Abstract boundary value problems modeling transport processes in semi-infinite geometry

Abstract

Since Hangelbroek's ¹¹ study of non-conservative neutron transport with isotropic scattering considerable effort has been made towards the solution of abstract half-space problems of the from

where T is an injective self-adjoint operator and A a positive self-adjoint Fredholm operator on a complex hilbert space H and Q₊ is the orthogonal projection onto the maximal positive Tinvariant subspace of H. concrete examples about in neutron transport theory⁶, radiative transfer^{8, 23, 13}, rarefied gas dynamics⁷, phonon transport¹⁷ and Brownian motion in liquids¹⁸. Substantial contributions tot he development of the abstract theory were made by Beals^{2, 3, 4, 4a}, Greenberg^{10, 9}, Hangelbroek^{11, 12}, Lekkerkerker^{12, 15}, Van der Mee^{16, 10, 9}, Protopoesucu⁴ and Zweifel¹⁰. In this article we review the abstract theory as presented by Greenberg et al.¹⁰, and work out the specific example of strongly evaporating liquids. Finally, we discuss some related and open half-space problems with reflective boundary conditions.