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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.28661 |
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| _version_ | 1866911553916239872 |
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| author | Sangalli, Giancarlo Terazzi, Davide Zanotti, Pietro |
| author_facet | Sangalli, Giancarlo Terazzi, Davide Zanotti, Pietro |
| contents | We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resonant solutions and (in)stability of the linear wave equation Sangalli, Giancarlo Terazzi, Davide Zanotti, Pietro Numerical Analysis 35L05, 35L20, 35B34, 35B35 We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs. |
| title | Resonant solutions and (in)stability of the linear wave equation |
| topic | Numerical Analysis 35L05, 35L20, 35B34, 35B35 |
| url | https://arxiv.org/abs/2603.28661 |