Numerical and asymptotic analyses of the Boltzmann equation-Their engineering and mathematical aspects-

1.在法国群(FG)的帮助下，日本群(JG)通过对Boltzmann方程的系统渐近分析，建立了描述具有有限马赫数的轻微稀薄气体流动的一般行为的渐近理论(流体动力学方程、滑移边界条件和边界附近的Knudsen层修正)。该理论还从Boltzmann方程出发，系统地推导了流体力学中的粘性边界层方程。这一结果启发了边界层方程中FG的数学研究。JG和FG共同致力于从多粒子系统(刘维尔方程)直接推导流体动力学方程(欧拉方程组)。使用不同的方法，每个小组都成功地进行了正式推导。此外，JG反思了从多粒子系统中形式化地推导玻尔兹曼方程，并为FG提供了一个清晰而系统的…更多的抽搐派生。这为玻尔兹曼方程的推导提供了一种新的数学证明。JG与FG合作，用数值分析的方法研究了边界温度不连续引起的稀薄气体流动，阐明了大范围稀薄气体流动的行为。在这种流动中，分子速度分布函数的不连续性从边界传播到气体中。作为数学研究的第一步，基于一个与Boltzmann方程具有相似性质但结构简单得多的线性输运方程，JG和FG以严谨的数学方法共同分析了介质中不连续的传播。JG建立了包括流体动力学方程(如标准欧拉方程和N-S方程)在内的一般守恒方程的动力学解格式的系统理论，为FG的进一步数学研究提供了理论基础。此外，JG在FG的帮助下，成功地获得了发生蒸发或冷凝的界面上流体动力学方程的严格边界条件。JG为FG提供了描述气体混合物幽灵效应的流体动力学类型方程组，FG研究了它的数学性质。较少

1. With the help of the French group (FG), the Japanese group (JG) has established the asymptotic theory (the fluid-dynamic equations, the slip boundary conditions, and the Knudsen-layer corrections near the boundary) describing the general behavior of slightly rarefied gas flows with finite Mach numbers by means of a systematic asymptotic analysis of the Boltzmann equation. This theory also contains the systematic derivation of the viscous boundary-layer equation in fluid dynamics from the Boltzmann equation. The results inspired the mathematical study of FG on the boundary-layer equation.2. JG and FG have jointly worked on the direct derivation of the fluid-dynamic equations (the Euler system of equations) from the many-particle system (the Liouville equation). Using different approaches, each group has succeeded in the formal derivation. In addition, JG reflected on the formal derivation of the Boltzmann equation from the many-particle system and provided FG with a clear and systema … More tic derivation. This leads to a possibility of a new mathematical proof of the derivation of the Boltzmann equation.3. Cooperating with FG, JG investigated the rarefied gas flow induced by the discontinuity in the boundary temperature by means of a numerical analysis and clarified the behavior of the flow for a wide range of gas rarefaction. In this flow, the discontinuity of the molecular velocity distribution function propagates from the boundary into the gas. As the first step of mathematical study, on the basis of a linear transport equation that possesses a similar property to the Boltzmann equation concerning the propagation of discontinuity but has a much simpler structure, JG and FG jointly analyzed, with mathematical rigor, the propagation of discontinuity in the medium.4. JG constructed a systematic theory for the kinetic solution scheme of the general conservation equations including fluid-dynamic equations (e.g., the standard Euler and Navier-Stokes equations) and provided FG with the theory for further mathematical study. In addition, JG, with the help of FG, succeeded in obtaining mathematically rigorous bounds of the boundary conditions for the fluid-dynamic equations on the interface where evaporation or condensation is taking place.5. JG provided FG with the system of fluid-dynamic type equations that describe the ghost effect for gaseous mixtures, and FG investigated its mathematical properties. Less