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

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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