Weak Diffusive Stability of Roll Solutions at the Zigzag Boundary
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917834073833472 |
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| author | Chowdhary, Abhijit Haberle, Mason Ofori-atta, William Wu, Qiliang |
| author_facet | Chowdhary, Abhijit Haberle, Mason Ofori-atta, William Wu, Qiliang |
| contents | Roll solutions at the zigzag boundary, typically selected by patterns and defects in numerical simulations, are shown to be nonlinearly stable. This result also serves as an example that linear decay weaker than the classical diffusive decay, together with quadratic nonlinearity, still gives nonlinear stability of spatially periodic patterns. Linear analysis reveals that, instead of the classical $t^{-1}$ diffusive decay rate, small perturbations of roll solutions at the zigzag boundary decay with a $t^{-3/4}$ rate along with time, due to the degeneracy of the quadratic term of the continuation of the translational mode of the linearized operator in the Bloch-Fourier spaces. The nonlinear stability proof is based on a decomposition of the neutral translational mode and the faster decaying modes in the Bloch-Fourier space, and a fixed-point argument, demonstrating the irrelevancy of the nonlinear terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_12365 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak Diffusive Stability of Roll Solutions at the Zigzag Boundary Chowdhary, Abhijit Haberle, Mason Ofori-atta, William Wu, Qiliang Pattern Formation and Solitons Analysis of PDEs Dynamical Systems 35B10, 35B35, 35B36, 35B40 Roll solutions at the zigzag boundary, typically selected by patterns and defects in numerical simulations, are shown to be nonlinearly stable. This result also serves as an example that linear decay weaker than the classical diffusive decay, together with quadratic nonlinearity, still gives nonlinear stability of spatially periodic patterns. Linear analysis reveals that, instead of the classical $t^{-1}$ diffusive decay rate, small perturbations of roll solutions at the zigzag boundary decay with a $t^{-3/4}$ rate along with time, due to the degeneracy of the quadratic term of the continuation of the translational mode of the linearized operator in the Bloch-Fourier spaces. The nonlinear stability proof is based on a decomposition of the neutral translational mode and the faster decaying modes in the Bloch-Fourier space, and a fixed-point argument, demonstrating the irrelevancy of the nonlinear terms. |
| title | Weak Diffusive Stability of Roll Solutions at the Zigzag Boundary |
| topic | Pattern Formation and Solitons Analysis of PDEs Dynamical Systems 35B10, 35B35, 35B36, 35B40 |
| url | https://arxiv.org/abs/2310.12365 |