摘要
本文综合运用泛函分析、算子理论和半群理论等现代分析方法,研究了迁移方程解的构造性理论和应用,获得了迁移算子的谱分析、迁移方程解的大时间渐近稳定性和展开理论、参数方程解的存在性和迁移方程的扩散近似理论等一系列新结果.主要结果述叙如下:
1.关于迁移算子的谱分析:
(1) 在非齐次(广义、反射等)边界条件下,(a)对具备向异性、连续能量、非均匀介质的迁移算子,获得了在区城Γ中仅有有限个具有限代数重数的离散本征值的存在性;(b)对具各向同性、(单能)连续能量、(非)均匀介质的迁移算子A,获得了在区城P_(as)(A)中仅有有限个具有限代数重数的实离散本征值;
(2) 在有界凸体情况,对具备向同性、连续能量、均匀介质的迁移算子A,得到了A仅有可数无限个向负无穷远点聚结的代数指标为1的实离散本征值。
(3) 在抽象(周期、完全反射)边界条件下,对具备向异性、连续能量、均匀介质的迁移算子A,它产生C_0群和该群的Dyson-Phillips展开式的二阶余项在L~2空间上是紧的;
(4) 在抽象(周期)边界条件下,对具备向异性、连续能量、均匀介质的奇异迁移算子A,它产生C_0半群和该半群的Dyson-Phillips展开式的二阶余项在L~1空间上是弱紧的。
2.关于迁移方程解的渐近性质:对最小速率可为零的板几何(齐次和非
The objective of this paper is using modern analytical method of the functional analysis, operator theory and semigroup theory, This enable us to investigate in details the constructive theory and application of the transport equations solution, and to get a series new results about the spectral analysis of transport operators, large time asymptotic stability of transport equations solution and expansion theory, the existence of the parameter equation solution and the diffusion approximate theory of the transport equation. Main results are as follows:1. About the spectral analysis of transport operators: (1) In nonzero (generalized, reflecting etc.)boundary conditions, (a) The transport operators A with anisotropic continuous energy inhomo-geneous only consist of finite isolated eigenvalues which have a finite algebraic multiplicity in trip Γ; (b) The transport operators A with isotropic (monoenergetic) continuous energy (inhomogeneous)homogeneous only consist of finite real isolated eigenvalues which have a finite algebraic
multiplicity in trip Pas (A);(2) In any bounded convex body, The transport operators A with isotropic continuous energy homogeneous, It is obtained that A has only countable infinite of real isolated eigenvalues which algebraic index is one accumulating at minus infinity;(3) In abstract (periodic, perfect reflecting) boundary conditions, For the transport operator A with anisotropic continuous energy homogeneous,It is to prove that A generates a strongly continuous Cq group and the compactness properties of the second-order remainder term of the Dyson-Pillips expansion for the Co group in I? space;(4) In abstract (periodic) boundary condition, For the transport operator A with anisotropic continuous energy homogeneous, It is to prove that A generates a strongly continuous Co semigroup and the weak compactness properties of the second-order remainder term of the Dyson-Pillips expansion for the Co semigroup in L1 space;2. About the asymptotic stability of transport equation solutions: For the transport equations that minimun velocity rate is zero in slab geometry (zero and nonzero boundary conditions) and any bounded convex body, We proved the existence of the strictly dominant eigenvalue for the transport operators A with anisotropic continuous energy inhomo-
geneous, and to get the large time asymptotic stability for the transport equations solution;3. About the expansion theory of the transport equations solution:(1) We discuss the completeness of generalized eigenfunction system for the Riese operator and Jorgens transport operator, and to obtain the some sufficient conditions and the necessary and sufficient conditions about the completeness;(2) For Jorgens transport equation with anisotropic continuous energy inhomogeneous, We are obtained results as follows:t> 6r;71=1 7 = 1/ = Tf0 = ]T E^ T(*))7Wo + TT(t)f0, t > 3r?;=iwhere A1? A2, ...is a series eigenvalue of the transport operator A, T(t) be a transport semigroup and A be its infinitesimal generator, P3 be the eigen-projection correspondingly Ay, E(Xt, T(t))T(t)be the generalized eigenspace,Tr(t) = lin-4. About the existence of the parameter equation solution: In complex plane,
(1) For the control critical eigenvalue equation with isotropic continuous energy inhomogeneous in slab geometry, If the equation has nonzero solution, then its parameter 5 is real number; If ||MJ|| < 1, there is not nonzero parameter 6 such that the equation has nonzero solution; If ||MJ|| > 1, there exist only infinitely many positive numbers 5n(n=0,l,2,.. .), such that the equation has nonzero solution, A necessary and sufficient condition for the existence of control critical eigenvalue 5q is ||MJ|| > 1.(2) In generalized boundary condition, For the control critical eigenvalue equation with isotropic continuous energy homogeneous in slab geometry, We obtain results as follows: (a) If ||T|| < 1, there is not nonzero parameter 5 such that the equation has nonzero solution; If ||T|| > 1, there exist only infinitely many positive numbers 5n(n=0,l,2,.. .), such that the equation has nonzero solution, A necessary and sufficient condition for the existence of control critical eigenvalue do is ||T|| > l,and 5q can be determined by the equation ||G(/5o)|| = Land it has estimates term:--------— / dv / \MS
\2dw- (1 - a)27T JE J-co5. About the diffusion approximate theory of transport equation:
(1) We derive the diffusion approximation of the transport equation, and to get the convergence and to give the error estimate for the diffusion approxition;(2) For unsteady neutron transport equation in multidimensional, We are using Galerkin finite element method, and to prove the convergence of the approximate solution and the existence of generalized solutions.
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