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

1. 在法国（FG）小组的帮助下，日本（JG）小组通过对玻尔兹曼方程的系统渐近分析，建立了描述有限马赫数下微稀薄气体流动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）。该理论还包含了由玻尔兹曼方程系统推导流体动力学中的粘性边界层方程。所得结果启发了边界层方程FG的数学研究。JG和FG共同致力于从多粒子系统（Liouville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形式化推导。此外，JG对从多粒子系统出发的玻尔兹曼方程的形式化推导进行了反思，为FG提供了清晰、系统的推导。这就有可能对玻尔兹曼方程的推导作出新的数学证明。JG与FG合作，通过数值分析研究了边界温度不连续引起的稀薄气体流动，阐明了大范围气体稀薄流动的行为。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，JG和FG基于一个与玻尔兹曼方程关于不连续传播的性质相似但结构更简单的线性输运方程，用数学的严谨性共同分析了介质中的不连续传播。JG为包括流体动力学方程在内的一般守恒方程（如标准欧拉方程和Navier-Stokes方程）的动力学解方案构建了系统的理论，并为FG进一步的数学研究提供了理论依据。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边界条件的数学上严格的边界。JG向FG提供了描述气体混合物幽灵效应的流体动力学型方程组，FG研究了其数学性质。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