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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.13175 |
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| _version_ | 1866916795435188224 |
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| author | Park, Jeongheon |
| author_facet | Park, Jeongheon |
| contents | We consider the interior Stefan problem under radial symmetry in two dimension. A water ball surrounded by ice undergoes melting or freezing. We construct a discrete family of global-in-time solutions, both melting and freezing scenarios. The evolution of the free boundary, represented by the radius of the water ball, $λ(t)$ exhibits exponential convergence to a limiting radius value $λ_\infty > 0$, characterized by the asymptotic expression \[ λ(t) = λ_\infty + (1 - λ_\infty)\, e^{-\frac{λ_k}{λ_\infty^2} t + o_{t \to \infty}(1)}, \] where $λ_k$ stands for the $k$-th Dirichlet eigenvalue of the Laplacian on the unit disk for any $k\in \mathbb{N}$. Our approach draws inspiration from the research conducted by Hadžić and Raphaël [24] concerning the exterior radial Stefan problem, which involves an ice ball is surrounded by water. In contrast, the bounded geometry in our setting leads to scenario results in a non-degenerate spectrum, leading to distinctly different long-term behavior. These solutions for each $k$ remain stable under perturbations of co-dimension $k - 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Melting and freezing rates of the radial interior Stefan problem in two dimension Park, Jeongheon Analysis of PDEs 35A01, 35K55, 35P15 We consider the interior Stefan problem under radial symmetry in two dimension. A water ball surrounded by ice undergoes melting or freezing. We construct a discrete family of global-in-time solutions, both melting and freezing scenarios. The evolution of the free boundary, represented by the radius of the water ball, $λ(t)$ exhibits exponential convergence to a limiting radius value $λ_\infty > 0$, characterized by the asymptotic expression \[ λ(t) = λ_\infty + (1 - λ_\infty)\, e^{-\frac{λ_k}{λ_\infty^2} t + o_{t \to \infty}(1)}, \] where $λ_k$ stands for the $k$-th Dirichlet eigenvalue of the Laplacian on the unit disk for any $k\in \mathbb{N}$. Our approach draws inspiration from the research conducted by Hadžić and Raphaël [24] concerning the exterior radial Stefan problem, which involves an ice ball is surrounded by water. In contrast, the bounded geometry in our setting leads to scenario results in a non-degenerate spectrum, leading to distinctly different long-term behavior. These solutions for each $k$ remain stable under perturbations of co-dimension $k - 1$. |
| title | Melting and freezing rates of the radial interior Stefan problem in two dimension |
| topic | Analysis of PDEs 35A01, 35K55, 35P15 |
| url | https://arxiv.org/abs/2506.13175 |