Abstract Riccati equations in an L1 space of finite measure and applications to transport theory

✍ Scribed by Jonq Juang; Paul Nelson

• Publisher: John Wiley and Sons
• Year: 1990
• Tongue: English
• Weight: 748 KB
• Volume: 12
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670120104

✦ Synopsis

Abstract

We consider operator‐valued Riccati initial‐value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R~0~. Here A to D and R~0~ have values as non‐negative bounded linear operators in L^1^ (μ), where μ is a finite measure, and T is a closed non‐negative operator in L^1^ (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.