Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913977031720960 |
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| author | Rodrigues, L. Miguel Wheeler, Aric |
| author_facet | Rodrigues, L. Miguel Wheeler, Aric |
| contents | We develop a stability theory for two-dimensional periodic traveling waves of general parabolic systems, possibly including conservation laws. In particular, we identify a diffusive spectral stability assumption and prove that it implies nonlinear stability for variously-(non)localized perturbations, including critically nonlocalized perturbations. Thus we extend the stability parts of Johnson et al., Invent. Math. 2014, to two-dimensional patterns and of Melinand-Rodrigues, preprint 2024, to systems with conservation laws. In doing so we need to bypass two kinds of low spectral regularity, explicitly conic-like singularities due to multidimensionality and Jordan-block like singularities due to conservation laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws Rodrigues, L. Miguel Wheeler, Aric Analysis of PDEs We develop a stability theory for two-dimensional periodic traveling waves of general parabolic systems, possibly including conservation laws. In particular, we identify a diffusive spectral stability assumption and prove that it implies nonlinear stability for variously-(non)localized perturbations, including critically nonlocalized perturbations. Thus we extend the stability parts of Johnson et al., Invent. Math. 2014, to two-dimensional patterns and of Melinand-Rodrigues, preprint 2024, to systems with conservation laws. In doing so we need to bypass two kinds of low spectral regularity, explicitly conic-like singularities due to multidimensionality and Jordan-block like singularities due to conservation laws. |
| title | Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.04023 |