We consider the problem of a one-dimensional elastic filament immersed in a two-dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. We then establish local well-posedness for this equation with initial data in low-regularity Hölder spaces. This is accomplished by first extracting the principal linear evolution by a small-scale decomposition and then establishing precise smoothing estimates on the nonlinear remainder. Higher regularity of these solutions is established via commutator estimates with error terms generated by an explicit class of integral kernels. Furthermore, we show that the set of equilibria consists of uniformly parametrized circles and prove nonlinear stability of these equilibria with explicit exponential decay estimates, the optimality of which we verify numerically. Finally, we identify a quantity that respects the symmetries of the problem and controls global-in-time behavior of the system.
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Take k sufficiently large, so that k…X .tk/kC1;ˇ is small enough to apply Theorem 1.6 with initial data X .tk/. This concludes the proof. □ Acknowledgments. The authors thank Peter Polácˇik whose lectures on evolution equations, attended by the first two authors, led to the use of semigroup theory in Hölder spaces for this paper. The authors thank Charlie Peskin for allowing us to name this problem after him, and the first author thanks him for introducing the immersed boundary method to the first author as a graduate student; the discussions then led to this project. The first author was supported in part by National Science Foundation Grant DMS-1516978 and DMS-1620316, and the third author by National Science Foundation Grant DMS-1516565. The authors also thank the hospitality of the IMA where most of this work was performed.
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