Convergence of linear solutions through convergence of periodic initial data
Harrison Gaebler, Wesley R Perkins
TL;DR
The paper establishes a rigorous link between the convergence of subharmonic initial data and the convergence to localized perturbations in linear evolution problems. By framing the problem in a general $A$-generated $C_0$ semigroup setting and employing a Bloch-transform representation, it proves that if $n$-periodic initial data $g_n$ converge over a period to a localized datum $g$, then the corresponding linear evolutions converge in $L^2(\mathbb{R})$ to the localized evolution for each $t$, with uniform-in-time convergence under bounded semigroups. This provides a solid linear-analytic counterpart to nonlinear results connecting subharmonic and localized perturbations and offers a robust methodology based on norm convergence over periods and Bloch theory. The work lays groundwork for potential extensions to weak convergence and informs nonlinear stability analyses by clarifying the precise mechanism by which periodic data approximate localized states in the linear regime.
Abstract
When studying the stability of $T$-periodic solutions to partial differential equations, it is common to encounter subharmonic perturbations, i.e. perturbations which have a period that is an integer multiple (say $n$) of the background wave, and localized perturbations, i.e. perturbations that are integrable on the line. Formally, we expect solutions subjected to subharmonic perturbations to converge to solutions subjected to localized perturbations as $n$ tends to infinity since larger $n$ values force the subharmonic perturbation to become more localized. In this paper, we study the convergence of solutions to linear initial value problems when subjected to subharmonic and localized perturbations. In particular, we prove the formal intuition outlined above; namely, we prove that if the subharmonic initial data converges to some localized initial datum, then the linear solutions converge.