Loeb solutions of the boltzmann equation

Existence problems for the Boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory. The present paper considers the spatially periodic case with L¹ initial data. The main result is that the Loeb subsolutions obtained in a preceding paper are shown to be true solutions. The proof relies on the observation that monotone entropy and finite energy imply Loeb integrability of non-standard approximate solutions, and uses estimates from the proof of the H-theorem. Two aspects of the continuity of the solutions are also considered.     

Parametric solutions of the eigenvalue equation for a convectively cooled slab

The convective cooling of a slab by an ambient fluid under the most general linear and homogeneous boundary conditions is considered. For the roots of the corresponding transcendental eigenvalue equation an explicit formula is written down in a parametric form. The practical consequences of this representation, among them certain “singular solutions” which cannot be obtained by a direct numerical treatment of the original eigenvalue equation, are discussed in detail.     

Scattering kernels for the boundary conditions of the boltzmann equation on the moving boundary of two-phase systems

Summary This paper deals with the problem of the boundary conditions for the Boltzmann equation on the moving boundary between a liquid and a vapour phase with condensation and vaporization in non-equilibrium conditions. A physical theory is proposed on the basis of the mathematical theory of the scattering kernels, of the kinetic theory and of the statistical theory of phase-transition. The mathematical result of this theory consists in a set of equations correlating, by means of integral operators, the distribution function of the molecules leaving the liquid surface to the one of the molecules arriving on the surface of the liquid phase.

---

Sommario Il presente lavoro studia il problema della condizione al contorno per l'equazione di Boltzmann sulla frontiera mobile fra la fase liquida e quella di vapore di un sistema bifasico in condizioni di disequilibrio termodinamico. Si propone, sulla base dei metodi matematici della teoria cinetica e della teoria degli «scattering kernels» e sulla base dei risultati precedenti di una teoria statistica di transizione di fase, una teoria fisico-matematica per lo studio del problema. In termini matematici si ottiene un operatore che correla la funzione di distribuzione delle molecole che lasciano la superficie liquida alla funzione di distribuzione delle molecole che arrivano sulla superficie. Tale operatore rappresenta la formulazione della condizione al contorno.

---

---

This work has been realized within the activities of the Italian Council for the Research, Gruppo Nazionale per la Fisica Matematica, and presented at the Euromech Colloquium 86 on the «Boltzmann Equation».     

Solution of the inverse problem of thermal conductivity for a melting slab with entrainment

This article proposes an approximate solution to the inverse problem of the Stefan type for a finite region with arbitrary boundary and initial conditions. A comparison with exact solutions is made.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 132–137, May–June, 1973.     

Study of stable and unstable jet flows on the basis of the boltzmann equation

Free supersonic underexpanded jets are studied using a direct method conservative splitting scheme for solving the Boltzmann equation. Numerical solutions for a jet flowing into a vacuum and into a fluid-filled space are presented for the following ranges of the parameters: Knudsen number 10⁻⁶<Kn<∞ and pressure ratio 10<n<∞. The solutions are compared with experimental data. Instabilities associated with free turbulence effects in the mixing layer are detected for low Kn numbers. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 153–157, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00829).     

An approximate solution to the integral radiative transfer equation in an optically thick slab

We consider the problem of solving the integral form of the radiative transfer equation in an atmosphere with optical thickness τ₀1. We propose a method transforming this problem in the same problem posed in an atmosphere with optical thickness τ₁τ₀. An error over- estimation is derived. To cite this article: A. Amosov et al., C. R. Mecanique 331 (2003).     

Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

It is proved that under a physically realistic truncation of the collision kernel, the Boltzmann equation in the one-dimensional slab [0,1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. This solution satisfies the boundary conditions almost everywhere, and has, at worst, exponentially growing total energy.     

Boundedness,continuity, positivity and dominance of the solution of the neutron boltzmann equation

Summary In this paper the behavior of the solution of the integral form of the neutron transport Boltzmann equation is investigated for slab and sphere geometry. The study of the properties of the integral transport operator K leads to interesting results concerning the following subjects: boundedness, continuity and positivity of the total flux; existence of a dominant critical eigenfunction; spectral properties of K; dependence of the critical eigenvalue on the dimensions of slab and sphere; uniform convergence of the Neumann series in subcritical conditions; approach to the criticality; summability of the solutions and neutron conservation.

