THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

Investigation of Acoustomagnetoelectric Effect in Bandgap Graphene by the Boltzmann Transport Equation

We study the acoustomagnetoelectric (AME) effect in two-dimensional graphene with an energy bandgap using the semiclassical Boltzmann transport equation within the hypersound regime, (where represents the acoustic wavenumber and is the mean free path of the electron). The Boltzmann transport equation and other relevant equations were solved analytically to obtain an expression for the AME current density, consisting of longitudinal and Hall components. Our numerical results indicate that both components of the AME current densities display oscillatory behaviour. Furthermore, geometric resonances and Weiss oscillations were each defined using the relationship between the current density and Surface Acoustic Wave (SAW) frequency and the inverse of the applied magnetic field, respectively. Our results show that the AME current density of bandgap graphene, which can be controlled to suit a particular electronic device application, is smaller than that of (gapless) graphene and is therefore, more suited for nanophotonic device applications.

Analysis of Two-Phase Cavitating Flowwith Two-Fluid Model Using Integrated Boltzmann Equations

In the present work,both computational and experimentalmethods are employed to study the two-phase flow occurring in a model pump sump.The twofluid model of the two-phase flow has been applied to the simulation of the threedimensional cavitating flow.The governing equations of the two-phase cavitating flow are derived from the kinetic theory based on the Boltzmann equation.The isotropic RNG k−ε−kca turbulence model of two-phase flows in the form of cavity number instead of the formof cavity phase volume fraction is developed.The RNG k−ε−kca turbulence model,that is the RNG k−εturbulence model for the liquid phase combined with the kca model for the cavity phase,is employed to close the governing turbulent equations of the two-phase flow.The computation of the cavitating flow through a model pump sump has been carried out with this model in three-dimensional spaces.The calculated results have been compared with the data of the PIV experiment.Good qualitative agreement has been achievedwhich exhibits the reliability of the numerical simulation model.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

Boltzmann Equations with Quantum Effects (2): Entropy Identity, Existence and Uniqueness of Spatial Decay Solutions

A previous study is continued by investigating the Boltzmann equation for particles with quantum effects (BQE). First, the corresponding entropy identity is proved, then if the initial data f(x,v,0) satisfies 0≤f(x,v,0)≤CΦ(x,v,0) for a constant 0<C<∞ and function Φ(x,v,t), we prove the existence and uniqueness of spatial decay solutions of the BQE within a given function space B(Φ) using fixed point theory. Moreover, if there is a continuous function F(x,v) which belongs to a function set, then there exists a mild solution f(x,v,t) of the BQE such that f ∞(x,v)= limt→∞f(x+vt,v,t)=F(x,v).

Boltzmann Equations with Quantum Effects (1):Long Time Behavior of Spatial Decay Solutions

The Boltzmann equations for Fermi-Dirac particles and Bose-Einstein particles, both in the absence of external force fields, are combined into a more general form called the Boltzmann equation with quantum effects (BQE). It is assumed that the initial data f(x,v,0) satisfies 0≤f(x,v,0)≤cΦ(x,v,0) for a positive constant c and certain types of control functions Φ(x,v,t). Then within a given function space B(Φ), we prove that f(x+tv,v,t) uniformly converges to f ∞(x,v) in a certain norm where f ∞(x,v)= limt→∞f(x+tv,v,t) and different initial data determines different long time limits.

Lattice Boltzmann Flux Solver:An Efficient Approach for Numerical Simulation of Fluid Flows

A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.

A novel incompressible finite-difference lattice Boltzmann equation for particle-laden flow

In this paper, we propose a novel incompressible finite-difference lattice Boltzmann Equation （FDLBE）. Because source terms that reflect the interaction between phases can be accurately described, the new model is suitable for simulating two-way coupling incompressible multiphase flow The 2-D particle-laden flow over a backward-facing step is chosen as a test case to validate the present method. Favorable results are obtained and the present scheme is shown to have good prospects in practical applications.

Investigation on the Dielectric Properties of CO2 and CO2-Based Gases Based on the Boltzmann Equation Analysis

In this paper, the dielectric properties of CO2, CO2/air, CO2/O2, CO2/N2, CO2/CF4, CO2/CH4, CO2/He, C02/H2, CO2/NH3 and CO2/CO were investigated based on the Boltzmann equation analysis, in which the reduced critical electric field strength （E/N）cr of the gases was derived from the calculated electron energy distribution function （EEDF） by solv- ing the Boltzmann transport equation. In this work, it should be noted that the fundamental data were carefully selected by the published experimental results and calculations to ensure the validity of the calculation. The results indicate that if He, H2, N2 and CH4, in which there axe high ionization coefficients or a lack of attachment reactions, are added into CO2, the dielectric properties will decrease. On the other hand, air, O2, NH3 and CFa （ranked in terms of （E/N）cr value in increasing order） have the potential to improve the dielectric property of CO2 at room temperature.

