The Research Of Transport Equations With General Boundary Conditions In Slab Geometry

Since 1950s the exploration work of Lehner-Wing and Jorgens, the constructive theory of transport equation solution is the subject of greatly interested in mathematics, physics and engineering technology. This paper researches the spectrum of transport operator A for anisotropic continuous energy and homogeneous or nonhomogeneous in slab geometry with periodic or general boundary conditions by the operator theory, the semigroup theory and the methods of modern analysis mathematics, and we get a series of good results about the spectral analysis of transport operator A relating to the transport equations. The main results are showed as follow:1. Part one, introduction, in this part, we introduce the development of the transport theory research;2. The second part, in L^p(1≤p＜∞) space, it researches transport equation for anisotropic continuous energy and nonhomogeneous with general boundary conditi -ons in slab geometry, it is to prove that the transport operator A generates a C₀ semigroup V(t)(t≥0) and the second-order remained term of the Dyson-Phillips expansion of the C₀ semigroup is compact in L^P(1＜p＜∞) space and weakly compact in L¹ space, and it is to obtain that the spectrum of the transport operator A only consists of finite isolated eigenvalues which have a finite algebraic multiplicity in tripΓ, moreover, it proves that the existence of the dominant eigenvalue;3. The third part, in L² space, it researches transport equation for anisotropic continuous energy and homogeneous with periodic boundary conditions in slab geometry, it is to prove that the transport operator A has no complex eigenvalue, and the spectrum of the transport operator A consists of finite real isolated eigenvalues which have a finite algebraic multiplicity in trip P_as(A).