---

Sommario In questo lavoro viene esaminato il comportamento della soluzione della forma integrale dell'equazione di Boltzmann del trasporto dei neutroni nel caso di una lastra piana e di una sfera. Lo studio delle proprietà dell'operatore integrale del trasporto K porta ad interessanti risultati riguardanti i seguenti argomenti: limitatezza, continuità e positività del flusso totale; esistenza di una autofunzione dominante critica; proprietà spettrali di K; dipendenza dell'autovalore critico dalle dimensioni della lastra piana e della sfera; convergenza uniforme della serie di Neumann in condizioni sottocritiche; avvicinamento alla criticità; sommabilità delle soluzioni e conservazione dei neutroni.

---

---

Work performed under contract C.N.R.     

Solution of a dual integral equation for crack and indentation problems

Using the Sonine integrals, some dual integral equations with the Bessel kernels were reduced to a single integral equation. Then the closed form solutions of these dual integral equations were obtained. Based on the method, the anti-plane shear of an elastic layer was solved exactly.     

Solution of a nonlinear Boltzmann equation for neutron transport in L1 space

The aim of this paper is to give a constructive method for the solution of the Boltzmann equation for neutron transport in a bounded space domain subject to typical boundary and initial conditions. Sufficient conditions are given to insure the existence of a unique solution. The method entails the use of a semiinner product in a Banach space together with successive approximations, and leads to a recursion formula for the calculation of approximate solutions and error estimates. The linearized Boltzmann equation for neutron transport is included as a special case.     

Solution to the form of Poisson equation by the boundary element methods

In this paper, the domain integral of the form of Poisson equation is translated into complete boundary integral by the fundamental solution of higher-order Laplace operator, the dimensions of the problem can be contracted into one. The numerical examples for Stokes equations show that this method is efficient.     

Solution for dissimilar elastic inclusions in a finite plate using boundary integral equation method

This paper studies the boundary value problem for a finite plate containing two dissimilar inclusions. The matrix and the two inclusions have different elastic properties. The loadings applied along the outer boundary are in equilibrium. The mentioned problem is decomposed into three boundary value problems (BVPs). Two of them are interior BVP for the elastic inclusions, while the other is a BVP for the triply-connected region. Three problems are connected together through the common displacements and tractions along the interface boundaries. Explicit form for the complex variable boundary integral equation (CVBIE) is derived. After discretization of relevant BIEs, the solutions are evaluated numerically. Three numerical examples for different elastic constant combinations are provided.     

Solution to radiative transfer equation in a 3-D rectangular enclosure due to discontinuous heat flow divergence

In this paper a new model and computer code is presented by considering singular and discontinuous heat flow divergence. A hybrid model including Smith’s WSGG model and Coppale and Vervish’s model is used for calculating gas radiative properties. Energy equation is solved simultaneously to reach temperature field which specify gas radiative properties. S ₈ order of discrete ordinate method is used to solve RTE. It is assumed that walls of enclosure are gray, diffuse and opaque with specified temperature. Boundary conditions are corrected in each iteration that change temperature field.     

Solution of the Chaplygin equation in plane-parallel transonic flow with a normal shock

In [1], for the case in which the exact Chaplygin equation of motion is replaced by the Tricomi equation, Frankl constructed an example of plane-parallel transonic flow with a normal shock terminating within the flow (later this example was extended by Biibosunov to obtain a curved shock [2]). This flow is an analog of the point vortex flow and has concentric circles as streamlines, and a unique dependence of the velocity vector on the Cartesian coordinates is provided everywhere. In the present paper this example is extended to the case of the Chaplygin equation, and the shock is considered to be normal.The author wishes to thank S. V. Fal'kovich for discussion of the study results and helpful advice.     

徐州矿区采空塌陷区公路工程地质灾害与防治对策分析

采用格子Boltzmann方法模拟可变形膜与周围流体的相互作用. 分析了格子Boltzmann 方法中的边界处理方法和边界受力的计算方法，并且用此方法计算流场中可变形膜的受力. 可将离散化后的膜看作一系列的质点，从而得到膜的动力学方程. 将可变形膜在流场中受到 的力引入方程中，可以计算膜的变形. 求解了几种不同情况下，膜的形状随时间的变化. 发现，如果可变形膜非常软或者非常硬，经过足够长的时间后，膜的形状会接近一 条直线，即回到初始状态. 模拟过程是二阶精度的.