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

Myeongju Chae, Soyeun Jung

TL;DR

The paper proves a rigorous nonlinear instability for roll solutions in the 2D generalized Swift–Hohenberg equation by leveraging a Grenier-type framework, Bloch–Floquet analysis, and precise semigroup bounds. It identifies a most unstable Bloch mode with maximal growth rate $\lambda_M$ and constructs an exponentially growing linear solution that dominates the dynamics, then builds a high-order nonlinear approximation whose remainder remains small up to a time $T^{\delta} ~ |\ln \delta|$. The result shows that arbitrarily small perturbations can drive the system away from the roll within finite time, establishing a robust transition from spectral to nonlinear instability in a genuinely two-dimensional setting. This work also clarifies the role of the unbounded Bloch parameter domain and sets the stage for extending Grenier-type analyses to dissipative pattern-forming PDEs beyond the classical one-dimensional context.

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 $σ$ ranges over an unbounded domain.

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

TL;DR

The paper proves a rigorous nonlinear instability for roll solutions in the 2D generalized Swift–Hohenberg equation by leveraging a Grenier-type framework, Bloch–Floquet analysis, and precise semigroup bounds. It identifies a most unstable Bloch mode with maximal growth rate and constructs an exponentially growing linear solution that dominates the dynamics, then builds a high-order nonlinear approximation whose remainder remains small up to a time . The result shows that arbitrarily small perturbations can drive the system away from the roll within finite time, establishing a robust transition from spectral to nonlinear instability in a genuinely two-dimensional setting. This work also clarifies the role of the unbounded Bloch parameter domain and sets the stage for extending Grenier-type analyses to dissipative pattern-forming PDEs beyond the classical one-dimensional context.

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 ranges over an unbounded domain.

Paper Structure

This paper contains 10 sections, 13 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Roll solutions
  3. Spectral instability
  4. Main result and strategy
  5. Discussion
  6. Linear instability of rolls
  7. Semigroup estimates
  8. Nonlinear instability
  9. Appendix A
  10. Appendix B

Key Result

Theorem 1.1

Assume $27s-38b^2>0$ and let $\omega$ be a parameter satisfying Then there exists an $\varepsilon_0>0$ such that for all $\varepsilon \in (0, \varepsilon_0]$ and all $\omega \in [-\frac{1}{2}, \frac{1}{2}]$ there is a unique (up to translation) stationary $2\pi$--periodic roll solution of 2DgSH_intro, which is even in $x$ and bifurcating from the uniform state $u\equiv 0$. These periodic solutio

Figures (1)

  • Figure 1: The four regions $\mathcal{R}_1$ -- $\mathcal{R}_4$. The blue dot indicates $\sigma^*=(-\frac{1}{2}, \frac{\sqrt{3}}{2})$. Figure reproduced from CJ.

Theorems & Definitions (28)

  • Theorem 1.1: Existence of the rolls, CJJ
  • Theorem 1.2: Spectral instability, CJ
  • Theorem 1.3: Nonlinear instability
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 18 more