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We study the problem where a one-dimensional elastic string is immersed in atwo-dimensional steady Stokes fluid. This is known as the Stokes immersedboundary problem and also as the Peskin problem. We consider the case withequal viscosities and with a fully non-linear tension law; this model has beencalled the fully nonlinear Peskin problem. In this case we prove local in timewell-posedness for arbitrary initial data in the scaling critical Besov space$\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$. We additionally prove theoptimal higher order smoothing effects for the solution. To prove this resultwe derive a new formulation of the boundary integral equation that describesthe parametrization of the string, and we crucially utilize a new cancellationstructure.

Critical local well-posedness for the fully nonlinear Peskin problem