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We consider the problem of a one dimensional elastic filament immersed in atwo dimensional steady Stokes fluid. Immersed boundary problems in which a thinelastic structure interacts with a surrounding fluid are prevalent in scienceand engineering, a class of problems for which Peskin has made pioneeringcontributions. Using boundary integrals, we first reduce the fluid equations toan evolution equation solely for the immersed filament configuration. We thenestablish local well-posedness for this equation with initial data inlow-regularity H\"older spaces. This is accomplished by first extracting theprincipal linear evolution by a small scale decomposition and then establishingprecise smoothing estimates on the nonlinear remainder. Higher regularity ofthese solutions is established via commutator estimates with error termsgenerated by an explicit class of integral kernels. Furthermore, we show thatthe set of equilibria consists of uniformly parametrized circles and provenonlinear stability of these equilibria with explicit exponential decayestimates, the optimality of which we verify numerically. Finally, we identifya quantity which respects the symmetries of the problem and controlsglobal-in-time behavior of the system.

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