Nonlinear semigroups with unbounded generators under Carleman linearization

TL;DR

This paper analyzes the convergence of Carleman linearization for nonlinear evolution equations using semigroup theory, especially addressing unbounded generators and applying Trotter-Kato approximation.

Contribution

It extends Carleman linearization analysis to unbounded operators and nonlinear PDEs, providing new convergence conditions and simplifying existing proofs.

Findings

Convergence of Carleman linearization can be analyzed via dissipativity constraints.

Trotter-Kato theorem links the linearized semigroup to finite-dimensional approximations.

Conditions are identified for polynomial nonlinearities to ensure convergence.

Abstract

We treat the convergence of Carleman linearization of nonlinear evolutionary equations through the approximation theory of strongly continuous semigroups, by Carleman embedding the underlying nonlinear semigroups as linear semigroups. Linear semigroup theory then lets one replace the norm constraint on the convergence of Carleman linearization in the form used by quantum algorithms for a class of semi-discretized evolution equations by a dissipativity constraint, simplifying arguments for convergence. Applying Trotter-Kato approximation theorem to the linearized semigroup realizes the semigroup as a limit finite dimensional operator exponentials, reducing the question of convergence rate of Carleman linearization to that of the Trotter-Kato approximation. We then examine convergence of the Carleman linearization as the operators become unbounded, treating the hyperviscuous Burger's equation as an example. Next we consider the perturbation theory of the Carleman semigroup and obtain conditions when polynomial nonlinearities correspond to the Carleman linearized semigroup being a $1$-integrated semigroup, so convergence is implied by variants of Trotter-Kato approximation for integrated semigroups.