Well-posedness and global behavior of the Peskin problem of an immersed elastic filament in Stokes flow

TL;DR

This paper analyzes the well-posedness and stability of a model describing an elastic filament in a Stokes fluid, establishing local solutions, stability of equilibria, and conditions for global behavior.

Contribution

It introduces a boundary integral approach to prove local well-posedness and nonlinear stability of equilibria for the Peskin problem in low-regularity spaces.

Findings

Local well-posedness in low-regularity H"older spaces.

Nonlinear stability of circular equilibria with exponential decay.

Identification of a key quantity controlling global behavior.

Abstract

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 H\"older 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 which respects the symmetries of the problem and controls global-in-time behavior of the system.