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Abstract We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space . We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure.
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This content will become publicly available on January 30, 2026
Well-posedness of the 3D Peskin problem
This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e. the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.
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- PAR ID:
- 10599175
- Publisher / Repository:
- World Scientific Publishing Company
- Date Published:
- Journal Name:
- Mathematical Models and Methods in Applied Sciences
- Volume:
- 35
- Issue:
- 01
- ISSN:
- 0218-2025
- Page Range / eLocation ID:
- 113 to 216
- Subject(s) / Keyword(s):
- Peskin problem 3D fluid–structure interaction immersed boundary problem stokes flow.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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