution to the boltzmann equation with diffusive effect at the boundary in this paper the stability of boundary layer solutions to the boltzmann equation with diffusive effect at the boundary for hard sphere model is considered when the mach number of the far field is less than 1 the exponential decay in time is proven for linearized operator first at last based on this property nonlinear stability of the boundary layers is obtained by bootstrap argument

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

existence of boundary layers to the boltzmann equation with cutoff soft potentials when the boltzmann equation is used to study a physical problem with boundary there usually exists a layer of the width in the order of the knudsen number along the boundary there have been extensive studies on the existence and stability of boundary layers to the boltzmann equation in different physical settings based on the previous work in this paper we consider the existence of boundary layer solutions to the boltzmann equation for the cutoff soft potentials when its parameter satisfying 2 0 the boundary condition is imposed on the incoming particles of dirichlet type and the solution is assumed to approach to a global maxwellian at the far field as for the hard sphere model and the cutoff hard potentials the existence of solutions is shown to depend on the mach number of the far field maxwellian moreover there is an implicit solvability condition on the boundary data which gives the codimension of the boundary data related to the number of the positive characteristic speeds in the fo

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

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

corrigendumcorrigendum to quasi asymptotic contractions set valued dynamic systems uniqueness of endpoints and generalized pseudodistances in uniform spaces nonlinear anal 70 2009 1059 1068 doi 10 1016 j na 2008 01 032

a boundary value problem for discrete velocity models on considere deux problemes aux valeurs limites non lineaires concernant un ecoulement stationnaire a une dimension entre des plaques paralleles

fluid dynamic limit to rarefaction wave for the boltzmann equation abstract the hydrodynamic limit for the boltzmann equation to the compressible euler equation as knudsen number e vanishes is a difficult and challenging problem in the mathematics when the corresponding compressible euler equation has a single rarefaction wave xin and zeng 2010 23 recently verified the hydrodynamic limit as e tends to zero with a convergence rate e 1 5 ln e in this paper the convergence rate of xin and zeng 2010 23 is improved to e 1 3 ln e 2 by different scaling arguments

nonlinear boundary layers of the boltzmann equation i existence we study the half space problem of the nonlinear boltzmann equation assigning the dirichlet data for outgoing particles at the boundary and a maxwellian as the far field we will show that the solvability of the problem changes with the mach number of the far maxwellian if 1 respectively we also show that the same is true for the linearized problem at the far maxwellian and the manifold is then a hyperplane the proof is essentially based on the macro micro or hydrodynamics kinetic decomposition of solutions combined with an artificial damping term and a spatially exponential decay weight

a half space problem for the boltzmann equation with specular reflection boundary condition there are many open problems on the stability of nonlinear wave patterns to the boltzmann equation even though the corresponding stability theory has been comparatively well established for the gas dynamical systems in this paper we study the nonlinear stability of a rarefaction wave profile to the boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut off the analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection and the decomposition on both the solution and the boltzmann equation introduced in 24 26 for energy method

exterior problem for the boltzmann equation with temperature difference asymptotic stability of steady solutions abstract the asymptotic stability of non equilibrium steady solutions to the exterior problem for the boltzmann equation was first proved by ukai and asano in 17 under the assumption that the temperature associating with the far field maxwellian and the one preserved by the kinetic boundary condition are the same in this paper we generalize ukai and asano s result in the sense that the two temperatures mentioned above are allowed to be different the proof of the main theorem is based on the ideas developed in 16 17 and 20 as well as some new observations

a discrete velocity stationary wigner equation this paper is concerned with the one dimensional stationary linear wigner equation a kinetic formulation of quantum mechanics specifically we analyze the well posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete velocity model we find that the problem is uniquely solvable if zero is not a discrete velocity otherwise one obtains a differential algebraic equation of index 2 and hence the inflow data make the system overdetermined

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

using string invariants for prediction searching for optimal parameters abstract we have developed a novel prediction method based on string invariants the method does not require learning but a small set of parameters must be set to achieve optimal performance we have implemented an evolutionary algorithm for the parametric optimization we have tested the performance of the method on artificial and real world data and compared the performance to statistical methods and to a number of artificial intelligence methods we have used data and the results of a prediction competition as a benchmark the results show that the method performs well in single step prediction but the method s performance for multiple step prediction needs to be improved the method works well for a wide range of parameters

