On blowup for gain‐term‐only classical and relativistic Boltzmann equations

We show that deletion of the loss part of the collision term in all physically relevant versions of the Boltzmann equation, including the relativistic case, will in general lead to blowup in finite time of a solution and hence prevent global existence. Our result corrects an error in the proof given (Math. Meth. Appl. Sci. 1987; 9 :251–259), where the result was announced for the classical hard sphere case; here we give a simpler proof which applies much more generally. Copyright © 2004 John Wiley & Sons, Ltd.     

Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman–Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.     

A lattice Boltzmann model for blood flows

A lattice Boltzmann model for blood flows is proposed. The lattice Boltzmann Bi-viscosity constitutive relations and control dynamics equations of blood flow are presented. A non-equilibrium phase is added to the equilibrium distribution function in order to adjust the viscosity coefficient. By comparison with the rheology models, we find that the lattice Boltzmann Bi-viscosity model is more suitable to study blood flow problems. To demonstrate the potential of this approach and its suitability for the application, based on this validate model, as examples, the blood flow inside the stenotic artery is investigated.     

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.     

Boltzmann方程的L~1稳定性

本文研究在小初值情况下Boltzmann方程经典解的L1稳定性.借助于Toscani等人所给的估计,对硬位势和软位势作了讨论,完善了[2]中关于硬球模型的结果.     

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.     

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.     

Lattice Boltzmann method for slip flow heat transfer in circular microtubes: Extended Graetz problem

Slip flow heat transfer in circular microtubes is of fundamental interest and practical importance. However, to the best knowledge of the present author, there is no open publication of developing simple and efficient lattice Boltzmann (LB) models on such topic. To bridge the gap, in this paper a simple LB model, which is based on our recent work [S. Chen, J. Tölke, M. Krafczyk, Simulation of buoyancy-driven flows in a vertical cylinder using a simple lattice Boltzmann model, Phys. Rev. E 79 (2009) 016704], is designed. In addition, the recently developed Langmuir slip model [S. Chen, Z.W. Tian, Simulation of thermal micro-flow using lattice Boltzmann method with Langmuir slip model, Int. J. Heat Fluid Flow 31 (2010) 227-235], which possesses a clear physical picture and keeps the Reynolds analogy, is extended to capture velocity slip as well as temperature jump in microtubes. The feasibility and capability of the present model are validated by the extended Graetz problem, which is a benchmark prototype for forced convection heat transfer in circular microtubes.     

关于弱非线性、弱涨落的Boltzmann方程链的经典解的存在性问题

本文按照Grad~([6])指出的方法具体地证明了弱非线性、弱涨落的Boltzmann方程链的经典解的存在性.     

Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

The Boltzmann equation with potential force in the whole space

In this paper, we consider the Cauchy problem of the Boltzmann equation with potential force in the whole space. When some more natural assumptions compared with those of the previous works are made on the potential force, we can still obtain a unique global solution to the Boltzmann equation even for the hard potential cases by energy method, if the initial data are sufficiently close to the steady state. Moreover, the solution is uniformly stable. Copyright © 2010 John Wiley & Sons, Ltd.     

Boltzmann方程的永久型解

该文讨论如下空间非均匀的Boltzmann方程\frac{\partial f}{\partial t} + \xi\cdot \nabla_{x}f(t,x,\xi) = Q(f, f).在角截断的硬位势情况下, 对初值接近行波Maxwell分布时,作者利用一种新的迭代方法, 证明了该方程存在一个非负的永久型解. 因此在空间区域无界的情形下,该文对Villani的猜测给出了否定的回答[12, 13].     

Lattice-Boltzmann Models for heat transfer

A multispeed heat transfer lattice Boltzmann model is presented. The model possesses the perfect gas state equation with arbitrary special heat ratio. The macroscopic conservation equations are derived by the Chapman-Enskog method. The one dimensional simulation for the sinusoidal energy distributions are compared with the theoretical results, showing good agreement. The theoretical conductivity in the energy equation is in accordance with the simulations.     

The stationary Boltzmann equation for a two‐component gas for soft forces in the slab

The stationary Boltzmann equation for soft forces in the context of a two‐component gas is considered in the slab. An existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries in a renormalized sense. Weak L¹ compactness is extracted from the control of the entropy production term. Trace at the boundaries is also controlled. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

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.     

Vortices in a Gas of Hard Spheres

We approximately describe the transition regime between two vortex-type flows in a gas of hard spheres. Such flows rotate as solid bodies about their axes, which in turn move translationally with arbitrary linear velocities. We study the asymptotic behavior of the integral norm of the discrepancy between the two sides of the Boltzmann equation under a special choice of hydrodynamic parameters of the distribution.     

Properties of stationary solutions of a generalized Tjon-Wu equation

A generalized version of the Tjon-Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ_∗ of the equation has the following property: Either μ_∗ is supported at one point or suppμ_∗=[0,∞). Moreover we show that in the second case the distribution function of μ_∗ is continuous. Some open questions are discussed.     

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.     

三维矩形槽道中颗粒沉降的数值模拟

采用三维格子Boltzmann方法对矩形通道中的颗粒沉降进行了模拟研究．单颗粒沉降的模拟结果表明,颗粒最终的稳定沉降位置沿槽道中心线,不受颗粒初始位置和直径的影响．颗粒和壁面之间的两体相互效应可以用无因次沉降速度定量描述,无因次沉降速度的模拟结果和实验结果定量上吻合一致．模拟分析了双颗粒沉降的DKT（drafting, kissing and tumbling）过程,探讨了颗粒直径比以及壁面效应对DKT过程的影响．模拟发现当颗粒直径相同时,双颗粒的沉降过程为周期性的DKT过程,从而形成双螺旋形式的沉降轨迹,此螺旋沉降轨迹的频率和振幅受颗粒初始位置影响．从模拟结果中还得到颗粒群的最终稳定构型,并进行了构型对比分析．最后对包含49个颗粒的颗粒群沉降行为进行了模拟,说明多体相互作用在对称性的情况下可以简化．