Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914014963957760 |
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| author | Bies, Piotr Michał Cieślak, Tomasz Fuest, Mario Lankeit, Johannes Muha, Boris Trifunović, Srdan |
| author_facet | Bies, Piotr Michał Cieślak, Tomasz Fuest, Mario Lankeit, Johannes Muha, Boris Trifunović, Srdan |
| contents | We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors (SIAM J.\ Math.\ Anal.\ \textbf{55} (2023), 7024--7038)) to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lamé operator are also stated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20794 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity Bies, Piotr Michał Cieślak, Tomasz Fuest, Mario Lankeit, Johannes Muha, Boris Trifunović, Srdan Analysis of PDEs 74A15 (Primary), 74H40, 35M30 (Secondary) We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors (SIAM J.\ Math.\ Anal.\ \textbf{55} (2023), 7024--7038)) to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lamé operator are also stated. |
| title | Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity |
| topic | Analysis of PDEs 74A15 (Primary), 74H40, 35M30 (Secondary) |
| url | https://arxiv.org/abs/2507.20794 |