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

cross correlation and the predictability of financial return series a b s t r a c t this paper examines whether we can improve the predictability of financial return series by exploiting the effect of cross correlations among different financial markets we forecast financial return series based on the support vector machines svm method which can surpass the random walk model consistently by comparing the mean absolute errors and the root mean squared errors we show that it is hard to improve the predictability of financial return series by incorporating correlated return series into svm based forecasting models even though there are granger causal relationships among them

quasi asymptotic contractions set valued dynamic systems uniqueness of endpoints and generalized pseudodistances in uniform spaces abstract for set valued dynamic systems in uniform spaces we introduce the concept of quasi asymptotic contractions with respect to some generalized pseudodistances describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints stationary points of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations in particular each dynamic process converges and the limit is an endpoint the definition result ideas and techniques are new for set valued dynamic systems in uniform locally convex and metric spaces and even for single valued maps

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

derivation of slip boundary conditions for the navier stokes system from the boltzmann equation rarefied gas flow behavior is usually described by the boltzmann equation the navier stokes system being valid when the gas is less rarefied slip boundary conditions for the navier stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level boltzmann equation these slip conditions are explicitly written in terms of asymptotic behavior of some linear half space problems the validity of this analysis is established in the simple case of the couette flow for which it is proved that the right boundary conditions are obtained

stability of contact discontinuity for the boltzmann equation the boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics that is euler equations and navier stokes equations as for a basic wave pattern to euler equations we consider the nonlinear stability of contact discontinuities to the boltzmann equation even though the stability of the other two nonlinear waves i e shocks and rarefaction waves has been extensively studied there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves its derivative has no definite sign and decays slower than a rarefaction wave moreover it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism based on the new definition of contact waves to the boltzmann equation corresponding to the contact discontinuities for the euler equations we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate in our analysis an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates

existence of nonlinear boundary layer to the boltzmann equation with reverse reflection boundary condition abstract many physical models have boundaries when the boltzmann equation is used to study a physical problem with boundary there usually exists a layer of width of the order of the knudsen number along the boundary hence the research on the boundary layer problem is important both in mathematics and physics based on the previous work in this paper we consider the existence of boundary layer solution to the boltzmann equation for hard sphere model with positive mach number the boundary condition is imposed on incoming particles of reverse reflection type and the solution is assumed to approach to a global maxwellian in the far field similar to the problem with dirichlet boundary condition studied in s ukai t yang s h yu nonlinear boundary layers of the boltzmann equation i existence comm math phys 3 2003 373 393 the existence of a solution is shown to depend on the mach number of the far field maxwellian moreover there is an implicit solvability condition on the boundary data according to the solvability condition the co dimension of the boundary data related to the number of the positive characteristic speeds is obtained

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

exact sequences in the cohomology of a group extension abstract in 1 the authors constructed a seven term exact sequence in the cohomology of a group extension 1 n g q 1 with coefficients in a g module m however they were unable to establish the precise link between the maps in that sequence and the corresponding maps arising from the spectral sequence associated to the group extension and the g module m in this paper we show that there is a close connection between 1 and the two earlier papers 6 and 7 in particular we show that the results in 6 and 7 entail that the maps of 1 other than the obvious inflation and restriction maps do correspond to the corresponding ones arising from the spectral sequence

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

the stationary nonlinear boltzmann equation in a couette setting this paper studies the stationary nonlinear boltzmann equation for hard forces in a couette setting between two coaxial rotating cylinders with given indata of maxwellian type on the cylinders a priori estimates are obtained mainly in l2 leading to multiple isolated solutions together with a hydrodynamic limit control based on asymptotic expansions together with a rest term

existence and uniqueness for nonlinear boundary value problems in kinetic theory a boundary value problem for the stationary nonlinear boltzmann equation in a slab has been examined in a weightedl space it has been proved that the problem possesses a unique solution for boundary data small enough the proof is based on the implicit function theorem it has also been shown that for the linearized problem the fredholm alternative applies

