TY - JOUR
T1 - Global solutions for operator Riccati equations with unbounded coefficients
T2 - A non‐linear semigroup approach
AU - Kuiper, Hendrik J.
PY - 1995/4/10
Y1 - 1995/4/10
N2 - Let 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 (Formula Presented.) 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.
AB - Let 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 (Formula Presented.) 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.
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U2 - 10.1002/mma.1670180406
DO - 10.1002/mma.1670180406
M3 - Article
AN - SCOPUS:84988163677
SN - 0170-4214
VL - 18
SP - 317
EP - 336
JO - Mathematical Methods in the Applied Sciences
JF - Mathematical Methods in the Applied Sciences
IS - 4
ER -