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.

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.

THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE

This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1-3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].

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.

Zakharov-Rubenchik方程组的格子Boltzmann方法

Zakharov-Rubenchik方程组常用于描述非线性介质中高、低频波间相互作用的波耦合现象。本文针对该方程组的数值求解问题,构建了一种格子Boltzmann方法的D1Q3演化模型,并利用Chapman-Enskog展开和多尺度分析技术,推导出了各个方向的平衡态分布函数和修正函数的具体表达式,从而将所建的演化模型准确恢复到宏观方程组。最后,通过数值算例证明了该方法的有效性。

非线性时间分数阶空气动力学方程的格子Boltzmann研究

针对非线性时间分数阶空气动力学方程,提出了一种新的格子Boltzmann模型.通过使用Chapman-Enskog展开和时间多尺度展开技术,得到了一系列不同时间尺度上的系列偏微分方程.选择合适的平衡态分布函数的矩,恢复出宏观方程,数值模拟出非线性时间分数阶空气动力学方程的解.数值实验表明,格子Boltzmann方法是研究非线性时间分数阶空气动力学方程的有效工具.

轴对称Boltzmann模型方程统一算法与空天飞行环境喷管流动模拟

拦截机动飞行器周围大范围区域存在主喷/侧喷流/羽流影响,而传统的纳维-斯托克斯(Navier-Stokes,N-S)方程不能很好模拟发动机喷管扩张段出口附近流动情况,需要一种新的方法来处理这种全流域流动问题。为解决该问题,针对特定轴对称喷管内流动,本文通过数学推导确立描述不同克努森数稀薄环境条件下的轴对称喷管内流动Boltzmann模型方程,初步建立适于该模型方程的数值格式与气体动理论统一算法。通过开展同轴圆筒间的定常/非定常旋转流动以及轴对称喷管内流动数值计算研究,发现统一算法计算流场与其他途径得到的结果吻合较好,验证了统一算法在全局克努森数喷管流动模拟的适应性和可靠性。通过与低密度风洞实验对比,喷管出口核心区羽流结构一致,羽流轴线压力分布一致,表明统一算法可以有效解决喷管入口压缩段到扩张段多流域混合,尤其是出口附近稀薄气体真空低压环境流动问题。

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.

Thermal equation of state for lattice Boltzmann gases

The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation （LBGKE） induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.

Derivation of lattice Boltzmann equation via analytical characteristic integral

A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.

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.

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.

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations （NPDEs） in the form ut ＋αuux ＋βu^nuz ＋γuxx ＋δuzxx ＋ζxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

基于元学习的气体放电等离子体电子Boltzmann方程数值求解

在气体放电等离子体中,电子的输运行为可由Boltzmann方程精确描述,该方程的解是许多等离子体仿真模型的基础。物理信息神经网络作为一种求解Boltzmann方程的新型方法,虽克服了传统数值方法网格剖分和方程离散的缺陷,但其参数空间规模大,在求解多任务时训练效率较低。为此,该文构建了一种基于元学习的双循环物理信息神经网络,在内循环中对多个Boltzmann方程求解任务进行优化训练,得到各任务优化后的元损失函数,用于在外循环中进行网络参数更新,从而提高网络在求解新任务时的计算效率。计算结果表明,基于元学习的双循环物理信息神经网络在求解新的Boltzmann方程时,网络损失函数值和L2误差值的下降速度均显著快于普通的物理信息神经网络。此外,该文还研究了网络容量和内循环迭代次数对Boltzmann方程多任务求解效率的影响,结果显示计算效率并不随网络容量的增大而提高,且受内循环迭代次数影响较小。

Numerical Solution of Nonlinear Klein-Gordon Equation Using Lattice Boltzmann Method

In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L2, L∞ and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.

线性Poisson-Boltzmann方程的虚单元L^(2)误差估计

针对一类线性Poisson-Boltzmann方程的虚单元L^(2)误差估计进行分析.首先引入正则化的线性Poisson-Boltzmann方程,将原问题转化为非奇性Poisson-Boltzmann方程.然后给出L^(2)范数的误差估计.最后在四边形和五边形混合多边形网格上进行数值实验,数值结果验证了理论分析的正确性.

INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS

We study the initial-boundary value problem for the one dimensional Euler-Boltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions （U△t,d, I△t,d） and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.

Stability of Global Solution to Boltzmann-Enskog Equation with External Force

In the presence of external forces depending only on the time and space variables, the Boltzmann-Enskog equation formally conserves only the mass of the . system, and its entropy functional is also nonincreasing. Corresponding to this type of equation, we first give some hypotheses of its bicharacteristic equations and then get some results about the stablity of its global solution with the help of two new Lyapunov functionals： one is to describe interactions between particles with different velocities and the other is to measure the L1 distance between two mild solutions. The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the veri-fication of the L1 stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.

A Self-Consistent Model on Cylindrical Monopole Plasma Antenna Excited by Surface Wave Based on the Maxwell-Boltzmann Equation

The paper analyzes the motion of electron in plasma antenna and the distribution of electromagnetic field power around the plasma antenna, and proposes a self-consistent model according to the structure of cylindrical monopole plasma antenna excited by surface wave;calculation of the model is based on Maxwell-Boltzmann equation and gas molecular dynamics theory. The calculation results show that this method can reflect the relationships between the external excitation power, gas pressure, discharge current and the characteristic of plasma. It is an accurate method to predicate and calculate the parameters of plasma antenna.

SHUBIN REGULARITY FOR THE RADIALLY SYMMETRIC SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH DEBYE-YUKAWA POTENTIAL

In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.