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

cohomology theory for financial time series abstract khovanov cohomology in time series data is used to model financial time in figure eight hyperbolic knotted time series we defined chern simons current from the interaction of behavior of traders over wilson loop with link eight states we build a new path integral for the market phase transitions as wilson loop between predictor and predictant paths in physiology of time series data the obtained results are presented in the form of proved theorems in the first theorem we prove the existences of chern simons current in the bid ask spread as arbitrage opportunity for market phase transition of the interaction of yang mills behavior field of the trader in the second theorem we prove the existence of market 8 states as a consequence of the theorems we classify the smallest subunit of market microstructure as 16 physiology cones according to all possibilities of the future outcome of endpoint state in time series data the financial market movements for up and down directions with the curvature as link number are obtain from a market phase transition with eigenvalue of new defined dirac operator for the financial market the market curvature value appears as knot and link properties in time series data

on a boundary layer problem for the nonlinear boltzmann equation this article deals with a boundary layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero the model considered here is the stationary nonlinear boltzmann equation in one dimension with a slightly perturbed reflection boundary condition we restrict our attention to the case of hard spheres collisions with grad s cutoff assumption existence uniqueness and asymptotic behavior are derived by means of energy estimates

existence of nonlinear boundary layer solution to the boltzmann equation with physical boundary conditions abstract in this paper the existence of boundary layer solutions to the boltzmann equation with two physical boundary conditions for hard sphere model is considered the boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends to a global maxwellian in the far field similar to the problem with dirichlet boundary condition studied in s ukai t yang s h yu nonlinear boundary layers of the boltzmann equation i existence comm math phys 236 3 2003 373 393 the existence of a solution is shown to depend on the mach number of the far field maxwellian and there is an implicit solvability conditions yielding the co dimensions of the boundary data at last the specular reflection boundary condition is considered and the similar conclusions are obtained

the nonlinear boundary layer to the boltzmann equation with mixed boundary conditions for hard potentials in this paper the existence of boundary layer solutions to the boltzmann equation for hard potential with mixed boundary condition i e a linear combination of dirichlet boundary condition and diffuse reflection boundary condition at the wall is considered the boundary condition is imposed on the incoming particles and the solution is supposed to approach to a global maxwellian in the far field as for the problem with dirichlet boundary condition chen et al 2004 5 the existence of a solution highly depends on the mach number of the far field maxwellian furthermore an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given

on the cauchy problem for the nonlinear boltzmann equation global existence uniqueness and asymptotic stability the analysis of the initial value problem for the nonlinear boltzmann equation is considered in this paper a theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in banach spaces the theorem also defines the asymptotic stability of the solutions

the grad limit for a system of soft spheres a model describing the dynamics of a system of soft spheres is introduced the model contains a random mechanism which mollifies the deterministic model of hard sphere molecules it allows a rigorous deduction of the corresponding boltzmann equation for a wide class of initial data and arbitrarily long times

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

the study of thai stock market across the 2008 financial crisis the cohomology theory for financial market can allow us to deform kolmogorov space of time series data over time period with the explicit definition of eight market states in grand unified theory the anti de sitter space induced from a coupling behavior field among traders in case of a financial market crash acts like gravitational field in financial market spacetime under this hybrid mathematical superstructure we redefine a behavior matrix by using pauli matrix and modified wilson loop for time series data we use it to detect the 2008 financial market crash by using a degree of cohomology group of sphere over tensor field in correlation matrix over all possible dominated stocks underlying thai set50 index futures the empirical analysis of financial tensor network was performed with the help of empirical mode decomposition and intrinsic time scale decomposition of correlation matrix and the calculation of closeness centrality of planar graph

