Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactions
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arXiv
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| Format: | Preprint |
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2023
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| author | Mel'nyk, Taras Rohde, Christian |
| author_facet | Mel'nyk, Taras Rohde, Christian |
| contents | This article completes the study of the influence of the intensity parameter $α$ in the boundary condition $\varepsilon \partial_{\boldsymbolν_\varepsilon} u_\varepsilon - u_\varepsilon \, \overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbolν_\varepsilon = \varepsilon^α φ_\varepsilon $ given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order $\mathcal{O}(\varepsilon^{-1})$ is considered. The novelty of this article is the case of $α<1,$ which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity $φ_\varepsilon$ (the cases $α=1$ and $α>1$ were previously studied by the same authors).
A complete Puiseux asymptotic expansion is constructed for the solution $u_\varepsilon$ as $\varepsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter $\varepsilon.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02387 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactions Mel'nyk, Taras Rohde, Christian Analysis of PDEs 35K20, 35R02, 35B40, 35B25, 35B45, 35K57, 35Q49 This article completes the study of the influence of the intensity parameter $α$ in the boundary condition $\varepsilon \partial_{\boldsymbolν_\varepsilon} u_\varepsilon - u_\varepsilon \, \overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbolν_\varepsilon = \varepsilon^α φ_\varepsilon $ given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order $\mathcal{O}(\varepsilon^{-1})$ is considered. The novelty of this article is the case of $α<1,$ which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity $φ_\varepsilon$ (the cases $α=1$ and $α>1$ were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution $u_\varepsilon$ as $\varepsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter $\varepsilon.$ |
| title | Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactions |
| topic | Analysis of PDEs 35K20, 35R02, 35B40, 35B25, 35B45, 35K57, 35Q49 |
| url | https://arxiv.org/abs/2307.02387 |