Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866917770867769344 |
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| author | Engel, Maximilian Kuehn, Christian |
| author_facet | Engel, Maximilian Kuehn, Christian |
| contents | We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the reaction term and viewing the slow variable as a dynamic bifurcation parameter, we analyze the passage through the fast subsystem bifurcation point for the spectral Galerkin approximation of the PDE. We characterize the invariant manifolds for the finite-dimensional Galerkin ODEs using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, we also make the domain dynamic during the blow-up analysis. Finally, we elaborate in which sense our results approximate the infinite-dimensional problem. Within our analysis, we find that the PDEs appearing in entry and exit blow-up charts are quasi-linear free boundary value problems, while in the central/scaling chart we obtain a PDE, which is often encountered in classical reaction-diffusion problems exhibiting solutions with finite-time singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_09973 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities Engel, Maximilian Kuehn, Christian Dynamical Systems 34E15, 35K57, 37G10, 37L15, 37L65 We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the reaction term and viewing the slow variable as a dynamic bifurcation parameter, we analyze the passage through the fast subsystem bifurcation point for the spectral Galerkin approximation of the PDE. We characterize the invariant manifolds for the finite-dimensional Galerkin ODEs using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, we also make the domain dynamic during the blow-up analysis. Finally, we elaborate in which sense our results approximate the infinite-dimensional problem. Within our analysis, we find that the PDEs appearing in entry and exit blow-up charts are quasi-linear free boundary value problems, while in the central/scaling chart we obtain a PDE, which is often encountered in classical reaction-diffusion problems exhibiting solutions with finite-time singularities. |
| title | Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities |
| topic | Dynamical Systems 34E15, 35K57, 37G10, 37L15, 37L65 |
| url | https://arxiv.org/abs/2007.09973 |