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

the solutions for the boundary layer problem of boltzmann equation in a half space we study the half space boundary layer problem for boltzmann equation with cut off potentials in all the cases 3 1 while the boundary condition is imposed on the incoming particles of dirichlet type and the solution is assumed to approach to a global maxwellian at the far field the same as for cut off hard sphere model there is an implicit solvability condition on the boundary data which gives the co dimensions of the boundary data in terms of positive characteristic speeds

hydrodynamic limit of the stationary boltzmann equation in a slab we study the stationary solution of the boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature we prove that when the force is sufficiently small there exists a solution which converges in the hydrodynamic limit to a local maxwellian with parameters given by the stationary solution of the corresponding compressible navier stokes equations with no slip boundary conditions corrections to this maxwellian are obtained in powers of the knudsen number with a controlled remainder

initial boundary value problems for the boltzmann equation global existence of weak solutions for an open set of 3 bounded or not we consider initial boundary value problems for the boltzmann equation for general gas surface interaction laws and for hard potentials we prove a global existence result for weak solutions the proof uses the regularization of the collision operator and the renormalization method for the regularized problem by using weak compactness in l1 and averaged stability ofq f f we prove the existence of weak solutions of our problem

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

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

a boundary value problem for the two dimensional broadwell model it is shown that a certain boundary value problem for the steady two dimensional broadwell model on a rectangle has a solution the boundary conditions specify the ingoing particle densities on each side of the rectangle

a new energy method for the boltzmann equation an energy method for the boltzmann equation was proposed by liu yang and yu physica d 188 178 192 2004 based on the decomposition of the boltzmann equation and its solution around the local maxwellian the main idea is to rewrite the boltzmann equation as a fluid type dynamics system with the nonfluid component appearing in the source terms coupled with an equation for the time evolution of the nonfluid component in this paper we will elaborate this method and our main observation is that the microscopic projection of the local maxwellian with respect to a given global maxwellian is not linear but quadratic based on this and by analyzing the fluid type system using the analytic techniques for the system of conservation laws we can indeed control the conserved quantities u and 12u2 e of the boltzmann equation by the microscopic projection of the solution of the boltzmann equation with respect to the global maxwellian which is sufficient to deduce the energy estimates for the solution of

loeb solutions of the boltzmann equation existence problems for the boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory the present paper considers the spatially periodic case with l1 initial data the main result is that the loeb subsolutions obtained in a preceding paper are shown to be true solutions the proof relies on the observation that monotone entropy and finite energy imply loeb integrability of non standard approximate solutions and uses estimates from the proof of the h theorem two aspects of the continuity of the solutions are also considered

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

boundary value problems for the steady boltzmann equation we discuss steady boundary value problems for the boltzmann equation with inflow and diffusive boundary conditions in one two and three dimensions with suitable truncations of the collision kernel general existence and uniqueness results are obtained if the domain is sufficiently small in one dimension the existence of solutions on general intervals is obtained by abstract fixed point theory

nonlinear stability of boundary layers for the boltzmann equation with cutoff soft potentials abstract the study on the boundary layer is important in both mathematics and physics this paper considers the nonlinear stability of boundary layer solutions for the boltzmann equation with cutoff soft potentials when the mach number of the far field is less than 1 unlike the collision frequency is strictly positive in the hard potential or hard sphere model the collision frequency has no positive lower bound for the cutoff soft potentials so the decay in time cannot be expected instead the present paper proves that the solution will always be in a small region around the boundary layer by noticing the decay property of collision operator in velocity

on nonlinear stationary half space problems in discrete kinetic theory we show that every steady discrete velocity model of the boltzmann equation on the real line i d dx fi ci f which satisfies anh theorem and for which all i 0 has solutions on the half line 0 which take prescribed non negativefi o if i 0 and approach a certain manifold of maxwellians asx such solutions give the density distribution in a knudsen boundary layer in the discrete velocity case

compressible navier stokes approximation to the boltzmann equation abstract even though the system of the compressible navier stokes equations is not a limiting system of the boltzmann equation when the knudsen number tends to zero it is the second order approximation by applying the chapman enskog expansion the purpose of this paper is to justify this approximation rigorously in mathematics that is if the difference between the initial data for the compressible navier stokes equations and the boltzmann equation is of the second order of the knudsen number so is the difference between two solutions for all time the analysis is based on a refined energy method for a fluid type system using the techniques for the system of viscous conservation laws

