Functional Analysis and its Applications

这个项目的目的在于应用泛函分析的方法研究数学分析的各个分支。取得的主要结果如下。[L]研究流体力学方程与气体运动论之间关系的结构。证明了当平均自由程趋于0时，Boltzmann方程的解收敛于可压缩欧拉方程的解。证明了当马赫数趋于0时，可压缩欧拉方程的解收敛于不可压缩欧拉方程的解；当流体速度与光速之比趋于0时，Vlasov-Maxwell方程的解收敛于Vlasov-Poisson方程的解。在李群表示理论的研究方面，在[2][3][4][5]中，我们研究了李超代数的最高权表示的结构，得到了类似于李群的离散级数表示的有趣表示，并在一种特殊情况下得到了一个超对偶对的分解。在[6]中，我们利用旗子流形上的轨道结构刻画了映射，将半单李群的任何表示嵌入到主级数表示中，并在经典单李群的情况下给出了旗子流形上轨道结构的符号图，我们在1990年京都举行的国际数学家大会上就这些主题(第七节：李群和表示)作了应邀的分会报告[7]。

The aim of this project consists in researching various branches of mathematical analysis by applying methods of functional analysis. Main results obtained are as follows. [l]concerns with the structure of relations between equations arising in fluid dynamics and kinetic theory of gases. We have proved that the solution of the Boltzmann equation converges to the solution of compressible Euler equation as the mean free path tends to 0. Moreover, we have proved that the solution of compressible Euler equation converges to that of the incompressible Euler equation as the Mach number tends to 0 and that the solution of the Vlasov-Maxwell equation converges to that of the Vlasov-Poisson equation as the ratio, fluid verocity/ light verocity, tends to 0. With respect to the research on the representation theory of the Lie groups, in[2][3][4][5]we have investigated the structure of the highest weight representation of Lie superalgebras and obtained interesting representations similar to the discrete series representations of Lie groups, and a decomposition of a super dual pair has been obtained in a special case. In[6]we have described the maps, embedding any representations of the semi-simple Lie group to the principal series representations, by using the 'orbit structure on the flag Manifolds, and have given symbolic diagrams of orbit structures on the flag manifolds in the case of the classical simple Lie groups. We have given an invited Section Lecture on these themes(the Section 7 : Lie Groups and Representations)in the International Congress of Mathematicians held in Kyoto 1990[7].