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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.27219 |
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| _version_ | 1866918474923638784 |
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| author | Kandalam, Achyuta Telekicherla Spirn, Daniel |
| author_facet | Kandalam, Achyuta Telekicherla Spirn, Daniel |
| contents | The Immersed Boundary Method has long served as a robust computational framework for fluid-structure interactions, yet the rigorous analysis of 1D Peskin filaments anchored to rigid boundaries remains sparse. In this paper, we generalize the classical Peskin problem to the half-plane by considering an elastic filament whose endpoints are anchored to a no-slip wall. Moving beyond the algebraic complexity of the traditional Blake image system, we utilize the boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni. This representation allows for a transparent decomposition of the hydrodynamic interactions into a free space principal part and a regularizing reflected component without resorting to hypersingular integral operators. Through this framework, we prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little Hölder spaces, we establish local well posedness and prove that the filament exhibits instantaneous $C^\infty$ regularization in both space and time. This work provides a rigorous analytical foundation for anchored filaments in bounded domains and suggests a spectrally accurate numerical path for simulating tethered biological structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27219 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anchored Peskin Problem Kandalam, Achyuta Telekicherla Spirn, Daniel Analysis of PDEs 35K58, 35Q35, 74F10, 76D07 The Immersed Boundary Method has long served as a robust computational framework for fluid-structure interactions, yet the rigorous analysis of 1D Peskin filaments anchored to rigid boundaries remains sparse. In this paper, we generalize the classical Peskin problem to the half-plane by considering an elastic filament whose endpoints are anchored to a no-slip wall. Moving beyond the algebraic complexity of the traditional Blake image system, we utilize the boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni. This representation allows for a transparent decomposition of the hydrodynamic interactions into a free space principal part and a regularizing reflected component without resorting to hypersingular integral operators. Through this framework, we prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little Hölder spaces, we establish local well posedness and prove that the filament exhibits instantaneous $C^\infty$ regularization in both space and time. This work provides a rigorous analytical foundation for anchored filaments in bounded domains and suggests a spectrally accurate numerical path for simulating tethered biological structures. |
| title | Anchored Peskin Problem |
| topic | Analysis of PDEs 35K58, 35Q35, 74F10, 76D07 |
| url | https://arxiv.org/abs/2604.27219 |