boundary layers and incompressible navier stokes fourier limit of the boltzmann equation in bounded domain i we establish the incompressible navier stokes fourier limit for solutions to the boltzmann equation with a general cutoff collision kernel in a bounded domain appropriately scaled families of diperna lions mischler renormalized solutions with maxwell reflection boundary conditions are shown to have fluctuations that converge as the knudsen number goes to 0 every limit point is a weak solution to the navier stokes fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the knudsen number the main new result of the paper is that this convergence is strong in the case of the dirichlet boundary condition indeed we prove that the acoustic waves are damped immediately namely they are damped in a boundary layer in time this damping is due to the presence of viscous and kinetic boundary layers in space as a consequence we also justify the first correction to the infinitesimal maxwellian that one obtains from the chapman enskog expansion with navier stokes scaling this extends the work of golse and saint raymond 20 21 and levermore and masmoudi 28 to the case of a bounded domain the case of a bounded domain was considered by masmoudi and saint raymond 34 for the linear stokes fourier limit and saint raymond 41 for the navier stokes limit for hard potential kernels neither 34 nor 41 studied the damping of the acoustic waves this paper extends the result of 34 41 to the nonlinear case and includes soft potential kernels more importantly for the dirichlet boundary condition this work strengthens the convergence so as to make the boundary layer visible this answers an open problem proposed by ukai 46 2016 wiley periodicals inc

the navier stokes limit of stationary solutions of the nonlinear boltzmann equation we consider the flow of a gas in a channel whose walls are kept at fixed different temperatures there is a constant external force parallel to the boundaries which may themselves also be moving the system is described by the stationary boltzmann equation to which are added maxwellian boundary conditions with unit accommodation coefficient we prove that when the temperature gap the relative velocity of the planes and the force are all sufficiently small there is a solution which converges in the hydrodynamic limit to a local maxwellian with parameters given by the stationary solution of the corresponding compressible navier stokes equations with no slip voundary conditions corrections to this maxwellian are obtained in powers of the knudsen number with a controlled remainder

on a taylor couette type bifurcation for the stationary nonlinear boltzmann equation this paper studies the stationary nonlinear boltzmann equation for hard forces in a taylor couette setting between two coaxial rotating cylinders with given indata of maxwellian type on the cylinders a priori l q estimates are obtained and used to prove a taylor type bifurcation with isolated solutions and a hydrodynamic limit control based on asymptotic expansions together with a rest term correction the positivity of such solutions is also considered

erratum to best proximity points for cyclic and noncyclic set valued relatively quasi asymptotic contractions in uniform spaces nonlinear anal 2008 doi 10 1016 j na 2008 04 037

cross correlation analysis of stock markets using emd and eemd abstract empirical mode decomposition emd is a data driven signal analysis method for nonlinear and nonstationary data since it is intuitive direct posterior and adaptive emd is widely applied to various fields of study in this paper emd and ensemble empirical mode decomposition eemd a modified method of emd are applied to financial time series through analyzing the intrinsic mode functions imfs of emd and eemd we find eemd method performs better on the orthogonality of imfs than emd with clustering the ordered frequencies of imfs the imfs obtained from eemd method are grouped into high medium and low frequency components representing the short medium and long term volatilities of the index sequences respectively with the cross correlation analysis of dcca cross correlation coefficient our findings allow us to gain further and detailed insight into the cross correlations of stock markets

convergence to rarefaction waves for the nonlinear boltzmann equation and compressible navier stokes equations abstract the main purpose of this paper is to study the asymptotic equivalence of the boltzmann equation for the hard sphere collision model to its corresponding euler equations of compressible gas dynamics in the limit of small mean free path when the fluid flow is a smooth rarefaction or centered rarefaction wave with finite strength the corresponding boltzmann solution exists globally in time and the solution converges to the rarefaction wave uniformly for all time or away from t 0 as 0 a decomposition of a boltzmann solution into its macroscopic fluid part and microscopic kinetic part is adopted to rewrite the boltzmann equation in a form of compressible navier stokes equations with source terms in this setting the same asymptotic equivalence of the full compressible navier stokes equations to its corresponding euler equations in the limit of small viscosity and heat conductivity depending on the viscosity is also obtained

