Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908682000793600 |
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| author | Chae, Myeongju Jung, Soyeun |
| author_facet | Chae, Myeongju Jung, Soyeun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation Chae, Myeongju Jung, Soyeun Analysis of PDEs 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. |
| title | Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.00764 |