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.     

Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force

For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.     

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.     

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.     

The Spatially Homogeneous Relativistic Boltzmann Equation with a Hard Potential

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in spatially flat Robertson-Walker spacetimes.     

Kac方程的解及其性质

研究Kac方程的初值问题．证明了该类方程存在唯一的全局分布解．并且使用一种新的线性化方法证明了该类方程的解具有相应的多项式衰减性．     

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].     

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.     

偏微分方程求解的一种新颖方法-格子Boltzmann模型

介绍了一种偏微分方程求解的一种新颖方法格子Boltzmann模型,详细分析了它的基本理论和基本原理.并通过不可压Navier-Stokes方程组和二维含源项扩散方程的数值模拟计算实例,说明格子Boltzmann方法的有效性,展示了广阔的应用前景,为今后更深入的研究和广泛应用提供参考.     

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Corrections are given to the above-mentioned article.     

组合KdV方程带修正函数的格子Boltzmann模型

采用一种带修正函数的新格子Boltzmann模型模拟了组合Kdv方程,分析了由此得出的迭代格式的单调性和稳定性,并且得到了格式的单调性条件.在文中的单调性条件下,迭代格式是L∞稳定的.文中的数值模拟结果表明该格式是可行的.     

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

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.     

Optimal time decay of the Boltzmann equation with frictional force

In this paper, we use the combination of energy method and Fourier analysis to obtain the optimal time decay of the Boltzmann equation with frictional force towards equilibrium. Precisely speaking, we decompose the equation into macroscopic and microscopic partitions and perform the energy estimation. Then, we construct a special solution operator to a linearized equation without source term and use Fourier analysis to obtain the optimal decay rate to this solution operator. Finally, combining the decay rate with the energy estimation for nonlinear terms, the optimal decay rate to the Boltzmann equation with frictional force is established.     

Thermal Creep Flow for the Boltzmann Equation

It is known that the Boltzmann equation has close relation to the classical systems in fluid dynamics. However, it provides more information on the microscopic level so that some phenomena, like the thermal creep flow, can not be modeled by the classical systems of fluid dynamics, such as the Euler equations. The author gives an example to show this phenomenon rigorously in a special setting. This paper is completely based on the author's recent work, jointly with Wang and Yang.     

Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.     

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.     

组合BBM-Burgers方程孤波解的稳定性

讨论了组合BBM-Burgers方程的孤立波解在Liapunov意义下的稳定性,证明了孤波解在初始微扰满足一定条件时具有条件稳定性.     

关于Boltzmann方程的空间均匀的ES模型

在Prandtl数Pr∈[2/3,∞)的情况下,我们讨论了Boltzmann方程的空间均匀的椭圆统计模型.首先,我们建立了解的存在唯一性.其次,我们证明了该解收敛到平衡态并给出了其Maxwell分布型的下界估计.最后,我们给出了熵等式从而证明了该方程的熵是衰减的.