nonlinear stability of boundary layers of the boltzmann equation for cutoff hard potentials many physical models have boundaries for the boltzmann equation the study on the boundary layer in the region of the width in the order of the kundsen number along the boundary is important both in mathematics and physics in this paper we consider the nonlinear stability of boundary layer solutions to the boltzmann equation for hard potentials with angular cut off the boundary condition is imposed on incoming particles of the dirichlet type and the solution tends to a global maxwellian in the far field for the existence of the boundary layer solutions it is proved by chen et al anal appl 2 337 363 2004 by introducing a weight function which is a function of both position and velocity to overcome the difficulty from the sublinear growth in the collision frequency unlike the hard sphere model even for stability in the case when the mach number of the far field is less than 1 exponential decay in time cannot be expected for the cutoff hard potentials instead an algebraic decay in time to

asymptotic behavior on the milne problem with a force term abstract this paper studies the existence uniqueness and asymptotic behavior of the solution for a half space linearized stationary boltzmann equation with an external force term in the case of a specified incoming distribution at the boundary and a given mass flux without the external force the solution of the stationary boltzmann equation has been proved to tend toward a constant state which is independent of the space variable due to the presence of the external force we show that the solution tends to some function which depends on the space variable

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

the half space problem for the boltzmann equation at zero temperature at zero temperature the maxwellian distribution is a delta function of velocity in this paper the boltzmann equation is linearized around a delta function and then analyzed by a comparison method using these results and similar bounds for the nonlinear collision operator a nonlinear boundary value problem at zero temperature is solved the results are applied to the asymptotic description at the cold end of the shock profile at infinite mach number all solutions f are assumed to have the form f x 1 a x f x in which a and f are regular functions

kinetic boundary layers on the adequate discretization of the boltzmann collision operator we develop criteria for the discretization of the boltzmann collision operator under which linearized kinetic boundary layers exhibit the same algebraic structure as their continuous counterparts these criteria are shown to be sufficient for the well posedness of kinetic boundary layers after the analysis of the discrete layer an example illustrates how to include models which lead to differential algebraic problems existence and uniqueness of nonlinear boundary layers adjacent to an equilibrium state are proven

behaviour of the fokker planck boltzmann equation near a maxwellian we consider the initial value problem for the fokker planck boltzmann equation namely viewed as the boltzmann equation with an additional diffusion term in velocity space to describe for instance the transport in thermal baths of binary elastic collisional particles the strong solution for initial data near an absolute maxwellian is proved to exist globally in time and tends asymptotically in the l infty y l xi 1 norm to another time dependent self similar maxwellian in large time the effect of the diffusion in phase space is investigated it produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid dynamic observable accelerating the convergence of solutions to the equilibrium of a self similar maxwellian at a faster time decay rate than the boltzmann equation this phenomena is also observed for homogeneous fokker planck boltzmann equations where the time decay rate in the l xi 1 norm to the self similar maxwellian is proved to be faster than exponential moreover the fokker planck boltzmann equation is shown to converge under an appropriate scaling strongly to the boltzmann equation in the process of the zero diffusion limit

best proximity points for cyclic and noncyclic set valued relatively quasi asymptotic contractions in uniform spaces abstract given a uniform space x and nonempty subsets a and b of x we introduce the concepts of some families v of generalized pseudodistances on x of set valued dynamic systems of relatively quasi asymptotic contractions t a b 2 a b with respect to v and best proximity points for t in a b and we describe the methods which we use to establish the conditions guaranteeing the existence of best proximity points for t when t is cyclic i e t a 2 b and t b 2 a or when t is noncyclic i e t a 2 a and t b 2 b moreover we establish conditions guaranteeing that for each starting point each generalized sequence of iterations of these contractions in particular each dynamic process converges and the limit is a best proximity point for t in a b these best proximity points for t are determined by unique endpoints in a b for a map t 2 when t is cyclic and for a map t when t is noncyclic the results and the methods are new for set valued and single valued dynamic systems in uniform locally convex metric and banach spaces various examples illustrating the ideas of our definitions and results and fundamental differences between our results and the well known ones are given