A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909602368454656 |
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| author | Tréton, Samuel Zhang, Mingmin |
| author_facet | Tréton, Samuel Zhang, Mingmin |
| contents | To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z $\ge$ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall-a ''piston''-that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) $\sim$ ct^$β$ with $β$ $\in$ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When $β$ goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when $β$ = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space Tréton, Samuel Zhang, Mingmin Analysis of PDEs To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z $\ge$ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall-a ''piston''-that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) $\sim$ ct^$β$ with $β$ $\in$ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When $β$ goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when $β$ = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary. |
| title | A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.03304 |