Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912727608328192 |
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| author | Yan, Qi |
| author_facet | Yan, Qi |
| contents | We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on $\mathbb{R}^2$ in this paper. We equivalently formulate them as a stochastic one-phase Stefan problem (or a stochastic free boundary problem) of the curvature, which is parameterized by the arc-length, and the length of the curves. After rewriting the stochastic Stefan problem as a quasilinear parabolic evolution equation, we apply the theory for quasilinear parabolic stochastic evolution equations developed by Agresti and Veraar in 2022 to get the existence and uniqueness of a local strong solution up to a maximal stopping time that is characterized by a blow-up alternative. When the solutions blow up, the corresponding stochastic curve flows either develop singularities or shrink to a point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$ Yan, Qi Probability 60H15, 60H30, 53E40, 80A22 We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on $\mathbb{R}^2$ in this paper. We equivalently formulate them as a stochastic one-phase Stefan problem (or a stochastic free boundary problem) of the curvature, which is parameterized by the arc-length, and the length of the curves. After rewriting the stochastic Stefan problem as a quasilinear parabolic evolution equation, we apply the theory for quasilinear parabolic stochastic evolution equations developed by Agresti and Veraar in 2022 to get the existence and uniqueness of a local strong solution up to a maximal stopping time that is characterized by a blow-up alternative. When the solutions blow up, the corresponding stochastic curve flows either develop singularities or shrink to a point. |
| title | Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$ |
| topic | Probability 60H15, 60H30, 53E40, 80A22 |
| url | https://arxiv.org/abs/2511.19924 |