Boundary Value Problems for Transport Equations - Valeri Agoshkov

. .

Sommaire:
1 Problems and equations of transport theory.- 1.1 Some notions of transport theory.- 1.2 Basic transport equations.- 1.3 Boundary conditions and statements of problems.- 1.4 Typical boundary value problems.- 1.5 Integral equations of transport theory.- 1.6 Adjoint problems.- 1.7 Correctness of statements and need of new functional spaces.- 2 Functional spaces, existence of traces, and extension of functions.- 2.1 Spaces Hp1(? ? D). Trace existence and extensions of functions.- 2.2 Spaces Hp1[-1,1] ? (0, H). Trace existence and extensions of functions.- 2.3 Spaces of periodic functions V and Hpt + ?, k Properties of operator ?.- 2.4 Spaces Hpt + ?, k(? ? D). Existence of traces and extensions of functions.- 3 Variational statements and generalized solutions of transport problems.- 3.1 The first variational problem. Necessary and sufficient conditions of solvability in Hp1.- 3.2 The second variational problem. Estimates of boundary values of solutions.- 3.3 General approach to symmetrization. The third variational problem. Existence of solutions.- 3.4 Reflection operators and fundamental functions.- 3.5 Existence of solutions for periodic problems.- 4 Regularity properties of generalized solutions.- 4.1 Regularity of periodic solutions in a plane-parallel geometry.- 4.2 Regularity of periodic solutions in three-dimensional geometry.- 4.3 Regularity for the boundary value plane-parallel problem.- 4.4 Three-dimensional boundary value problems.- 4.5 Regularity of solutions in (x, y)-geometry.- 5 Applications to analysis of transport problems and numerical algorithms.- 5.1 Operators L-1S, SL*-1.- 5.2 Fundamental functions of reflection operators and inverse problems.- 5.3 Convergence of domain decomposition methods for transport problems.- 5.4 Some applications fornumerical algorithms.- 5.5 Energy dependent problems.- 5.6 Justification of a perturbation algorithm for a nonlinear transport equation.- Appendix. Main notations and functional spaces.