Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

TL;DR

This paper rigorously proves the nonlinear instability of roll solutions in the 2D generalized Swift-Hohenberg equation, demonstrating how small perturbations grow and cause deviations from the initial pattern within finite time.

Contribution

It extends the spectral-to-nonlinear instability analysis to a two-dimensional setting with unbounded Bloch parameters, providing a rigorous proof for the instability of rolls.

Findings

Small initial perturbations grow over time.

Deviations from roll solutions occur within finite time.

Spectral information near the maximally unstable mode is key.

Abstract

Within the framework developed in \cite{Gr, JLL, RT1}, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter $\sigma$ ranges over an unbounded domain.