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

Abstract

AbstractLet X be a Banach space of real‐valued functions on [0, 1] and let ℒ(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ℒ(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the Bi(t)'s are bounded linear integral operators on X. 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 on L1(0, 1). Conditions on X, 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 satisfy R(t) ϵ ℒ(L1(0,1)) for all t ≥ 0.