On equations of continuity and transport type on metric graphs and fractals
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| Format: | Preprint |
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2024
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| _version_ | 1866917865922232320 |
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| author | Hinz, Michael Schefer, Waldemar |
| author_facet | Hinz, Michael Schefer, Waldemar |
| contents | We study first order equations of continuity and transport type on metric spaces of martingale dimension one, including finite metric graphs, p.c.f. self-similar sets and classical Sierpiński carpets. On such spaces solutions of the continuity equation in the weak sense are generally non-unique. We use semigroup theory to prove a well-posedness result for divergence free vector fields and under suitable loop and boundary conditions. It is the first well-posedness result for first order equations with scalar valued solutions on fractal spaces. A key tool is the concept of boundary quadruples recently introduced by Arendt, Chalendar and Eymard. To exploit it, we prove a new domain characterization for the relevant first order operator and a novel integration by parts formula, which takes into account the given vector field and the loop structure of the space. We provide additional results on duality and on metric graph approximations in the case of periodic boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On equations of continuity and transport type on metric graphs and fractals Hinz, Michael Schefer, Waldemar Analysis of PDEs Functional Analysis 28A80, 31C25, 35F10, 35F16, 35R02, 47A07, 47B44, 47D06 We study first order equations of continuity and transport type on metric spaces of martingale dimension one, including finite metric graphs, p.c.f. self-similar sets and classical Sierpiński carpets. On such spaces solutions of the continuity equation in the weak sense are generally non-unique. We use semigroup theory to prove a well-posedness result for divergence free vector fields and under suitable loop and boundary conditions. It is the first well-posedness result for first order equations with scalar valued solutions on fractal spaces. A key tool is the concept of boundary quadruples recently introduced by Arendt, Chalendar and Eymard. To exploit it, we prove a new domain characterization for the relevant first order operator and a novel integration by parts formula, which takes into account the given vector field and the loop structure of the space. We provide additional results on duality and on metric graph approximations in the case of periodic boundary conditions. |
| title | On equations of continuity and transport type on metric graphs and fractals |
| topic | Analysis of PDEs Functional Analysis 28A80, 31C25, 35F10, 35F16, 35R02, 47A07, 47B44, 47D06 |
| url | https://arxiv.org/abs/2412.07988 |