Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.     

On space-time estimates for the Vlasov-Poisson system

The motion of a collisionless plasma is described by the Vlasov-Poisson system. A space-time identity for one-dimensional plasmas with two species of charge is here generalized to higher dimensions with spherical symmetry. Some estimates on large time behavior are obtained.     

LESSER KNOWN MIRACLES OF BURGERS EQUATION Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.     

The Kinetic Transport Equation in the Case of Compton Scattering

We improve the well-known form of the transport equation accounting for Compton scattering. We pose and study the direct problem of finding the radiation density distribution for given characteristics of a medium and known density of exterior sources. We prove existence and uniqueness theorems for a solution to the boundary value problem under consideration. The character of constraints corresponds mostly to the process of photon migration in a substance whose characteristics vary continuously with the space and energy variables. Unlike similar results, the assertions are proven without using the traditional inequalities for the coefficients of the transport equation.     

Global weak solutions to the relativistic BGK equation

Abstract

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of stationary solutions in kinetic models with Gaussian thermostats

The thermostatted kinetic framework has been recently proposed in [C. Bianca, Nonlinear Analysis: Real World Applications 13 (2012) 2593‐2608] for the modeling of complex systems in the applied sciences under the action of an external force field that moves out of equilibrium the system. The framework consists in an integro‐differential equation with quadratic nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with the existence of stationary solutions proof. The main result is gained by fixed point and measure theory arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

A differential equation‐based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.     

Plasma diffusion and relaxation in a magnetic field

A Fokker–Planck-type kinetic equation modeling a test-particle weakly interacting with an electrostatic plasma, in the presence of a magnetic field B, is analytically solved. Explicit expressions are obtained for variable moments and particle density as a function of time. Diffusion in space is classical, characterized by a particle MSD varying as B⁻², in agreement with previous results.     

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

The stationary Boltzmann equation for hard forces in the context of a two‐component gas is considered in the slab. An L¹ existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries. Weak L¹ compactness is extracted from the control of the entropy production term. Trace at the boundaries are also controlled. Copyright © 2007 John Wiley & Sons, Ltd.     

Conservative numerical method for solving the averaged Boltzmann equation

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step.     

A KINETIC APPROACH TO SCALAR MULTI-DIMENSIONAL CONSERVATION LAWS

ＡＫＩＮＥＴＩＣＡＰＰＲＯＡＣＨＴＯＳＣＡＬＡＲＭＵＬＴＩ－ＤＩＭＥＮＳＩＯＮＡＬＣＯＮＳＥＲＶＡＴＩＯＮＬＡＷＳ￥ＨａＪｉａｘｉｎ（胡家信）（ＹｏｕｎｇＳｃｉｅｎｔｉｓｔｓＬａｂｏｒａｔｏｒｙｏｆＷｕｈａｎＩｎｓｔ．ｏｆＭａｔｈ．Ｓｃｉ．，Ａｃａｄ...     

On the Linearized Balescu–Lenard Equation

The Balescu–Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation.     

Kinetic model of the Boltzmann equation with limiting gas flow regimes at low Knudsen numbers

A new model of the Boltzmann kinetic equation is constructed that describes both slow nonisothermal and Navier-Stokes continuum gas flows. The model is used to compute the slow nonisothermal flow past a circular cylinder. It is shown that the force exerted by the gas on the cylinder is affected by thermal stresses.     

非线性玻尔兹曼方程的傅里叶谱方法

玻尔兹曼方程作为空气动理学中最基本的方程之一,是连接微观牛顿力学和宏观连续介质力学的重要桥梁.该方程描述了一个由大量粒子组成的复杂系统的非平衡态时间演化:除了基本的输运项,其最重要的特性是粒子间的相互碰撞由一个高维,非局部且非线性的积分算子来描述,从而给玻尔兹曼方程的数值求解带来非常大的挑战.在过去的二十年间,基于傅里叶级数的谱方法成为了数值求解玻尔兹曼方程的一种很受欢迎且有效的确定性算法.这主要归功于谱方法的高精度及它可以被快速傅里叶变换加速的特质.本文将回顾玻尔兹曼方程的傅里叶谱方法,具体包括方法的导出,稳定性和收敛性分析,快速算法,以及在一大类基于碰撞的空气动理学方程中的推广.     

关于非线性双曲波动变分方程的动力学形式

受到[3],[4]和[5]的启发,本文对应于某种波动变分方程的弱解,给出了该方程的动力学 形式,此方程来源于长原子液晶运动双极链中的长波以及其它邻域的研究.     

Numerical analysis of the spiral Couette flow of a rarefied gas between coaxial cylinders

An implicit quasi-monotone second-order accurate method is proposed for analyzing the spiral Couette flow of a rarefied gas between coaxial cylinders. The basic advantages of the method over the conventional method of stationry iterations are that the former is conservative with respect to the collision integral, has a simple software implementation for any types of boundary conditions, and applies to a wide range of Knudsen numbers.     

Fluid-dynamic equations for reacting gas mixtures

Starting from the Grad 13-moment equations for a bimolecular chemical reaction, Navier-Stokes-type equations are derived by asymptotic procedure in the limit of small mean paths. Two physical situations of slow and fast reactions, with their different hydrodynamic variables and conservation equations, are considered separately, yielding different limiting results.This work was performed in the frame of the activities sponsored by MIUR (Project Mathematical Problems of Kinetic Theories), by INdAM, by GNFM, by the University of Parma (Italy), and by the European TMR Network Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis.This revised version was published online in April 2005 with a corrected issue number.     

Discrete velocity models with general boundary conditions in a slab

We consider a gas between parallel plates, described by a discrete velocity model. At the boundary, we impose the most general linear boundary conditions which preserve mass. Using a fixed‐point theorem we prove the existence of at least one parameter family of solutions, continuous in x. The velocities are assumed to have a non‐zero component in the direction orthogonal to the boundaries. Copyright © 2001 John Wiley & Sons, Ltd.