Boltzmann Equation and Multi-Dimensional Shock Interactions in Gas Dynamics

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The goal of this project is to study nonlinear waves and boundary phenomena in gas flow. The project has two parts: (1) The study of boundary effects from the point of view of kinetic theory, in particular, the Boltzmann shock layer, initial layer, and their interactions, and the thermal, curvature, and condensation effects of the solid boundary; (2) Study of multidimensional gas flow with shocks, in particular, the effects of shock reflections from a solid boundary on the overall flow patterns for the compressible Euler equations. To analyze boundary effects from the point of view of kinetic theory is physically natural, as the inclusion of the microscopic velocity in the Boltzmann solutions allows for physically realistic modeling of the solid boundary conditions. It is mathematically challenging, as the boundary condition, such as the Maxwell type of the interpolation of specular and diffusion reflection conditions, makes possible the study of the rich interactions of particles and fluid waves. This project will pursue local analysis of the physical phenomena. Multi-dimensional gas flow with shocks is the consequence of the strongly nonlinear effect of compression and the global interaction of the gas with the solid boundary. Mathematical analysis thereby becomes highly nonlinear and global. New extremal principles are needed for the construction of solutions with free boundary for nonlinear partial differential equations of mixed types. Already, an Ellipticity Principle has helped to explain that, with given boundary condition at far field, the uniqueness is possible only for self-similar flow, and not for stationary flows. The study of the boundary effects on gas flow is of great importance for engineering practices. Mathematical analysis is needed for quantitative and qualitative understanding of the physical process. A classical example is the vacuum pump based on the phenomenon of thermal gradient flow, rather than on mechanical devices, to generate pressure differences. Preliminary analytical studies show that the thermal gradient flow is stronger for more rarefied gases, an important consideration in the design of vacuum pumps. The study of Prandtl's conjecture and the von Neumann paradox in compressible gas flows provides basic understanding for the shock structure of supersonic flight. For instance, mathematical analysis quantifies the difference between the shock structure at the nose and that inside the engine of an aircraft. This project aims to advance understanding of the mathematical models of these important processes.