Functional Analysis and its Applications

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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].