  We consider a simplified extensible version of a dynamic free boundary
problem for a thin filament with radius $\epsilon>0$ immersed in 3D Stokes
flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel
type of angle-averaged Neumann-to-Dirichlet operator for the Stokes equations,
and much of the difficulty in the analysis lies in understanding this operator.
Here we show that the main part of this angle-averaged NtD map about a closed,
curved filament is the corresponding operator about a straight filament, for
which we can derive an explicit symbol. Remainder terms due to curvature are
lower order with respect to regularity or size in $\epsilon$. Using this
operator decomposition, it is then possible to show that the simplified free
boundary evolution is a third-order parabolic equation and is locally well
posed. This establishes a more complete mathematical foundation for the myriad
computational results based on slender body approximations for thin immersed
elastic structures.
