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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.17175 |
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| _version_ | 1866915035458043904 |
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| author | Giga, Yoshikazu Gösswein, Michael Katayama, Sho |
| author_facet | Giga, Yoshikazu Gösswein, Michael Katayama, Sho |
| contents | We consider a surface diffusion flow of the form $V=\partial_s^2f(-κ)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $κ$ denotes the curvature and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$. We consider this equation for the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)κ$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $κ=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large time behavior of exponential surface diffusion flows on $\mathbb{R}$ Giga, Yoshikazu Gösswein, Michael Katayama, Sho Analysis of PDEs We consider a surface diffusion flow of the form $V=\partial_s^2f(-κ)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $κ$ denotes the curvature and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$. We consider this equation for the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)κ$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $κ=0$. |
| title | Large time behavior of exponential surface diffusion flows on $\mathbb{R}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.17175 |