The goal of this project is to develop new mathematical methods for the study of multi-dimensional gas flows with shocks and the kinetic theory. For the study of multi-dimensional gas flows, the mathematical problem of nonlinear equations of mixed types and free boundary that one needs to address are most challenging. Recent tremendous progress on the study of planary waves as well as the extensive numerical experiments allows one to identify the basic key problems. One such problem is the time-dependent, self-similar formulation of the Prandtl-Meyer reflection off a ramp. The solid boundary has the crucial stabilizing effect on the domain of ellipticity. The resolution of these basic, tractable problems would provide the necessary analytical tools for other problems in nonlinear partial differential equations.
As modern devices are getting smaller and as one travels further from the earth surface, the kinetic aspects of the gases become more pronounced. The dual property of the particle and fluid aspects of the Boltzmann equation is the main issue of the proposed research. For the study of the fluid aspect, the ideas from conservation laws are being generalized to bear on the Boltzmann equation. The particle aspect will be studied based on the construction and estimates of the Boltzmann Green function, for which there is the recent progress based on the combination of spectral analysis and the construction of the kinetic-types waves. Important, long-standing open problems such as the thermal creep phenomena are being understood with the exact analysis of the Boltzmann equation. New mathematical methods are being developed and should prove useful also for other problems in the kinetic theory.
