Perturbation of nilpotent semigroups and application to heat exchanger equations (Q550470)

The authors describe a stability property for nilpotent semigroups. The main result says that, if \(A\) is the generator of a nilpotent semigroup \(T(t)\) on a Banach space \(X\) and if \(B\) is a closed linear operator in \(X\), then \(A+B\) also generates a nilpotent semigroup \(S(t)\) on \( X\), under some hypotheses on \(B\), which imply that \(B\) is a Miyadera-Voigt perturbation of \(A\). The main idea of the proof is to use the representation of \(S(t)\) as a Dyson-Phillips series and to prove some properties of each term of these series using its integral representation in terms of \(T\) and \(B \). The paper ends with an application of this result to a two-stream parallel-flow heat exchanger equation. The authors prove that the \(C_{0}\)-semigroup generated by the operator associated to this flow is nilpotent and they consider some perturbation of this flow which preserves the nilpotent property of the semigroup.