CAUCHY PROBLEM FOR LINEARIZED NON-CUTOFF BOLTZMANN EQUATION WITH DISTRIBUTION INITIAL DATUM

In this article, we study the Cauchy problem for the linearized spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. By using the spectral decomposition, we solve the Cauchy problem with initial datum in the sense of distribution, which contains the dual space of a Gelfand-Shilov class. We also prove that this solution belongs to the Gelfand-Shilov space for any positive time.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of （x, t） in the region R3 × （0, ＋∞）. More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not hap- pen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

On the validity of the Boltzmann–BGK model through relaxation evaluation

The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical comparison between the BGK and direct simulation Monte Carlo（DSMC） results for three specifically designed relaxation problems. In these problems, one or half component of the velocity distribution is characterized by another Maxwellian distribution with a different temperature. It is analyzed that the relaxation time in the BGK model is unequal to the molecular mean collision time. Relaxation of component distribution fails to involve enough contribution from other component distributions, which makes the BGK model unable to capture details of velocity distribution, especially when discontinuity exists in distribution. The BGK model,however, predicts satisfactory results including fluxes during relaxation when the temperature difference is small. Particularly, the model-induced error in the BGK model increases with the temperature difference, thus the model is more reliable for low-speed rarefied flows than for hypersonic flows.

Investigation of turbulence and skin friction modification in particle-laden channel flow using lattice Boltzmann method

Interaction between turbulence and particles is investigated in a channel flow. The fluid motion is calculated using direct numerical simulation(DNS) with a lattice Boltzmann(LB) method, and particles are tracked in a Lagrangian framework through the action of force imposed by the fluid. The particle diameter is smaller than the Kolmogorov length scale, and the point force is used to represent the feedback force of particles on the turbulence. The effects of particles on the turbulence and skin friction coefficient are examined with different particle inertias and mass loadings. Inertial particles suppress intensities of the spanwise and wall-normal components of velocity, and the Reynolds shear stress. It is also found that, relative to the reference particle-free flow,the overall mean skin-friction coefficient is reduced by particles. Changes of near wall turbulent structures such as longer and more regular streamwise low-speed streaks and less ejections and sweeps are the manifestation of drag reduction.

BOUNDED SOLUTION OF THE RELATIVISTIC BOLTZMANN EQUATION

In either a periodic box Td or Rd （1 ≤ d ≤ 3）, we obtain global bounded solution of the relativistic Boltzmann equation near global relativistic Maxwellian, in terms of natural mass, energy conservation and the entropy inequality.

Improving Solution of Euler Equations by a Gas-Kinetic BGK Method

Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in order to obviate the use of dissipation term and obtain,we believe,an improved solution.Section 1 deals essentially with three things:(1) as analytical solution of molecular probability density function at the cell interface has been obtained by the Boltzmann equation with BGK model,we can compute the flux term by integrating the density function in the phase space;eqs.(8) and(11) require careful attention;(2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution;eqs.(9) and(10) require careful attention;(3) the discrete equation by finite volume method can be solved using the time marching method.Computations are performed by the BGK method for the Sod′s shock tube problem and a two-dimensional shock reflection problem.The results are compared with those of the conventional JST scheme in Figs.1 and 2.The BGK method provides better resolution of shock waves and other features of the flow fields.

Spin-dependent balance equations in spintronics

It is commonly known that the hydrodynamic equations can be derived from the Boltzmann equation. In this paper, we derive similar spin-dependent balance equations based on the spinor Boltzmann equation. Besides the usual charge current, heat current, and pressure tensor, we also explore the characteristic spin accumulation and spin current as well as the spin-dependent pressure tensor and heat current in spintronics. The numerical results of these physical quantities are demonstrated using an example of spin-polarized transport through a mesoscopic ferromagnet.

EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy. More precisely, under the assumptions on the bicharacteristic generated by external force, we prove the global existence of solution for small initial data compared to the local Maxwellian exp（-p｜x - v｜^2）, which has infinite mass and energy.

Multi-relaxation-time lattice Boltzmann simulations of lid driven flows using graphics processing unit

Large eddy simulation （LES） using the Smagorinsky eddy viscosity model is added to the two-dimensional nine velocity components （D2Q9） lattice Boltzmann equation （LBE） with multi-relaxation-time （MRT） to simulate incompressible turbulent cavity flows with the Reynolds numbers up to 1 × 10^7. To improve the computation efficiency of LBM on the numerical simulations of turbulent flows, the massively parallel computing power from a graphic processing unit （GPU） with a computing unified device architecture （CUDA） is introduced into the MRT-LBE-LES model. The model performs well, compared with the results from others, with an increase of 76 times in computation efficiency. It appears that the higher the Reynolds numbers is, the smaller the Smagorinsky constant should be, if the lattice number is fixed. Also, for a selected high Reynolds number and a selected proper Smagorinsky constant, there is a minimum requirement for the lattice number so that the Smagorinsky eddy viscosity will not be excessively large.

GLOBAL EXISTENCE FOR THE RELATIVISTIC ENSKOG EQUATIONS

This article extends the results of Arkeryd and Cercignani [6]. It is shown that the Cauchy problem for the relativistic Enskog equation in a periodic box has a global mild solution if the mass, energy and entropy of the initial data are finite. It is also found that the solutions of the relativistic Enskog equation weakly converge to the solutions of the relativistic Boltzmann equation in L^1 if the diameter of the relativistic particle tends to zero.