Dawan Mustafa, Bernt Wennberg
The spatially homogeneous BGK equation is obtained as the limit if a model of a many particle system, similar to Mark Kac's charicature of the spatially homogeneous Boltzmann equation.
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In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and that the one-particle distribution function converges to the solution of the BGK equation.
We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $α\in [0,1]$, and the linearinteraction with the reservoirs by $(1-α)$, we prove that for all $α$ close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is \emph{unique}, and there are no spatially non-uniform NESS with a spa…
We consider a stochastic $N$-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the …
From particle systems to the BGK equation
本文研究DSMC方法对应的非均匀Kac随机粒子系统，通过修改动力学规则推导得出BGK方程，为BGK方程的原始论证提供了数学阐释，补充了相关领域的已有研究结果。
A space-dependent Boltzmann-BGK model for gas mixtures and its hydrodynamic limits
提出空间依赖的混合Boltzmann-BGK模型，保证守恒与熵耗散，并在三种流体动力学极限下分析其行为。
Hydrodynamic Limit of a Viscous BGK Model of the Boltzmann Equation
本文研究Boltzmann方程粘性BGK模型的流体动力学极限，利用空间Fisher信息导出界证明向粘性Euler方程的严格收敛，并分析单/多物种情形下带熵不等式的Euler方程形式推导，与对称双曲热力学相容理论比较。
Exact Hydrodynamic Manifolds for the Linear Boltzmann BGK Equation I: Spectral Theory
本研究对三维线性Boltzmann BGK算子做完整谱分析，得特征值显式超越方程，证实临界波数存在，证明有限维良好分离线性流体动力学流形存在，结果可作相关近似理论验证基准并为谱闭合算子提供基础。
In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body-attitude of an agent is modelled by a rotation matrix in $\mathbb{R}^3$ as in [Degond, Frouvelle, Merino-Aceituno, 2017]. The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit…
Asymptotic Relaxation of Moment Equations for a Multi-Species, Homogeneous BGK Model
本文研究了空间均匀多组分BGK模型的宏观速度、能量和温度矩方程，通过建立组分温度的正下界，证明了解的全局存在唯一性及指数衰减至稳态，并给出数值验证。
Ellipsoidal BGK model near a global Maxwellian
本文研究了椭圆BGK模型在初始数据为全球麦克斯韦分布小扰动时经典解的存在性，通过分析发现该模型退化程度与原始BGK模型或玻尔兹曼方程在-1/2<ν<1范围内相似。
Binary particle collisions with mass exchange
本文研究了具有质量交换的二元粒子碰撞动力学模型，推导了满足H定理的Boltzmann碰撞算子，并在流体力学极限下计算了粘性修正，证明其符合非平衡态热力学形式体系及Onsager倒易关系。
Limit Consistency of Lattice Boltzmann Equations
该研究提出极限一致性概念作为LBE与PDE系统一致性证明的模块化部分，以NSE为范例，通过嵌套泰勒展开与有限差分离散BGK玻尔兹曼方程，追踪离散状态与截断误差，证明各方程族的极限一致性，给出保持连续极限的LBM数值格式通用自上而下推导方法。
The Fourier-transformed version of the BGK model in one-dimension is solved in order to determine the general solution's asymptotics. The ultimate goal of this paper is to demonstrate that the solution to the model Boltzmann possesses a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial first moment of the gas. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case's method of elementary solutions, we find analytic time-dependent solu…
This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-va…
In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium d…
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