Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

In this paper Green’s functions for the Boltzmann equation around a global Maxwellian are used to construct the non-characteristic nonlinear Knudsen layers as well as their time-asymptotic stability. Furthermore, the detailed pointwise structures, nonlinear wave couplings, and wave interactions with boundary are studied.     

Boundary Singularity of Moments for the Linearized Boltzmann Equation

We study the boundary singularity for stationary solutions of the linearized Boltzmann equation with hard-sphere potential. An asymptotic estimate for the gradient of the moments is established, which shows the logarithmic singularity near the boundary. Our formula holds for the solutions of the Milne and Kramers problems obtained by Bardos-Caflish-Nicolaenko (Commun. Pure Appl. Math. 49:323–452, 1986). Our theorem requires the Hölder continuity of the boundary data. In particular, it applies to the complete condensation problem for half space.     

Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation

In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

General Properties of Collision Integral Kernels of the Nonlinear Boltzmann Equation

Technical Physics - We consider general properties of kernels of the inverse collision integral, as well as kernels of integral operators appearing in the expansion of the collision integral in...     

Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions

By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudyński and Ekiel-Jeżewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satisfying finite mass, energy and entropy. This is in fact an extension of the result of Dudyński and Ekiel-Jeżewska to the case of the relativistic Boltzmann equation with hard interactions. This work was supported by NSFC 10271121 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China, and sponsored by joint grants of NSFC 10511120278/10611120371 and RFBR 04-02-39026.     

应用于非线性热传导方程的格子玻尔兹曼方法

将格子玻尔兹曼方法应用于非线性热传导方程的求解,详细推导一种新的Lattice Boltzmann模型,并给出新方法所对应的多尺度方案和宏观量形式.导热系数与温度之间满足多项式函数关系,计算中模拟了不同的参数情况,并与线性热传导方程的理论解进行比较.新的Lattice Boltzmann方法展现出极大的灵活性和普适性,具有很好的应用前景.     

Generalized Boundary Conditions in an Existence and Uniqueness Proof for the Solution of the Non-Stationary Electron Boltzmann Equation by Means of Operator-Semigroups

Recently, based on the semigroup approach a new proof was presented of the existence of a unique solution of the non-stationary Boltzmann equation for the electron component of a collision dominated plasma. The proof underlies some restrictions which should be overcome to extend the validity range to other problems of physical interest. One of the restrictions is the boundary condition applied. The choice of the boundary condition is essential for the proof because it determines the range of definition of the infinitesimal generator and thus the operator semigroup itself. The paper proves the existence of a unique solution for generalized boundary conditions, this solution takes non-negative values, which is necessary for a distribution function from the physical point of view.     

On a Taylor-Couette Type Bifurcation for the Stationary Nonlinear Boltzmann Equation

This paper studies the stationary nonlinear Boltzmann equation for hard forces, in a Taylor-Couette setting between two coaxial, rotating cylinders with given indata of Maxwellian type on the cylinders. A priori L ^q -estimates are obtained, and used to prove a Taylor type bifurcation with isolated solutions and a hydrodynamic limit control, based on asymptotic expansions together with a rest term correction. The positivity of such solutions is also considered.     

A Half-space Problem for the Boltzmann Equation with Specular Reflection Boundary Condition

There are many open problems on the stability of nonlinear wave patterns to the Boltzmann equation even though the corresponding stability theory has been comparatively well-established for the gas dynamical systems. In this paper, we study the nonlinear stability of a rarefaction wave profile to the Boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut-off. The analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection, and the decomposition on both the solution and the Boltzmann equation introduced in [24, 26] for energy method.Research supported by the RGC Competitive Earmarked Research Grant, CityU 1142/01P.Research supported by the JSPS Research Fellowship for Foreign Researchers, the National Natural Science Foundation of China (10329101, 10431060), the National Key Program for Basic Research of China under grant 2002CCA03700, and the grant from the Chinese Academy of Sciences entitled Yin Jin Guo Wai Jie Chu Ren Cai Ji Jin.     

Existence of a Stationary Wave for the Discrete Boltzmann Equation in the Half Space

We study the existence and the uniqueness of stationary solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation. Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models. Received: 7 December 1998 / Accepted: 27 April 1999     

On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.     

A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

Abstract

A two-phase free boundary problem associated with nonlinear heat conduction is considered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution is proved for small times and a particular exact solution is discussed.     

Global Existence for the Einstein–Boltzmann Equation in the Flat Robertson–Walker Spacetime

The initial value problem for the Einstein–Boltzmann equation in the spatially homogeneous and isotropic case is considered. The global in time mild solution is obtained. Received: 18 May 1998 / Accepted: 23 November 1998     

Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation II: The Low Density Regime

In this paper we analyse the asymptotic dynamics of a system of N identical quantum particles in a low-density regime. Our approach follows the strategy introduced by the authors in a previous work,⁽²⁾ to treat the simpler weak coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula, in the spirit of the paper by Lanford.⁽³²⁾ For short times and small interaction potential, we rigorously prove that a subseries of the complete perturbative series, converges to the solution of the nonlinear Boltzmann equation that is physically relevant in the context. An important point is that we completely identify the cross-section entering the limiting Boltzmann equation, as being the Born series expansion of quantum scattering.As in ref. 2, our convergence result is only partial, in that we merely characterize the asymptotic behaviour of a subseries of the complete original perturbative expansion. We only have plausibility arguments in the direction of proving that the terms we neglect, when going from the original series to its associated subseries, are indeed vanishing in the limit.The present study holds in any dimension d ≥ 3.     

On the Quantum Boltzmann Equation

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreenessin the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.     

Existence, Stability, and Convergence of Solutions of Discrete Velocity Models to the Boltzmann Equation

We prove the convergence of finite-difference approximations to solutions of the Boltzmann equation. An essential step is the proof of convergence of discrete approximations to the collision integral. This proof relies on our previous results on the consistency of this approximation. For the space-homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the Boltzmann equation. In the space-dependent case we prove weak convergence to DiPerna–Lions solutions.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.