Interior and Exterior Problems for Kinetic Equations
DescriptionIn physics and applications, it is important to consider flows of fluid(gas) and clouds of particles outside obstacles or inside vessels. Typical examples are the airfoil problem and the re-entry problem for space shuttles. In most of the interesting cases, there are shock formations and transonic regions. However, the mathematical problems in this direction are extremely challenging and rigorous theories are absent for most of the general cases.For this, the mathematical theory in the level of fluid dynamics has been attempted by many outstanding researchers with a lot of impressive results. In this regard, the researchers should mention the important works and discussions on the subsonic, supersonic and transonic flows by Bers, Courant-Friedrichs, Mowawetz, Finn-Gilbarg, Gilbarg-Shiffman, Dong, etc. On the other hand, in the mescroscopic level, that is, for kinetic equations, the mathematical understanding is very limited. The startup contributions were made by Hilbert and the Girttingen school, Chapman and Enskog, Grad, Kogan and Maslova.Further, for the exterior problem, in the context of DiPerna and Lions renormalized solution, there are some investigation by Hamdache, Arkeryd-Cercignani. For the pertubation of an equilibrium, Ukai-Asano gave a rigorous but technical analysis on the existence and stability for the exterior problem when the far field Mach number is sufficiently small. Recently, the researchers made some improvement on Ukai-Asano's result by crucially using the velocity averaging argument for the compactness so that some general boundary conditions can be included in this framework. However, the existence of solutions far from an equilibrium for the exterior problem is still unsolved even though some preliminary discussion with some artificial assumptions can be found in the work by Arkeryd-Cercignani-Illner. There are also some important works on the interior problem, bounded domain problems for the Boltzmann equation, for example, by Arkeryd-Nouri, Arkeryd-Maslova, Arkeryd-Heintz, Asano, Mischler, Palczewski, Shizuta-Asano and Sone, etc.The main purpose of this research project is to study the interior and exterior problem in some general settings and hopefully, the researchers can relax some restrictions imposed on the boundary and the far field states in the previous works so that a better understanding on the mathematical theories can be obtained.
|Effective start/end date||1/10/08 → 17/04/12|