Global solutions for operator Riccati equations with unbounded coefficients: A non‐linear semigroup approach

摘要

AbstractLet Xbe a Banach space of real‐valued functions on 0, 1 and let ℒ(X) be the space of bounded linear operators onX. We are interested in solutionsR:(0, ∞) → ℒ(X) for the operator Riccati equationdocumentclass{article}pagestyle{empty}begin{document}$$ R#x02032; + TR + RT = TB_1 (t) + TB_2 (t)R + RTB_3 (t) + RTB_4 (t)R, $$end{document}whereTis an unbounded multiplication operator inXand theBi(t)'s are bounded linear integral operators onX. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators onL1(0, 1). Conditions onX, R(0),T, and the coefficients are found such that the theory of non‐linear semigroups may be used to prove global existence of strong solutions in ℒ(X) that also satisfyR(t) ϵ ℒ(L1(0